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  Differential and Integral Equations
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\def\leftmark{\sc Fausto Gozzi, Roberto Monte, and Vincenzo Vespri}
 \def\rightmark{\sc generation of analytic semigroups}




\begin{document}


\title{Generation of analytic semigroups
and domain characterization for degenerate elliptic operators with
unbounded coefficients arising in financial mathematics, part I}
\date{}
\maketitle


\vspace{ -1.5\baselineskip}





{\small
\begin{center}
{\sc Fausto Gozzi}\footnote{The authors thank I.A.C. - C.N.R. and
the C.N.R. for support.}\\ Dipartimento di Matematica per le
Decisioni,
 Economiche Finanziarie e Assicurative,
 Facolt\`a di Economia, Universit\`a di Roma ``La Sapienza"\\
via del Castro Laurenziano 9, 00161--Roma, Italy\\[7pt] {\sc
Roberto Monte$^1$}\\
 Dipartimento di Studi Economici, Finanziari e Metodi Quantitativi\\
Facolt\`a di Economia, Universit\`a di Roma ``Tor Vergata"\\ Via
di Tor Vergata, snc, 00133 - Roma, Italy\\[7pt] {\sc Vincenzo
Vespri$^1$}\\
 Dipartimento di Matematica ``U. Dini,"
Universit\`a di Firenze\\ Viale Morgagni 67/A - 50134 Firenze,
Italy  \\[7pt]
 (Submitted by: G. Da Prato)
\end{center}
}


\numberwithin{equation}{section} \allowdisplaybreaks


\renewcommand{\thefootnote}{}
\footnote{Accepted for publication: December 2000.} \footnote{AMS
Subject Classifications:  47D06, 35J70, 35K65, 47N10.}











\newtheorem{Theorem}{Theorem}[section]
\newtheorem{Definition}[Theorem]{Definition}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Remark}[Theorem]{Remark}
\newtheorem{Example}[Theorem]{Example}
\newtheorem{Hypothesis}[Theorem]{Hypothesis}
\newtheorem{Assumption}[Theorem]{Assumption}


\def\A{{\mathcal A}}
\def\ds{\displaystyle}


\newcommand{\reals}{\mathbb R}
\newcommand{\complex}{\mathbb C}
\newcommand{\nat}{\mathbb N}



\def\proof{\noindent{\bf Proof.} }


 \def\endproof{\qed}


 \def\bydef{\stackrel{def}{=}}


 \def\essinf{\hbox{\rm ess~inf}}


 \def\Rp{{\rm Re}~}


 \def\loc{\hbox{\scriptsize loc}}




  \begin{quote}
\footnotesize {\bf Abstract.} In this paper we study the
generation of analytic semigroup in the space $L^2(\reals^d)$, and
the characterization of the domain, for a family of degenerate
elliptic operators with unbounded coefficients, which includes
some well-known operators arising in Mathematical Finance. To
prove the generation of analytic semigroups, the operators of the
family are assumed to satisfy suitable growth and compensation
conditions. Under stronger assumptions, we obtain also the
characterization of the domain. Finally, various consequences of
the obtained results are considered in connection with some
applications (see, e.g., \cite{BARGOVE} for financial
applications). In a forthcoming paper, part II of this work, we
shall examine the problem of the generation of analytic semigroup
in $L^p(\reals^d)$, where $p\in(2,+\infty]$, and the
characterization of the domain, for the same family of operators.


\end{quote}



\section{Introduction}


In this paper we study the generation of analytic semigroup in the
space $L^2(\reals^d)$, and the characterization of the domain, for
a family of degenerate elliptic operators with unbounded
coefficients, which includes some well-known operators arising in
Mathematical Finance, and also in Mathematical Physics. These
results can be employed to obtain existence, uniqueness, and
regularity estimates for the solutions of the associated (linear
or semilinear) parabolic problems, through the well-known theory
of analytic semigroups (e.g., \cite{Lun}).


This has been done partly in \cite{BARGOVE} for some so-called
\lq\lq no-\-ar\-bi\-tra\-ge\rq\rq operators arising in contingent
claim pricing.


We consider the following differential operator in $\reals^d$
\begin{equation}
\label{SECA}
\A\equiv\sum_{i,j=1}^{d}\psi_{i}(x)\psi_{j}(x)a_{i,j}(x)D_{i,j}
+\sum_{i=1}^{d}b_{i}(x)D_{i}-\gamma^2(x),
\end{equation}
where $\{a_{i,j}(x)\}_{i,j=1}^{d}$ is a uniformly elliptic bounded
and measurable real-valued d-order matrix on $\reals^d$, and the
coefficients $\psi_i(x)$'s, $b_i(x)$'s and $\gamma(x)$ are
measurable (not bounded) real-valued functions on $\reals^d$, with
the $\psi_i(x)$'s allowed to vanish in a negligible set $Z$ at
most.


It is well known that, in general, these operators do not generate
analytic or strongly continuous semigroup (e.g. consider the
one-dimensional Ornstein--Uhlenbeck operator $\sigma D^{2}u+xDu$,
where $\sigma$ is a positive constant). Indeed, the main
difficulty to overcome, to obtain some generation result, is to
manage both the possible unboundedness of all coefficients, and
the possible degeneration of the ones of the second-order terms.
However, under suitable growth and compensation conditions on the
coefficients, we can obtain the generation result and the
characterization of the domain. The settlement of these conditions
plays a central role in this paper. Actually they come out as a
compromise between our attempt of covering, in a unitary approach,
several operators interesting for the applications, and to avoid
as many technicalities as possible in the proofs.


To give the reader an insight on the above-mentioned conditions
and their application, we anticipate here the simple case of
dimension $d=1$. In this case we shall ask for the existence of
two suitable constants $B_1$ and $B_2$ with $B_1+B_2<2$ such that
\begin{equation}
\label{Non-Var.-Case-Golden-Cond.-I-INTRO} \left|b(x)\right|\leq
B_1 E^{1/2}\left|\psi(x)\right|\gamma(x) \quad\forall x\in\reals,
\end{equation}
and
\begin{equation}
\label{Non-Var.-Case-Golden-Cond.-II-INTRO}
\left|D\left(\psi(x)^{2}a(x)\right)\right|\leq
B_2E^{1/2}\left|\psi(x)\right|\gamma(x) \quad\forall x\in\reals,
\end{equation}
where $E$ is the modulus of ellipticity of
$\{a_{i,j}(x)\}_{i,j=1}^{d}$. For example,
(\ref{Non-Var.-Case-Golden-Cond.-I-INTRO}) and
(\ref{Non-Var.-Case-Golden-Cond.-II-INTRO}) are easily verified by
the second-order differential operator
\[
x^{2}D^{2}+xD-(x^{2}+4),
\]
by choosing $B_1=1/2$ and $B_2=1$. This clearly implies the
generation result for the whole class of the modified Bessel
operators
\[
x^{2}D^{2}+xD-(x^{2}+\nu^{2}),
\]
where $\nu\in\reals$.


The above simple example also shows how to exploit the results of
this paper. Namely, the verification of our assumptions for a
suitably chosen operator actually yields the desired generation
result, and the characterization of the domain, for a wider family
of operators, which is deducible from the chosen one by some
analytic perturbation (see \cite{KA}).


We shall show that other classical operators, as the generalized
Sch\-r\"o\-din\-ger operator, and a wide class of diffusion
generators, among which the Black and Scholes operator, can be
handled by the approach outlined above.


Elliptic operators, that possibly degenerate somewhere in the
domain, or have unbounded coefficients, have been studied by many
authors. Among them we recall Baouendi and Goulaouic \cite{BG1},
\cite{BG2}, Clement and Timmerman \cite{CT}, and Vespri
\cite{VVannali}, who have proved the generation of analytic
semigroups for some operators whose coefficients are strongly
elliptic in the interior of their bounded domain, but possibly
degenerating in the boundary. We recall also Aronson and Besala
\cite{AB}, \cite{B}, and Cannarsa, Lunardi and Vespri \cite{CV1},
\cite{CV2}, \cite{LV1}, \cite{LV2}, who have obtained the
generation of analytic or strongly continuous semigroups for wide
families of operators with unbounded coefficients in $\reals^d$,
but satisfying everywhere the strong ellipticity condition.
Finally, we recall Cerrai \cite{Sandra}, who uses stochastic
methods to handle operators similar to ours, but with different
conditions and results, Campiti and Metafune \cite{CM}, \cite{M},
who also studied operators similar to ours, but in the
one-dimensional case and with the Ventcel's boundary conditions,
and Colombo, Giuli and Vespri \cite{CGV}, who obtained the
generation of strongly continuous semigroup for differential
operators in $\reals^d$, very similar to ours, but in divergence
form and with different compensation conditions.


In this paper we assume that the coefficients of the operator
$\A$, given by $(\ref{SECA})$, are defined in all $\reals^d$, and
we prove first the generation of analytic semigroups in the space
$L^2(\reals^d)$. This is obtained by a straightforward application
of standard Hilbert-space techniques, relying on suitable
preliminary a priori estimates, which are allowed by a natural
choice of the compensation conditions on the coefficients of $\A$.
Then we obtain the characterization of the domain of $\A$ by a
localization procedure adapted to the growth rate of the weights
$\psi_i(x)$'s at infinity and close to $Z$.


In a forthcoming paper, part II of this work, we show how to pass
from $L^2(\reals^d)$ to $L^p(\reals^d)$, for $p>2$, by applying a
suitable modification of Stewart's method \cite{S1,S2,VVannali}.


Our main aim is to provide a unitary framework in which we obtain
the generation of analytic semigroups and the characterization of
the domain for some elliptic operators arising in Mathematical
Finance. Indeed, in the no-arbitrage pricing theory, which plays a
central role in modern Mathematical Finance, it is important to
provide general existence, uniqueness and regularity results for
some parabolic equations. The semigroup techniques are then useful
tools to get this kind of results, provided one can establish the
semigroup generation and the characterization of the domain for
the elliptic operator associated to the parabolic equation
considered. In addition, a remark after Garman (see \cite{GAR85})
lightens the financial role played by the semigroup property in
the framework of no-arbitrage prices. This gives one more
motivation for applying the semigroup theory to problems coming
from Mathematical Finance (see Section \ref{APPLICATIONS}).


Finally, we point out that the analysis of several operators
connected with financial applications calls for a restriction of
the domain of the coefficients in $\reals^d_{+}$, or even in a
subset of it. Thus, an appropriate setting of the Cauchy problem
associated to the operator $\A$ is usually the positive orthant
$\reals^d_{+}$. In the above-mentioned forthcoming paper we will
show how to overcome this difficulty in some interesting cases, by
imposing suitable boundary conditions. This will allows us to
apply our results to prove existence and uniqueness of the
solution of the Cauchy problem associated to the operator $\A$ and
to give estimates of the solution of the no-arbitrage price and of
its derivatives for a large class of contingent-claim contracts
(see \cite{BARGOVE}).


The paper is organized as follows. In Section \ref{PRE} we
introduce the notation and recall some results about weighted
spaces and analytic semigroups. Section \ref{L2} is the heart of
the paper and is divided into four parts: the first two on the
generation of analytic semigroup, the third on the
characterization of the domain, and the fourth on the weighted
case. Finally, in Section \ref{APPLICATIONS} we show some examples
of operators which fit our setting.


\section{Preliminary material and notation}


\label{PRE}


Let $\Omega$ be an open subset of the $d$-dimensional Euclidean
space $\reals^d$. We denote by $C^\infty(\Omega)$ the linear space
of all infinitely differentiable complex-valued functions on
$\Omega$, and we write $C_c^\infty(\Omega)$ for the linear
submanifold of $C^\infty(\Omega)$ of all functions with compact
support in $\Omega$. For every integer $k\geq 0$, and every real
$1\leq p<\infty$, we denote by $C^{\infty,n,p}(\Omega)$ the linear
submanifold of $C^\infty(\Omega)$ of all functions $\varphi$ such
that $\sum_{\left|\alpha\right|\leq n}
(\int_\Omega\left|D^\alpha\varphi(x)\right|^pdx)^{1/p}<\infty$.
Here we are employing the usual multi-index differential notation,
$\alpha\equiv(\alpha_1,\dots,\alpha_d)$,
$\left|\alpha\right|\equiv\sum_{i=1}^d\alpha_i$, $D^\alpha\equiv
D^{\alpha_1}\cdots D^{\alpha_d}$,
$D^{\alpha_i}\equiv\partial^{\alpha_i}/\partial x_i^{\alpha_i}$.
Such a notation will be dropped in the sequel, writing
$\int_\Omega\left|\varphi(x)\right|^pdx$,
$\sum_{i=1}^d\int_\Omega\left|D_i\varphi(x)\right|^pdx$, and
$\sum_{i,j=1}^d\int_\Omega\left|D_{i,j}\varphi(x)\right|^pdx$,
rather than
$\sum_{\left|\alpha\right|=n}\int_\Omega\left|D^\alpha\varphi(x)\right|^pdx$,
when $n=0,1,2$, respectively.


We denote by $W^{n,p}(\Omega)$ the usual {\em Sobolev space} (see,
e.g., \cite{Ad}), defined as the completion of
$C_c^\infty(\Omega)$ with respect to the norm
\[
\Vert u\Vert_{W^{n,p}(\Omega)} \bydef
\sum_{\left|\alpha\right|\leq n} \Big (\int_{\Omega}\left|D^\alpha
u(x)\right|^pdx\Big )^{1/p},
\]
writing $L^p(\Omega)$ [resp. $H^n(\Omega)$] rather than
$W^{0,p}(\Omega)$ [resp. $W^{n,2}(\Omega)$], and using the
shorthands $W^{n,p}$, $L^p$ and $H^n$ for $W^{n,p}(\reals^d)$,
$L^p(\reals^d)$ and $H^n(\reals^d)$, respectively.


We denote by $W_{\loc}^{n,p}$ [resp. $L_{\loc}^p$, $H_{\loc}^n$]
the linear space of all measurable complex-valued functions on
$\reals^d$ belonging to $W^{n,p}(\Omega)$ [resp. $L^p(\Omega)$,
$H^n(\Omega)$] for every open subset $\Omega$ of $\reals^d$ having
compact closure, and, for any fixed real-valued function $\xi\in
W_{\loc}^{n,p}$, we define the {\em weighted Sobolev space}
$W_\xi^{n,p}$ as the completion of $C_c^\infty(\reals^d)$ with
respect to the {\em weighted norm}
\[
\Vert u\Vert_{W_\xi^{n,p}} \bydef \Vert \xi u\Vert_{W^{n,p}},
\]
It is well known that $W_\xi^{n,p}$ can also be defined as the
space of all measurable functions $u$ such that $\xi u\in
W^{n,p}$. Similarly, for any choice of the functions $\alpha$,
$\beta_i$, $i=1,\dots,d$, $\delta_{i,j}$, $i,j=1,\dots,d$
belonging to $L^p_{\loc}$, with $\essinf\left|\alpha\right|>0$, we
introduce the weighted Sobolev spaces $W_{(\alpha,\beta)}^{1,p}$
and $W_{(\alpha,\beta,\delta)}^{2,p}$ defined as the completion of
$C_c^\infty(\reals^d)$ with respect to the {\em weighted norm}
\[
\Vert u\Vert_{W_{(\alpha,\beta)}^{1,p}} \bydef \|\alpha
u\|_{L^p}+\sum_{i=1}^{d}\|\beta_i D_i u\|_{L^p}
\]
and
\[
\Vert u\Vert_{W_{(\alpha,\beta,\delta)}^{2,p}} \bydef \|\alpha
u\|_{L^p}+\sum_{i=1}^{d}\|\beta_i D_i u\|_{L^p}
+\sum_{i,j=1}^{d}\|\delta_{i,j}D_{i,j} u|_{L^p},
\]
respectively, and we introduce also the spaces
$W_{\xi,(\alpha,\beta)}^{1,p}$ [resp.
$W_{\xi,(\alpha,\beta,\delta)}^{2,p}$] of all measurable functions
$u$ such that $\xi u\in W_{(\alpha,\beta)}^{1,p}$ [resp. $\xi u\in
W_{(\alpha,\beta,\delta)}^{2,p}$], endowed with the norms
\[
\Vert u\Vert_{W_{\xi,(\alpha,\beta)}^{1,p}} \bydef \Vert\xi
u\Vert_{W_{(\alpha,\beta)}^{1,p}} \quad[\hbox{resp. } \Vert
u\Vert_{W_{\xi,(\alpha,\beta,\delta)}^{2,p}} \bydef \Vert\xi
u\Vert_{W_{(\alpha,\beta,\delta)}^{2,p}}].
\]


Finally, we shall also write $L_\xi^p$, $H_\xi^n$,
$H_{(\alpha,\beta)}^n$, $H_{(\alpha,\beta,\delta)}^n$,
$H_{\xi,(\alpha,\beta)}^n$ and $H_{\xi,(\alpha,\beta,\delta)}^n$
in conformity with the notation introduced above.


\section{Generation of analytic semigroups in $L^2(\reals^d)$}
\label{L2} In this section we consider the realization of the
operator $\A$ in $L^2$. For clearness we divide the analysis into
four subsections:
\begin{itemize}
\item
first, we consider the easiest case, in which the operator is
written in variational form, and we prove the generation of
analytic semigroups, finding also the consequent estimates for the
first derivatives;
\item
second, we examine the nonvariational case, again proving the
generation of analytic semigroups and the estimates for the first
derivatives;
\item
third, we characterize the domain of the realization $\A_2$ of
$\A$ in $L^2$;
\item
fourth, we extend the previous results to the case of a weighted
space $L^2_\xi$, for a suitable weight $\xi$.
\end{itemize}



\subsection{Operators in variational form}
\label{VARIATIONAL}


Let us consider the second-order differential operator in
variational form
\begin{equation}
\widehat{\A}u \bydef
\sum_{i,j=1}^dD_j(\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu)
+\sum_{i=1}^db_i(x)D_iu -\gamma^2(x)u, \label{widehat(A)}
\end{equation}
which satisfies the assumption below.
\begin{Assumption}
\label{hp:variational} Suppose that the following conditions hold
true
\begin{enumerate}
\item[]
\item
%\label{hp:variational-1}
For all $i,j=1,\dots,d$, the coefficients $a_{i,j}(x)$ are bounded
measurable real-valued functions on $\reals^d$ such that
$a_{i,j}(x)=a_{j,i}(x)$, and satisfying the strong ellipticity
condition
\begin{equation}
\Rp\sum_{i,j=1}^da_{i,j}(x)z_i\overline{z}_j\geq E\left|z\right|^2
\quad\forall z\in\complex^d, \label{hp:variational-1}
\end{equation}
for a suitable modulus of ellipticity $E>0$ independent of
$x\in\reals^d;$
\item
%\label{hp:variational-2}
for every $i=1,\dots,d$, the coefficients $b_i(x)$ are measurable
real-valued functions on $\reals^d$, while $\gamma(x)$ and
$\psi_i(x)$ are real-valued functions in $L_{\loc}^2$ with
$\essinf\gamma\geq 1$; $^1$\footnote{$^1$This condition could be
replaced by the seemingly more general $\essinf\gamma>0$, provided
to employ a standard normalization procedure.}
\item
for every $i=1,\dots,d$, we have
\begin{equation}
\label{Var.-Case-Golden-Cond.} \left|b_i(x)\right|\leq
BE^{1/2}\eta_i(x)\left|\psi_i(x)\right|\gamma(x) \quad\forall
x\in\reals^d,
\end{equation}
for a suitable constant $B<2$ and measurable real-valued functions
$\eta_i(x)$ on $\reals^d$ such that $\sum_{i=1}^d\eta_i^2(x)=1$.
\end{enumerate}
\end{Assumption}


\begin{Remark}
{\rm Note that in the case $b_i(x)=0$, for every $i=1,\dots,d$,
the operator $\widehat{\A}u$ reduces to a classical generalization
of the Schr\"odinger operator (see \cite{KA}). In this case
$\widehat{\A}u$ is clearly formally selfadjoint and it is easily
checked that it is also negative. Hence, it is a classical problem
to ask whether a suitable extension of $\widehat{\A}$ generates an
analytic semigroup. On the other hand, in this particular case,
(\ref{Var.-Case-Golden-Cond.}) is trivially satisfied by choosing
$B=0$. Therefore, as a byproduct of our approach, we obtain the
generation and the characterization of the domain for the
classical family of generalized Schr\"odinger operators. }
\end{Remark}


\begin{Remark}
\label{rm:hp1} \rm As mentioned in the introduction, we could also
consider more general hypotheses on the coefficients $\gamma$ and
$\psi_i$. In particular, by minor modifications of the arguments
below, we could deal with coefficients having isolated
nonintegrable singularities. In addition, the arguments exploited
in the proofs clearly show that Assumption \ref{hp:variational}.3,
and the forthcoming \ref{hp:nonvariational}.2, could be relaxed by
requiring that they hold true, but a permutation of the occurring
indices. Namely, we could rewrite \ref{hp:variational}.3 as
follows:
\begin{enumerate}
\item[$3^{\prime}$.]
There exists a permutation $\sigma$ of $\{1,\dots,d\}$ such that,
for every $i=1,\dots,d$, we have
\[
\left|b_i(x)\right|\leq
BE^{1/2}\eta_{\sigma(i)}(x)\left|\psi_{\sigma(i)}(x)\right|\gamma(x)
\quad\forall x\in\reals^d.
\]
\end{enumerate}
However, for the sake of simplicity, we have preferred to avoid
such a more general formulation, except for the exhibition of a
simple example when treating the nonvariational case.
\end{Remark}


In order to obtain a realization
$\widehat{\A}_2:D(\widehat{\A}_2)\to L^2$ of the formal operator
$\widehat{\A}$ which generates an analytic semigroup, we begin by
defining a suitable domain.


Write $\psi\equiv(\psi_1,\dots,\psi_n)$ and consider the weighted
Sobolev space $H_{(\gamma,\psi)}^1$ endowed with the norm
$\Vert\cdot\Vert_{H_{(\gamma,\psi)}^1}$. This space becomes a
dense linear submanifold of $L^2$ thanks to the condition
$\essinf\gamma\geq 1$. Then define $$ D(\widehat{\A}_2)\! \bydef
\! \{u\in H_{(\gamma,\psi)}^1
 : \exists\, K(u) \!>0\!\;\hbox{s.t.}\; |\widehat{a}(u,\varphi) |
\leq K(u)\Vert\varphi\Vert_{L^2}\;\forall\varphi\!\in \!C_c^\infty
(\reals^d) \}, $$ where $\widehat{a}(\cdot,\cdot)$ is the
sesquilinear form associated to $\widehat{\A}_2$ given by
\begin{eqnarray*}
\widehat{a}(u,v) &\bydef
&-\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{v}(x)dx\\
&&+\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{v}(x)dx
-\int_{\reals^d}\gamma^2(x)u(x)\overline{v}(x)dx,
\end{eqnarray*}
for all $u,v\in L^2$ such that the above integrals make sense.
Since $C_c^{\infty}(\reals^d)$ is dense in $L^2$, for each $u\in
D(\widehat{\A}_2)$ the antilinear functional
\[
\varphi\in
C_c^\infty(\reals^d)\to\widehat{a}(u,\varphi)\in\complex
\]
may be continuously extended to the whole of $L^2$. Therefore, by
virtue of Riesz' theorem, there exists a unique $f\in L^2$ such
that
\[
\widehat{a}(u,\varphi)=\left\langle f,\varphi\right\rangle,
\]
for every $\varphi\in L^2$. This implies that if we choose
\[
\widehat{\A}_2u\bydef f,
\]
then the operator $\widehat{\A}_2:D(\widehat{\A}_2)\to L^2$ is
well defined. It follows that we can characterize a function $u\in
D(\widehat{\A}_2)$ as a weak solution of the resolvent equation
\begin{equation}
(\lambda-\widehat{\A}_2)u=f, \label{var.-case-res.-equ.}
\end{equation}
for any fixed $\lambda\in\complex$, if and only if we have
\begin{align*}
\int_{\reals^d}f(x)\overline{\varphi}(x)dx &=
\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{\varphi}(x)dx\\
& -\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{\varphi}(x)dx
+\int_{\reals^d}(\lambda+\gamma^2(x))u(x)\overline{\varphi}(x)dx
\end{align*}
for every $\varphi\in C_c^\infty(\reals^d)$.


Following Lunardi \cite[Chapter 3]{Lun}, we establish the
announced generation result for the differential operator
$\widehat{\A}_2$ in two steps.


First we show that, for each $\lambda\in\complex$ such that
$\Rp\lambda>0$, the resolvent equation (\ref{var.-case-res.-equ.})
has a unique solution in $D(\widehat{\A}_2)$ for every $f\in L^2$.
$^1$\footnote{Notice that, as a consequence of this result, the
resolvent set of the operator $\widehat{\A}_2$ contains the half
plane $\left\{\lambda\in\complex : \Rp\lambda>0\right\}$. This
implies no loss of generality, thanks to the already-mentioned
analytic perturbation techniques.} Then we prove that, for every
such $\lambda$, the $\lambda$-resolvent of the operator
$\widehat{\A}_2$ satisfies
\[
\Vert\lambda R(\lambda;\widehat{\A}_2)\Vert\leq K,
\]
for a suitable $K>0$. The above procedure will be performed by
applying some preliminary lemmas.


>>>From now on we write
$A$ for $\max_{i,j=1,\dots,d}\Vert a_{i,j}\Vert_\infty$, where
\[
\Vert a_{i,j}\Vert_\infty
\equiv\sup_{x\in\reals^d}\left|a_{i,j}(x)\right| , \ \ \text{for
all $i,j=1,\dots,d$.}
\]


\begin{Lemma}
\label{Var.-Case-Cont.-Est.} Under Assumption
\ref{hp:variational}, for all $u,v\in H_{(\gamma,\psi)}^1$, we
have
\begin{align}
\label{Var.-Case-I-Cont.-Est.} &
\int_{\reals^d}\sum_{i,j=1}^d\left|\psi_i(x)\psi_j(x)a_{i,j}(x)
D_iu(x)D_j\overline{v}(x)\right|dx
\\
& \leq A\Big (\sum_{i=1}^d\Vert \psi_iD_iu\Vert_{L^2}\Big)
\Big(\sum_{i=1}^d\Vert \psi_iD_iv\Vert_{L^2}\Big), \nonumber
\end{align}
\begin{equation}
\label{Var.-Case-II-Cont.-Est.}
\int_{\reals^d}\sum_{i=1}^d\left|b_i(x)D_iu(x)\overline{v}(x)\right|dx
\leq BE^{1/2}\Vert \gamma v\Vert_{L^2} \Big(\sum_{i=1}^d\Vert
\psi_iD_iu\Vert_{L^2}\Big)
\end{equation}
and
\begin{equation}
\label{Var.-Case-III-Cont.-Est.}
\int_{\reals^d}\left|(\lambda+\gamma^2(x))u(x)\overline{v}(x)\right|dx
\leq(\left|\lambda\right|+1)\Vert\gamma u\Vert_{L^2}\Vert\gamma
v\Vert_{L^2}.
\end{equation}
\end{Lemma}
\proof The first claim follows by Schwarz's inequality, which
gives $$ \int_{\reals^d}\sum_{i,j=1}^d
\left|\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{v}(x)\right|dx
\leq A\sum_{i,j=1}^d\Vert\psi_iD_iu\Vert_{L^2}
\Vert\psi_jD_j\overline{v}\Vert_{L^2}. $$ Then, by virtue of
(\ref{Var.-Case-Golden-Cond.}), we have
\begin{align*}
& {\sum_{i=1}^d\int_{\reals^d}
\left|b_i(x)D_iu(x)\overline{v}(x)\right|dx}
 \\
& \leq BE^{1/2}\sum_{i=1}^d\int_{\reals^d}
\left|\eta_i(x)\psi_i(x)\gamma(x)D_iu(x)\overline{v}(x)\right|dx\\
& \leq BE^{1/2}\sum_{i=1}^d \Big
(\int_{\reals^d}\eta_i^2(x)\left|\gamma(x)\overline{v}(x)\right|^2dx\Big
)^{1/2} \Big (\int_{\reals^d}\left|\psi_i(x)D_iu(x)\right|^2dx\Big
)^{1/2}\\ & \leq BE^{1/2}\Big
(\int_{\reals^d}\left|\gamma(x)\overline{v}(x)\right|^2dx\Big
)^{1/2} \sum_{i=1}^d \Big
(\int_{\reals^d}\left|\psi_i(x)D_iu(x)\right|^2dx\Big )^{1/2},
\end{align*}
which is the desired (\ref{Var.-Case-II-Cont.-Est.}). Finally,
since $\essinf\gamma\geq 1$ we can write
\begin{eqnarray*}
\left|(\lambda+\gamma^2(x))u(x)\overline{v}(x)\right|
\leq(\left|\lambda\right|+1)\left|\gamma(x)u(x)\right|\left|\gamma(x)v(x)\right|,
\end{eqnarray*}
for every $x\in\reals^d$, and the latter implies
(\ref{Var.-Case-III-Cont.-Est.}), by applying again Schwarz's
inequality. \hfill\qed


\begin{Lemma}
\label{Var.-Case-Coerc.-Est.} Under Assumption
\ref{hp:variational}, for every $u \in H_{(\gamma,\psi)}^1$, we
have
\begin{eqnarray}
\hbox{\rm Re}\int_{\reals^d}\sum_{i,j=1}^d
\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{u}(x)dx \geq
E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2
\label{Var.-Case-I-Coerc.-Est.}
\end{eqnarray}
and
\begin{equation}
\int_{\reals^d}\sum_{i=1}^d\left|b_i(x)D_iu(x)\overline{u}(x)\right|dx
\leq\frac B2\Big (\Vert\gamma
u\Vert_{L^2}^2+E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2\Big).
\label{Var.-Case-II-Coerc.-Est.}
\end{equation}
\end{Lemma}
\proof Estimate (\ref{Var.-Case-I-Coerc.-Est.}) follows
immediately from (\ref{hp:variational-1}). Furthermore,
(\ref{Var.-Case-Golden-Cond.}) allows us to write
\begin{align*}
&
\lefteqn{\sum_{i=1}^d\int_{\reals^d}\left|b_i(x)D_iu(x)\overline{u}(x)\right|dx}
 \\
& \leq B\sum_{i=1}^d\int_{\reals^d}
\left|E^{1/2}\eta_i(x)\psi_i(x)\gamma(x)D_iu(x)\overline{u}(x)\right|dx\\
& \leq B\sum_{i=1}^d
 \Big (\int_{\reals^d}\eta_i^2(x)\left|\gamma(x)\overline{u}(x)\right|^2dx \Big
)^{1/2}
 \Big (\int_{\reals^d}E\left| \psi_i(x)D_iu(x)\right| ^2dx \Big )^{1/2}\\
& \leq B\sum_{i=1}^d\frac{1}{2}
 \Big (\int_{\reals^d}\eta_i^2(x)\left|\gamma(x)\overline{u}(x)\right|^2dx
+\int_{\reals^d}E\left|\psi_i(x)D_iu(x)\right|^2dx \Big )\\ &
=\frac B2 \Big (\int_{\reals^d} \Big (\sum_{i=1}^d\eta_i^2(x) \Big
) \left|\gamma(x)\overline{u}(x)\right|^2dx
+E\sum_{i=1}^d\int_{\reals ^d}\left|\psi_i(x)D_iu(x)\right|^2dx
\Big ),
\end{align*}
and this completes the proof. \hfill\qed


We are now in a position to carry on our procedure.
\begin{Proposition}
\label{Var.-Case-Ex.-Inv.} Under Assumption \ref{hp:variational},
for each $\lambda\in\complex$ such that $\Rp\lambda>0$, Equation
$(\ref{var.-case-res.-equ.})$ has a unique solution $u\in
D(\widehat{\A}_2)$ for every $f\in L^2(\reals^d)$.
\end{Proposition}
\proof For each $\lambda\in\complex$, we introduce the
sesquilinear form $\widehat{a}_\lambda(\cdot,\cdot)$ on the
Hilbert space $H_{(\gamma,\psi)}^1$ given by
\begin{eqnarray*}
\widehat{a}_\lambda(u,v)
&\bydef&\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{v}(x)dx\\
&&-\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{v}(x)dx
+\int_{\reals^d}(\lambda+\gamma^2(x))u(x)\overline{v}(x)dx.
\end{eqnarray*}
 Then (\ref{Var.-Case-I-Cont.-Est.})--(\ref{Var.-Case-III-Cont.-Est.})
imply that, for all $u,v\in H_{(\gamma,\psi)}^1$, we have
\begin{align*}
\left|\widehat{a}_\lambda(u,v)\right| & \leq
A(\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2})
(\sum_{i=1}^d\Vert\psi_iD_iv\Vert_{L^2})\\ & +BE^{1/2}\Vert\gamma
v\Vert_{L^2}(\sum_{i=1}^d
\Vert\psi_iD_iu\Vert_{L^2})+(\left|\lambda\right|+1) \Vert\gamma
u\Vert_{L^2}\Vert\gamma v\Vert_{L^2}
\\
& \leq  H\Vert u\Vert_{H_{(\gamma,\psi)}^1}\Vert
v\Vert_{H_{(\gamma,\psi)}^1},
\end{align*}
for a suitable $H>0$. Moreover, if $\Rp\lambda >0$, thanks to
(\ref{Var.-Case-I-Coerc.-Est.}) and
(\ref{Var.-Case-II-Coerc.-Est.}), we can write
\begin{align*}
&{\Rp\widehat{a}_\lambda(u,u)}
  \geq\Rp\int_{\reals^d}\sum_{i,j=1}^d
\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{u}(x)dx\\ & -\Big
|\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{u}(x)dx\Big|
+\Rp\lambda\int_{\reals^d}\left|u(x)\right|^2dx
+\int_{\reals^d}\gamma^2(x)\left|u(x)\right|^2dx\\ & \geq \!
 E\sum_{i=1}^d\Vert \psi_iD_iu\Vert_{L^2}^2
-\frac B2(\Vert \gamma u\Vert_{L^2}^2 +E\sum_{i=1}^d\Vert
\psi_iD_iu\Vert_{L^2}^2) +\Rp\lambda\Vert u\Vert_{L^2}^2
+\Vert\gamma u\Vert_{L^2}^2\\ & \geq K\Vert
u\Vert_{H_{(\gamma,\psi)}^1}^2,
\end{align*}
for a suitable $K>0$. Therefore, since for each $f\in L^2$ the map
$ v\in H_{(\gamma,\psi)}^1\to\left\langle f,v\right\rangle $
defines a continuous antilinear functional on
$H_{(\gamma,\psi)}^1$, a standard application of the Lax--Milgram
theorem allows us to conclude that for every $f\in L^2$ there
exists a unique $u\in H_{(\gamma,\psi)}^1$ satisfying the equation
\[
\widehat{a}_\lambda(u,\varphi)=\left\langle
f,\varphi\right\rangle,
\]
for every $\varphi \in C_c^\infty(\reals^d)$. Moreover, since $$
\widehat{a}(u,\varphi) =\lambda\left\langle
u,\varphi\right\rangle-\widehat{a}_\lambda(u,\varphi), $$ it is
clear that such a $u$ belongs to $D(\widehat{\A}_2)$. Therefore,
for each $\lambda \in \complex$ such that $\Rp\lambda
>0$, the operator $\lambda-\widehat{\A}_2:D(\widehat{\A}_2)\to
L^2 $ is invertible, as desired. \hfill\qed
\begin{Proposition}
\label{Var.-Case-Bound.-Inv.} Under Assumption
\ref{hp:variational}, for every $\lambda\in\complex$ such that
$\Rp\lambda >0$, the $\lambda$-resolvent
$R(\lambda;\widehat{\A}_2)$ of $\widehat{\A}_2$ satisfies
\[
\Vert\lambda R(\lambda;\widehat{\A}_2)\Vert\leq K,
\]
for a suitable $K>0$ independent of $\lambda$.
\end{Proposition}
\proof Given any fixed $\lambda\in\complex$ such that
$\Rp\lambda>0$, let $u\in D(\widehat{\A}_2)$ be the solution of
(\ref{var.-case-res.-equ.}) corresponding to some $f\in L^2$.
Since
\begin{equation}
\widehat{a}_\lambda(u,u)=\left\langle f,u\right\rangle
\label{Var.-Case-Fund.-Equality}
\end{equation}
holds true, we have
\begin{align*}
& \Vert u\Vert_{L^2}\Vert f\Vert_{L^2}
 \geq \Rp\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)
\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{u}(x)dx\\ &
-\Rp\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{u}(x)dx
+\Rp\int_{\reals^d}(\lambda+\gamma^2(x))u(x)\overline{u}(x)dx\\
&\geq  E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2 -\Big
|\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{u}(x)dx\Big |
+\Rp\lambda\Vert u\Vert_{L^2}^2 +\Vert\gamma u\Vert_{L^2}^2.
\end{align*}
Therefore, thanks to (\ref{Var.-Case-II-Coerc.-Est.}), we obtain
\begin{equation}
\label{Var.-Case-Fund.-Interm.-Est.} \frac{2}{2-B}\Vert
u\Vert_{L^2}\Vert f\Vert_{L^2} \geq\Vert\gamma u\Vert_{L^2}^2
+E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2.
\end{equation}
On the other hand, (\ref{Var.-Case-Fund.-Equality}) also yields
\begin{align*}
& \int_{\reals^d}\lambda\left|u(x)\right|^2dx
 =
-\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_iuD_j\overline{u}(x)dx\\
& +\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{u}(x)dx
-\int_{\reals^d}\gamma^2(x)\left|u(x)\right|^2dx
  +\int_{\reals^d}f(x)\overline{u}(x)dx,
\end{align*}
and the latter, by virtue of (\ref{Var.-Case-I-Cont.-Est.}), (\ref
{Var.-Case-II-Coerc.-Est.}) and
(\ref{Var.-Case-Fund.-Interm.-Est.}), implies
\begin{align*}
|\lambda|\|u\|_{L^2}^2 &\leq
A(\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2})^2
+\frac{B}{2}(\Vert\gamma
u\Vert_{L^2}^2+E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2)\\ &
+\Vert\gamma u\Vert_{L^2}^2+\Vert f\Vert_{L^2}\Vert u\Vert_{L^2}\\
\leq Ad & \sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2+\Vert\gamma
u\Vert_{L^2}^2 +\frac{2}{2-B}\Vert u\Vert_{L^2}\Vert f\Vert_{L^2}
\leq K\Vert u\Vert_{L^2}\Vert f\Vert_{L^2},
\end{align*}
for a suitable $K>0$ independent of $\lambda$. It then follows
\begin{equation}
|\lambda|\Vert u\Vert_{L^2}\leq K\Vert f\Vert_{L^2},
\label{Fund.-Final-Est.}
\end{equation}
which is our claim. \hfill\qed


As a consequence of Propositions \ref{Var.-Case-Ex.-Inv.} and \ref
{Var.-Case-Bound.-Inv.} it follows (see, e.g., \cite[p. 37]{Lun})


\begin{Theorem}
Under Assumption \ref{hp:variational}, the operator
$\widehat{\A}_2:D(\widehat{\A}_2)\to L^2$ generates an analytic
semigroup on $L^2$.
\end{Theorem}
In addition, we have
\begin{Corollary}
Under Assumption \ref{hp:variational}, every solution $u\in
D(\widehat{\A}_2)$ of $(\ref{var.-case-res.-equ.})$ satisfies
\[
\begin{array}{lll}
\left|\lambda\right|^{1/2}\Vert\gamma u\Vert_{L^2} \leq
K^{\prime}\Vert f\Vert_{L^2} &\hbox{and}&
\left|\lambda\right|^{1/2}\Vert\psi_iD_iu\Vert_{L^2}\leq
K^{\prime\prime}\Vert f\Vert_{L^2},
\end{array}
\]
for suitable $K^{\prime}, K^{\prime\prime}>0$ independent of
$\lambda$.
\end{Corollary}
\proof The claim immediately follows combining (\ref
{Fund.-Final-Est.}) with (\ref{Var.-Case-Fund.-Interm.-Est.}).
\hfill\qed


\subsection{Operators in nonvariational form}
Let us now consider the formal second-order differential operator
\begin{equation}
\A u \bydef\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_{i,j}u
+\sum_{i=1}^db_i(x)D_iu-\gamma^2(x)u, \label{operator(A)}
\end{equation}
whose coefficients satisfy conditions similar to those on the
coefficients of the operator (\ref{widehat(A)}), except for some
slight modifications and some additional requirements. Namely we
suppose
\begin{Assumption}
\label{hp:nonvariational} Conditions (2) and (3) of Assumption
\ref{hp:variational} hold true. In addition,
\begin{enumerate}
\item
\label{hp:nonvariational-1} for all $i,j=1,\dots,d$ the
coefficients $a_{i,j}(x)$ are differentiable on $\reals^d$;
\item
for all $i=1,\dots,d$ the coefficients $\psi_i(x)$ are
differentiable, and we have
\begin{equation}
\label{Non-Var.-Case-I-Golden-Cond.}
\begin{array}{c}
\ds{\left|b_i(x)\right|\leq
B_1E^{1/2}\eta_{1,i}(x)\left|\psi_i(x)\right|\gamma(x)
\quad\forall x\in\reals^d},
\\
\\
\ds{\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))\right| \leq
B_2E^{1/2}\eta_{2,i,j}(x)\left|\psi_i(x)\right|\gamma(x)
\quad\forall x\in\reals^d,}
\end{array}
\end{equation}
for suitable constants $B_1$ and $B_2$ such that $B_1+B_2<2$ and
measurable positive functions $\eta_{1,i}(x)$ and
$\eta_{2,i,j}(x)$ satisfying $$
\sum_{i=1}^d\eta_{1,i}^2(x)=d\sum_{i,j=1}^d\eta_{2,i,j}^2(x)=1. $$
\end{enumerate}
\end{Assumption}


\begin{Remark}
 \rm
As anticipated in the Introduction, in the one-dimensional case
(\ref{Non-Var.-Case-I-Golden-Cond.}) calls for the existence of
constants $B_1$ and $B_2$ with $B_1+B_2<2$ such that
\begin{equation}
\label{Non-Var.-Case-I-Golden-Cond.-One-Dim.-Case}
\begin{array}{cc}
\left|b(x)\right|\leq B_1E^{1/2}\left|\psi(x)\right|\gamma(x)
\quad\forall x\in\reals^d,\\
\\
\left|D(\psi^2(x)a(x))\right|\leq
B_2E^{1/2}\left|\psi(x)\right|\gamma(x) \quad\forall x\in\reals^d.
\end{array}
\end{equation}
The latter can be easily verified for several well-known
operators, but a trivial analytic perturbation. Besides the
{\em\lq\lq modified\rq\rq Bessel operator,} we mention also the
infinitesimal generator of the geometric Brownian motion
\[
\tfrac{1}{2}\alpha^2x^2D^2+rxD
\]
where $\alpha,r\in\reals_+$.


\end{Remark}


\begin{Remark}
 \rm
The simplest nontrivial example in higher dimensions is perhaps
the infinitesimal generator of the process which is often used to
model the evolution in time of a basket of correlated risky
assets. Suppose we have a basket of $d>1$ correlated risky assets
whose evolution in time is described by the $d$-dimensional
process $\big(X_t\big )_{t\geq 0}\equiv\big (\big (X_t^{(1)}\big
)_{t\geq 0},\dots,\big (X_t^{(d)}\big )_{t\geq 0}\big )$ which
satisfies the system of differential equations
\[
dX_t^{(k)}=r_kX_t^{(k)}dt+X_t^{(k)}\sum_{j=1}^d\alpha_{j,k}\,dW_t^{(j)},
\quad k=1,\dots\,d,
\]
where $r_{k},\alpha_{j,k}\in\reals_{+}$, for all $j,k=1,\dots\,d$,
and $\left(W_t^{(j)}\right)_{t\geq 0}$, $j=1,\dots\,d$, are
independent Wiener processes. Then, it is well known that the
infinitesimal generator of the process $\big(X_t\big )_{t\geq 0}$
is the differential operator
\[
\frac{1}{2}\sum_{j,k=1}^{d}x_{j}x_{k} \Big
(\sum_{i=1}^{d}\alpha_{j,i}\alpha_{i,k}\Big )x_{j}x_{k}D_{j,k}+
\sum_{k=1}^{d}r_{k}x_{k}D_{k}.
\]
This operator clearly satisfies Assumption
\ref{hp:nonvariational}, but a trivial analytic perturbation.


\end{Remark}


\begin{Remark}
 \rm
As already mentioned, Condition (2) of \ref{hp:nonvariational}
could be relaxed by requiring that it holds true, but a
permutation of the occurring indices. For instance this applies to
the infinitesimal generator of the {\em Brownian motion in the
unit circle}, which is the bidimensional stochastic process $\big
(X_t^{(k)}\big )_{t\geq 0}$, $k=1,2$ satisfying the system of
differential equations
\[
dX_t^{(1)}=-\tfrac{1}{2}X_t^{(1)}dt-X_t^{(2)}dW_t,
\]
\[
dX_t^{(2)}=-\tfrac{1}{2}X_t^{(2)}dt+X_t^{(1)}dW_t,
\]
where $\left(W_t\right)_{t\geq 0}$ is a standard Wiener process.
The infinitesimal generator of the Brownian motion in the unit
circle is then given by
\[
\tfrac{1}{2}
\left(x_{2}^2D_{1}^2-2x_{1}x_{2}D_{1,2}+x_{1}^2D_{2}^2-x_{1}D_{1}-x_{2}D_{2}\right),
\]
and it satisfies Assumption \ref{hp:nonvariational}, but a
permutation of the occurring indices, and a trivial analytic
perturbation.
\end{Remark}


Assumption \ref{hp:nonvariational} allows us to reduce the
analysis of the nonvariational case to the analysis of the
variational one. Indeed, introducing the sesquilinear form
$a(\cdot,\cdot)$ associated to the operator $\A$, given by
\[
a(u,v) \bydef\widehat{a}(u,v)
-\int_{\reals^d}\sum_{i,j=1}^dD_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{v}(x)dx,
\]
for all $u,v\in H_{(\gamma,\psi)}^1$, and writing
\[
D(\A_2)\bydef\big \{u \!\in \! H_{(\gamma,\psi)}^1 : \exists
K(u)>0\;\hbox{s.t.}\;\left|a(u,\varphi)\right| \leq K(u)\Vert
\varphi\Vert_2\;\forall\varphi\in \!C_c^\infty(\reals^d)\big \},
\]
we can study the realization $\A_2:D(\A_2)\to L^2$ of $\A$
employing the results obtained for the operator $\widehat{\A}_2$.
To this end, following the pattern of the previous section, we
establish some preliminary lemmas.


\begin{Lemma}
\label{Non-Var.-Case-Cont.-Lemma} Under Assumption
\ref{hp:nonvariational}, for all $u,v\in H_{(\gamma,\psi)}^1$, we
have
\begin{align}
& \int_{\reals^d}\sum_{i,j=1}^d
\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{v}(x)\right|dx
\nonumber \\ & \leq B_2E^{1/2}d^{1/2}\Vert\gamma
v\Vert_{L^2}\sum_{i=1}^d \Vert\psi_iD_iu\Vert_{L^2}.
\label{Non-Var.-Case-Cont.-Est.}
\end{align}
\end{Lemma}
\proof Thanks to (\ref{Non-Var.-Case-I-Golden-Cond.}), a
straightforward computation similar to that in the proof of Lemma
\ref{Var.-Case-Cont.-Est.} gives
\begin{align*}
& {\sum_{i,j=1}^d\int_{\reals^d}
\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{v}(x)\right|dx}
 \\
& \leq B_2E^{1/2}\sum_{i,j=1}^d \Big
(\int_{\reals^d}d\eta_{2,i,j}^2(x) |\gamma(x)\overline{v}(x)
|^2dx\Big)^{\frac 12 }
\Big(\frac{1}{d}\int_{\reals^d}\left|\psi_i(x)D_iu(x)\right|^2dx\Big)^{\frac
12 }\\ & =B_2E^{1/2}d^{1/2}\Vert\gamma v\Vert_{L^2}
\sum_{i=1}^d\Vert\psi_i D_iu\Vert_{L^2},
\end{align*}
which is our claim. \hfill\qed


Note that, if $\gamma(x)$ is locally bounded, then we have
\begin{eqnarray*}
\lefteqn{ \Big |\int_{\reals^d}\sum_{i,j=1}^d
D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{\varphi}(x)dx
\Big |} \hspace{1in}\\ && \leq
B_2E^{1/2}d^{1/2}\sup_{x\in\mbox{supp}(\varphi)}\left|\gamma(x)\right|
\Vert\varphi\Vert_{L^2}\sum_{i=1}^d\Vert \psi_i D_iu\Vert_{L^2},
\end{eqnarray*}
for every $\varphi\in C_c^\infty(\reals^d)$, and the above result
yields $ D(\A_2)=D(\widehat{\A}_2). $
\begin{Lemma}
\label{Non-Var.-Case-Coerc.-Lemma} Under Assumption
\ref{hp:nonvariational}, for all $u\in H_{(\gamma,\psi)}^1$, we
have
\begin{align}
& \int_{\reals^d}\sum_{i,j=1}^d
\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{u}(x)\right|dx
\nonumber
\\
& \leq\frac{B_2}{2}(\Vert\gamma u\Vert_{L^2}^2
+E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2).
\label{Non-Var.-Case-Coerc.-Est.}
\end{align}
\end{Lemma}
\proof By virtue of (\ref{Non-Var.-Case-I-Golden-Cond.}), an
analysis similar to that in the proof of Lemma
\ref{Var.-Case-Coerc.-Est.} yields
\begin{align*}
& {\sum_{i,j=1}^d\int_{\reals^d}
\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{u}(x)\right|}
 \\
& \leq B_2\sum_{i,j=1}^d\int_{\reals^d}
\left|E^{1/2}\eta_{2,i,j}(x)\psi_i(x)\gamma(x)D_iu(x)\overline{u}(x)\right|dx\\
& \leq\frac{B_2}2 \Big (\int_{\reals^d}d\Big
(\sum_{i,j=1}^d\eta_{2,i,j}^2(x)\Big )
\left|\gamma(x)\overline{u}(x)\right|^2dx +\frac
Ed\sum_{i,j=1}^d\int_{\reals^d}\left|\psi_i(x)D_iu(x)\right|^2\Big
),
\end{align*}
from which the desired result easily follows. \hfill\qed


Now we can prove
\begin{Proposition}
\label{Non-Var.-Case-Ex.-Inv.} Under Assumption
\ref{hp:nonvariational}, for each $\lambda\in\complex$ such that
$\Rp\lambda>0$, the equation
\begin{equation}
(\lambda-\A_2)u=f \label{non-var.-case-res.-equ.}
\end{equation}
has a unique solution $u\in D(\A_2)$ for every $f\in
L^2(\reals^d)$.
\end{Proposition}
\proof For each $\lambda\in\complex$, we introduce the
sesquilinear form $a_\lambda(\cdot,\cdot)$ on
$H_{(\gamma,\psi)}^1$ given by
\[
a_\lambda(u,v)\bydef\widehat{a}_\lambda(u,v)
+\int_{\reals^d}\sum_{i,j=1}^dD_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{v}(x)dx.
\]
Then, combining (\ref{Var.-Case-I-Cont.-Est.}), (\ref
{Var.-Case-II-Cont.-Est.}), (\ref{Var.-Case-II-Coerc.-Est.}) and
(\ref {Non-Var.-Case-Cont.-Est.}), for all $u,v\in
H_{(\gamma,\psi)}^1$, we have
\begin{eqnarray*}
\left|a_\lambda(u,v)\right| &\leq& H\Vert
u\Vert_{H_{(\gamma,\psi)}^1} \Vert v\Vert_{H_{(\gamma,\psi)}^1},
\end{eqnarray*}
for a suitable constant $H>0$. Moreover, if $\Rp\lambda >0$, from
(\ref{Var.-Case-I-Coerc.-Est.}), (\ref{Var.-Case-II-Coerc.-Est.})
and (\ref {Non-Var.-Case-Coerc.-Est.}) it follows that
\begin{align*}
& \Rp a_\lambda(u,u)
 \geq \Rp\int_{\reals^d}\sum_{i,j=1}^d
\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{u}(x)dx\\ & -\Big
|\int_{\reals^d}\sum_{i,j=1}^d
D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{u}(x)dx \Big |\\
& - \Big |\int_{\reals^d}\sum_{i=1}^d
b_i(x)D_iu(x)\overline{u}(x)dx \Big |
+\Rp\int_{\reals^d}(\lambda+\gamma^2(x))\left|u(x)\right|^2dx\\
&\geq  E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2
-\frac{(B_1+B_2)}2
 \Big (\Vert\gamma v\Vert_{L^2}^2+E\sum_{i=1}^d\Vert \psi_iD_iu\Vert_{L^2}^2
\Big )\\ &\quad
 +\Rp\lambda\Vert u\Vert_{L^2}^2+\Vert\gamma u\Vert_{L^2}^2
 \geq  K\Vert u\Vert_{H_{(\gamma,\psi)}^1}^2,
\end{align*}
for a suitable constant $K>0$. Therefore, the same argument as
that in the proof of Proposition \ref{Var.-Case-Ex.-Inv.} shows
that, for each $\lambda\in\complex$ such that $\Rp\lambda>0$, the
operator $\lambda-\A_2:D(\A_2)\to L^2$ is invertible. \hfill\qed


We have also
\begin{Proposition}
\label{Non-Var.-Case-Bound.-Inv.} Under Assumption
\ref{hp:nonvariational}, for each $\lambda\in\complex$ such that
$\Rp\lambda >0$, the $\lambda$-resolvent $R(\lambda;\A_2)$ of the
operator $\A_2$ satisfies
\[
\Vert\lambda R(\lambda;\A_2)\Vert\leq K,
\]
for a suitable $K>0$ independent of $\lambda$.
\end{Proposition}
\proof For a fixed $\lambda\in\complex$ such that $\Rp\lambda>0$,
let $u\in D(\A_2)$ be the solution of equation
(\ref{non-var.-case-res.-equ.}) corresponding to some $f\in L^2$.
Since, as in the variational case, the equality
\begin{equation}
a_\lambda(u,u)=\left\langle f,u\right\rangle
\label{Non-Var.-Case-Fund.-Equality}
\end{equation}
holds true, we have
\begin{align*}
& \Vert u\Vert_2\Vert f\Vert_2
 \geq \Rp\int_{\reals^d}\sum_{i,j=1}^d
\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{u}(x)dx\\ &
-\Rp\int_{\reals^d}\sum_{i,j=1}^d
D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\overline{u}(x)dx\\ &
-\Rp\int_{\reals^d}\sum_{i=1}^d b_i(x)D_iu(x)\overline{u}(x)dx
+\Rp\int_{\reals^d}(\lambda+\gamma^2(x))\left|u(x)\right|^2dx\\
&\geq  E\sum_{i=1}^d\Vert \psi_iD_iu\Vert_{L^2}^2 - \Big
|\int_{\reals^d}\sum_{i,j=1}^dD_j(\psi_i(x)\psi_j(x)a_{i,j}(x))
D_iu(x)\overline{u}(x)dx \Big |\\ &  - \Big
|\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\overline{u}(x)dx \Big |
+\Rp\lambda\Vert u\Vert_{L^2}^2+\Vert \gamma u\Vert_{L^2}^2.
\end{align*}
Hence, from (\ref{Non-Var.-Case-Coerc.-Est.}) and (\ref
{Var.-Case-II-Coerc.-Est.}), it follows that
\[
\Vert u\Vert_{L^2}\Vert f\Vert_{L^2} \geq \Rp\lambda \Vert
u\Vert_{L^2}^2 +\frac{2-(B_1+B_2)}2
 \Big (\Vert\gamma u\Vert_{L^2}^2+E\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2}^2 \Big
),
\]
which implies
\begin{equation}
\frac 2{2-(B_1+B_2)}\Vert u\Vert_{L^2}\Vert f\Vert_{L^2}
\geq(\Vert\gamma u\Vert_{L^2}^2 +E\sum_{i=1}^d\Vert
\psi_iD_iu\Vert_{L^2}^2). \label{Non-Var.-Case-Fund.-Interm.-Est.}
\end{equation}
Finally, a computation similar to that in the proof of Proposition
\ref{Var.-Case-Bound.-Inv.}, gives
\begin{equation}
\left|\lambda\right|\Vert u\Vert_{L^2}^2\leq K\Vert
u\Vert_{L^2}\Vert f\Vert_{L^2},
\label{non-var.-case-fund.-final-est.}
\end{equation}
for a suitable $K>0$ independent of $\lambda$, and this is the
desired result. \hfill\qed


>>>From the above results it follows
\begin{Theorem}
\label{th:genL2nonvar} Under Assumption \ref{hp:nonvariational},
the operator $\A_2:D(\A_2)\to L^2$ generates an analytic semigroup
on $L^2$.
\end{Theorem}
Moreover
\begin{Corollary}
Under Assumption \ref{hp:nonvariational}, for every solution $u\in
D(\A_2)$ of $(\ref{non-var.-case-res.-equ.})$, we have
\[
\begin{array}{lll}
\left|\lambda\right|^{1/2}\Vert \gamma u\Vert_{L^2} \leq
K^{\prime}\Vert f\Vert_{L^2} &\hbox{and}&
\left|\lambda\right|^{1/2}\Vert \psi_iD_iu\Vert_{L^2} \leq
K^{\prime\prime}\Vert f\Vert_{L^2},
\end{array}
\]
for suitably chosen $K^{\prime},K^{\prime\prime}>0$ independent of
$\lambda$.
\end{Corollary}
\proof As in the variational case, combining
(\ref{non-var.-case-fund.-final-est.}) with (\ref
{Non-Var.-Case-Fund.-Interm.-Est.}), the desired result
immediately follows. \hfill\qed


\subsection{Domain characterization}
\label{DOMCAR}


In this subsection we introduce some additional conditions on the
coefficients of the operator $\A_2:D(\A_2)\to L^2$ that allow us
to characterize its domain. We divide the domain characterization
in two parts: a first part in which we obtain the suitable
estimates for the first-order derivatives and a second part
devoted to the estimates for the second-order derivatives. This
division is justified for better collating the assumptions needed
to handle the two cases.


\subsubsection{Domain Characterization: first derivatives}
\label{DOMCAR1}


Throughout the sequel we make the following assumption:
\begin{Assumption}
\label{hp:domcar} Under (1) of Assumption \ref{hp:variational},
and (2) and (3) of Assumption \ref{hp:variational-1}, suppose in
addition that $\gamma$ is continuously differentiable and that,
for all $i,j=1,\dots,d$, we have
\begin{equation}
\label{Dom.-Char.-I-Golden-Cond.}
\begin{array}{ccc}
\left|b_i(x)\right|\leq
B_1E^{1/2}\eta_{1,i}(x)\left|\psi_i(x)\right|\gamma(x),
\quad\forall x\in\reals^d,\\
\\
\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))\right| \leq
B_2E^{1/2}\eta_{2,i,j}(x)\left|\psi_i(x)\right|\gamma(x),
\quad\forall x\in\reals^d,\\
\\
2\left|\psi_j(x)D_j\gamma(x)a_{i,j}(x)\right| \leq
B_3E^{1/2}\eta_{3,i,j}(x)\gamma^2(x), \quad\forall x\in\reals^d,
\end{array}
\end{equation}
for suitable constants $B_1,$ $B_2$ and $B_3$ such that
$B_1+B_2+B_3<2$ and suitable measurable functions $\eta_{1,i}(x),$
$\eta_{2,i,j}(x)$ and $\eta_{3,i,j}(x)$ on $\reals^d$ satisfying
$\sum_{i=1}^d\eta_{1,i}(x)=d\sum_{i,j=1}^d\eta_{2,i,j}^2(x)=$
$d\sum_{i,j=1}^d\eta_{3,i,j}^2(x)=1$.
\end{Assumption}


\begin{Remark}
\label{rm:hp3} {\rm Notice that
\begin{enumerate}
\item[(i)]
Assuming $\gamma$ continuously differentiable implies that the
bounded measurable functions $\gamma_n$ given by
\[
\gamma_n(x)\bydef \left\{\begin{array}{ll}
\gamma(x)&\hbox{if}\;\gamma(x)<n\\ n & \mbox{elsewhere}
\end{array}
\right.
\]
for every $n\geq 1$, are differentiable almost everywhere in
$\reals^d$. Indeed, the set
\[
L_n\bydef\left\{x\in\reals^d : \gamma(x)\geq n\right\}
\]
is closed for every $n\geq 1$, and $\gamma_n$ is clearly
differentiable in $\stackrel{\circ}{L}_n \cup L_n^c$. Furthermore,
for every $x$ belonging to the boundary $\partial L_n$ of $L_n$ we
have $\gamma(x)=n$, and it is easily seen that $\gamma_n$ is still
differentiable in those $x\in\partial L_n$ where $(d\gamma)_x=0$.
Finally, the set $\left\{x\in\partial L_n : (d\gamma)_x\neq
0\right\}$, where $\gamma_n$ is not differentiable, is a
continuously differentiable $(d-1)$-dimensional submanifold of
$\reals^d$, which is negligible.
\item[(ii)]
The assumption of continuous differentiability of $\gamma$ could
be weakened by recalling that the addition of a bounded measurable
zero-order term does not influence the generation property of our
operators. In fact, exploiting the same arguments, we could treat
the case in which the function $\gamma(x)$ is written as
$\gamma_0+\widetilde{\gamma}$, where $\gamma_0$ is measurable and
bounded, and $\widetilde{\gamma}$ is continuously differentiable
in $\reals^d$.
\end{enumerate}
}
\end{Remark}


We have
\begin{Proposition}
Under Assumption \ref{hp:domcar}, both $\gamma^2u$ and
$\psi_i\gamma D_iu$ belong to $L^2$, for every $i=1,\dots,d$. More
precisely, $u$ belongs to $H_{(\gamma^2,\gamma\psi)}^1$, and
\begin{equation}
\Vert u\Vert_{H_{(\gamma^2,\gamma\psi)}^1}\le K\Vert f\Vert_{L^2}
\label{L^2-mid.-est.}
\end{equation}
holds true for a suitable $K>0$. In particular, for every
$i=1,\dots,d$, also $b_iD_iu$ belongs to $L^2,$ and we have
\begin{equation}
\sum_{i=1}^d\Vert b_iD_iu\Vert_{L^2} \leq
d^2B_1E^{1/2}\sum_{i=1}^d\Vert \psi_i\gamma D_iu\Vert_{L^2}.
\label{L^2-after-mid.-est.}
\end{equation}
\end{Proposition}
\proof Let $u\in D(\A_2)$ be the solution of equation
(\ref{non-var.-case-res.-equ.}) corresponding to some $f\in L^2$.
Choosing $u_n\bydef\gamma_n^2u$ we have clearly $u_n\in L^2$, and
\begin{equation}
a_\lambda (u,u_n)=\left\langle f,u_n\right\rangle_2,
\label{Char.-Dom.-I-Fund.-Equality}
\end{equation}
for every $n\geq 1$. Thanks to the derivation rule we can rewrite
(\ref{Char.-Dom.-I-Fund.-Equality}) as
\begin{align}
\label{Char.-Dom.-II-Fund.-Equality} &
 \int_{\reals^d}(\lambda +\gamma^2(x))\gamma_n^2(x)\left|u(x)\right|^2dx
\\
& +\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)
\gamma_n^2(x)D_iu(x)D_j\overline{u}(x)dx \nonumber
 \\
& = \int_{\reals^d}f(x)\gamma_n^2(x)\overline{u}(x)dx
+\int_{\reals^d}\sum_{i=1}^db_i(x)D_iu(x)\gamma_n^2(x)\overline{u}(x)dx
\nonumber \\ &
-2\int_{\reals^d}\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)
\gamma_n(x)D_j\gamma_n(x)D_iu(x)\overline{u}(x)dx \nonumber \\ &
-\int_{\reals^d}\sum_{i,j=1}^dD_j (\psi_i(x)\psi_j(x)a_{i,j}(x))
D_iu(x)\gamma_n^2(x)\overline{u}(x)dx,\nonumber
\end{align}
which clearly implies
\begin{align}
\label{Char.-Dom.-III-Fund.-Equality} &
{\Rp\lambda\int_{\reals^d}\gamma^2(x)\left|u(x)\right|^2dx
+\int_{\reals^d}\gamma_n^2(x)\gamma^2(x)\left|u(x)\right|^2dx}
\nonumber
 \\
 &+\Rp\int_{\reals^d}\sum_{i,j=1}^d
\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)\gamma_n^2(x)D_j\overline{u}(x)dx
\nonumber\\ &\leq
\int_{\reals^d}\left|f(x)\gamma_n^2(x)u(x)\right|dx
+\int_{\reals^d}\sum_{i=1}^d\left|
b_i(x)D_iu(x)\gamma_n^2(x)\overline{u}(x)\right|dx \nonumber\\ &
+2\int_{\reals^d}\sum_{i,j=1}^d\left|\psi_i(x)\psi_j(x)a_{i,j}(x)
\gamma_n(x)D_j\gamma_n(x)D_iu(x)\overline{u}(x)\right|dx
\nonumber\\ &
+\int_{\reals^d}\sum_{i,j=1}^d\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))
D_iu(x)\gamma_n^2(x)\overline{u}(x)\right|dx.
\end{align}
On the other hand, we have
\begin{align}
\label{Char.-Dom.-I-Part.-Est.} &
{\Rp\int_{\reals^d}\sum_{i,j=1}^d
\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)\gamma_n^2(x)D_j\overline{u}(x)dx}
\nonumber
 \\
&= \int_{\reals^d}\gamma_n^2(x)\;
\Rp(\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_iu(x)D_j\overline{u}(x))dx
\nonumber\\ &\geq
E\int_{\reals^d}\gamma_n^2(x)\sum_{i=1}^d\left|\psi_i(x)D_iu(x)\right|^2dx,
\end{align}
and
\begin{align}
\label{Char.-Dom.-II-Part.-Est.} &
\int_{\reals^d}\left|f(x)\gamma_n^2(x)u(x)\right|dx
 \leq  (\int_{\reals^d}\left|f(x)\right|^2dx)^{1/2}
(\int_{\reals ^d}\gamma_n^4(x)\left|u(x)\right| ^2dx)^{1/2}
\nonumber \\ &\leq
\frac1{2\varepsilon}\int_{\reals^d}\left|f(x)\right|^2dx
+\frac{\varepsilon}2\int_{\reals^d}\gamma_n^4(x)\left|u(x)\right|^2dx
\nonumber \\ &\leq \frac{1}{2\varepsilon}\Vert f\Vert_{L^2}^2
+\frac{\varepsilon}{2}\int_{\reals^d}\gamma_n^2(x)\gamma^2(x)\left|u(x)\right|^2dx,
\end{align}
for every $\varepsilon >0$. Furthermore, thanks to
(\ref{Dom.-Char.-I-Golden-Cond.}), computations similar to those
in the proof of Lemma \ref{Non-Var.-Case-Coerc.-Lemma} give
\begin{align}
\label{Char.-Dom.-III-Part.-Est.} &
{\int_{\reals^d}\sum_{i=1}^d\left|b_i(x)D_iu(x)\gamma_n^2(x)\overline{u}(x)\right|dx}
 \nonumber\\
&\leq  B_1\int_{\reals^d}\sum_{i=1}^dE^{1/2}\eta_{1,i}(x)\gamma(x)
\gamma_n^2(x)\left|\psi_i(x)D_iu(x)u(x)\right|dx
 \\
&\le
\frac{B_1}2(\int_{\reals^d}\gamma_n^2(x)\gamma^2(x)\left|u(x)\right|^2dx
+E\int_{\reals^d}\gamma_n^2(x)\sum_{i=1}^d\left|\psi_i(x)D_iu(x)\right|^2dx),
\nonumber
\end{align}
and
\begin{align}
\label{Char.-Dom.-IV-Part.-Est.} &
 {\int_{\reals^d}\sum_{i,j=1}^d
\left|D_j(\psi_i(x)\psi_j(x)a_{i,j}(x))D_iu(x)\gamma_n^2(x)\overline{u}(x)\right|dx}
 \nonumber\\
&\leq  B_2\int_{\reals^d}\sum_{i,j=1}^dE^{1/2}\eta_{2,i,j}(x)
\gamma(x)\gamma_n^2(x)\left|\psi_i(x)D_iu(x)u(x)\right|dx
 \\
&\le
\frac{B_2}2(\int_{\reals^d}\gamma_n^2(x)\gamma^2(x)\left|u(x)\right|^2dx
+E\int_{\reals^d}\gamma_n^2(x)\sum_{i=1}^d\left|\psi_i(x)D_iu(x)\right|^2dx).
\nonumber
\end{align}
Finally, since (\ref{Dom.-Char.-I-Golden-Cond.}) clearly implies
\[
2\left| \psi_j(x)D_j\gamma_n(x)a_{i,j}(x)\right| \leq
B_3E^{1/2}\eta_{3,i,j}(x)\gamma_n(x)\gamma(x),
\]
for every $x\in\reals^d$, we have also
\begin{align}
\label{Char.-Dom.-V-Part.-Est.} & {2\int_{\reals^d}\sum_{i,j=1}^d
\left|\psi_i(x)\psi_j(x)a_{i,j}(x)\gamma_n(x)D_j\gamma_n(x)D_iu(x)\overline{u}(x)\right|dx}
 \nonumber\\
&\leq  B_3\int_{\reals^d}\sum_{i,j=1}^dE^{1/2}\eta_{3,i,j}(x)
\gamma_n^2(x)\gamma(x)\left|\psi_i(x)D_iu(x)u(x)\right|dx
 \\
&\leq\frac{B_3}2(\int_{\reals^d}\gamma_n^2(x)
\gamma^2(x)\left|u(x)\right|^2dx
+E\int_{\reals^d}\gamma_n^2(x)\sum_{i=1}^d\left|
\psi_i(x)D_iu(x)\right|^2dx). \nonumber
\end{align}
Hence, combining
(\ref{Char.-Dom.-I-Part.-Est.})--(\ref{Char.-Dom.-V-Part.-Est.})
with (\ref{Char.-Dom.-III-Fund.-Equality}), it follows that
\begin{align*}
&
 \Rp\lambda\int_{\reals^d}\gamma^2(x)\left|u(x)\right|^2dx
+\int_{\reals^d}\gamma_n^2(x)\gamma^2(x)\left|u(x)\right|^2dx
\\
& +E\int_{\reals^d}\gamma_n^2(x)\sum_{i=1}^d
\left|\psi_i(x)D_iu(x)\right|^2dx \\ &\leq
\frac1{2\varepsilon}+\Vert
f\Vert_{L^2}^2\frac{\varepsilon}2\int_{\reals ^d}
\gamma_n^2(x)\gamma^2(x)\left|u(x)\right|^2dx\\ &
+\frac{(B_1+B_2+B_3)}2(\int_{\reals^d}\gamma_n^2(x)
\gamma^2(x)\left|u(x)\right|^2dx \\ &
+E\int_{\reals^d}\gamma_n^2(x)
\sum_{i=1}^d\left|\psi_i(x)D_iu(x)\right|^2dx),
\end{align*}
and this yields
\begin{align}
\label{Char.-Dom.-final-Fund.-Equality} &
 \Rp
\! \lambda \int_{\reals^d}\gamma^2(x) |u(x) |^2dx
+\frac{2-(B_1+B_2+B_3+\varepsilon
)}{2}\int_{\reals^d}\gamma_n^2(x) \gamma^2(x) |u(x) |^2dx
 \nonumber\\
& +\frac{2-(B_1+B_2+B_3)}2E\int_{\reals^d}\gamma_n^2(x)
\sum_{i=1}^d\left|\psi_i(x)D_iu(x)\right|^2dx
 \leq\frac{1}{2\varepsilon}\Vert f\Vert_{L^2}^2.
\end{align}
Now, choosing $\varepsilon$ small enough, it is easily seen that
each term on the left side of
(\ref{Char.-Dom.-final-Fund.-Equality}) is positive. Therefore,
letting $n\rightarrow\infty$, the monotone convergence theorem
allows us to conclude that both $\gamma^2(x)u(x)$ and
$\gamma(x)\psi_i(x)D_iu(x)$ belong to $L^2$, and
(\ref{L^2-mid.-est.}) holds true. Then, by virtue of
(\ref{Dom.-Char.-I-Golden-Cond.}), we can easily complete the
proof. \hfill\qed


As an obvious consequence of the above proposition we have that
the sum
\begin{equation}
\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_{i,j}u(x)
\label{whole-sum}
\end{equation}
belongs to $L^2$.


\subsubsection{Domain Characterization: second derivatives}
\label{DOMCAR2}


Aiming to show that for all $i,j=1,\dots,d$ the single summand
$\psi_i(x)\psi_j(x)D_{i,j}u(x)$ belongs to $L^2$, we need to
strengthen our hypotheses on the coefficients $\psi_i(x)$'s.
Therefore, having in mind our examples, we will assume then the
negligibility of the set
\[
Z\equiv\left\{x\in\reals^d : \psi_i(x)=0,\;\hbox{for
some}\;i=1,\dots,d\right\},
\]
of all zeros of the $\psi_i(x)$'s, and the existence of a suitable
countable covering of $\reals^d-Z$ which allows us to perform a
localization procedure. Such a covering will be made by rectangles
of the type
\[
R(x_0,r\psi)\equiv \left\{x\in\reals^d : |x_i-x_i^{(0)}|\leq
r\vert\psi_i(x_0)\vert,\;i=1,\dots,d\right\},
\]
for $x_0\equiv(x_1^{(0)},\dots,x_d^{(0)})\in\reals^d-Z$ and $r>0$.
More precisely, we assume
\begin{Assumption}
\label{hp:psidomcar1} Under Assumption \ref{hp:domcar}, suppose in
addition that


\begin{enumerate}
\item[(i)]
for every $i=1,\dots,d$, the differentiable function $\psi_i(x)$
belongs to $H_{\loc}^1$ and the set $Z$ is negligible;
\item[(ii)]
there exist real numbers $r_1>0$ and $L>0$ such that for every
$0<r\leq r_1$ we can find a countable set $N_r\subset\reals^d-Z$
such that
\begin{enumerate}
\item[(a)]
the family ${\mathcal F}_{1}\equiv\left\{R(x,r\psi)\right\}_{x\in
N_r}$ is a covering of $\reals^d-Z$;
\item[(b)]
 each rectangle of the family ${\mathcal
F}_2\equiv\left\{R(x,2r\psi)\right\}_{x\in N_r}$ does not contain
any element of $Z$ and has a nonempty intersection with at most a
fixed number $n_r$ of other rectangles of $\mathcal{F}_2$ itself;
\item[(c)]
we have
\[
\frac{1}{L} \leq\min_{i=1,\dots,d}\inf_{x\in R(x_0,2r\psi)}
\frac{\vert\psi_i(x)\vert}{\vert\psi_i(x_{0})\vert}
\leq\max_{i=1,\dots,d}\sup_{x\in R(x_0,2r\psi)}
\frac{\vert\psi_i(x)\vert}{\vert\psi_i(x_0)\vert} \leq L,
\]
for each $x_0\in N_r$.
\end{enumerate}
\end{enumerate}
\end{Assumption}
\begin{Remark}
\rm Notice that
\begin{enumerate}
\item[(i)]
Part (i) of Assumption \ref{hp:psidomcar1} is necessary to get
estimates for the second-order derivatives of the solution $u$ of
the problem $\A u=f$. In fact, if some of the $\psi_i(x)$'s vanish
on some open set, then it is not possible, in general, to get
estimates for $D_{i,i}u$ on this~set.
\item[(ii)]
Part (ii) of \ref{hp:psidomcar1} allows us, via a change of
variable that will be introduced later, to transform our
degenerate global characterization problem into a set of
nondegenerate local ones (one for every rectangle $R(x,r\psi)$).
We will show that we can obtain local estimates for the
$\psi_i(x)\psi_j(x)a_{i,j}(x)D_{i,j}u$'s, which turn out to be
independent of the particular rectangle considered. Finally,
thanks to the properties of the covering given in part
(ii)--(a)/(b), these local estimates can be summed up giving the
desired global result.
\item[(iii)]
We point out that the real difficulty to overcome in this part is
to get estimates in some open neighborhood ${\mathcal  D}$ of $Z$
and $\infty$. In fact, one can split our problem into two parts,
one related to the neighborhood ${\mathcal  D}$ and the other to
the compact set $\reals^d-{\mathcal  D}$. The latter is a very
classical problem, and it can be handled with well-known
techniques (see \cite{GT}). For this reason we will concentrate on
getting estimates on the neighborhood ${\mathcal  D}$, while
skipping this problem on $\reals^d-{\mathcal  D}$.
\end{enumerate}


 \end{Remark}


Assumption \ref{hp:psidomcar1} is convenient for proving our
characterization result (Proposition \ref{L^2-domain-char}).
However, it is not easy to check it for given operators.
Therefore, before showing the characterization result, we will
establish Lemma \ref{psi-estimates-lemma}, which shows that
\ref{hp:psidomcar1} is verified under a more treatable assumption,
befitted with our examples coming from Financial Mathematics (see
Section \ref{APPLICATIONS}). More precisely, we are show that the
Assumption \ref {hp:psidomcar1bis} below implies
\ref{hp:psidomcar1}.
\begin{Assumption}
\label{hp:psidomcar1bis} Under Assumption \ref{hp:domcar}, suppose
in addition that
\begin{enumerate}
\item[(i)]
Part (i) of Assumption \ref{hp:psidomcar1} holds true;
\item[(ii)]
there exist $r_0>0$ (small), $R_0>0$ $($large$),$ and $\alpha>0$
such that for every $x\in \left\{x\in\reals^d :
dist(x,Z)<r_0\;\hbox{\rm or}\;dist(x,0)>R_0\right\}
\equiv{\mathcal  D}\left(r_0,R_0\right)$ and every $i=1,\dots,d$
we have
\[
|D_j\psi_i(x)|\leq\alpha;
\]
\item[(iii)]
for every $i=1,\dots,d$ the function $\psi_i(x)$ depends only on
the variable $x_i$.
\end{enumerate}
\end{Assumption}
\begin{Remark}
\rm Notice that
\begin{enumerate}
\item[(i)]
Part (ii) of Assumption \ref{hp:psidomcar1bis} states essentially
that there exists a neighborhood of $Z$ and of $\infty$ where all
derivatives $D_{j}\psi _{i}$ are uniformly bounded. This yields a
sublinear behavior of the $\psi _{i}$'s close to $Z$ and to
$\infty$.
\item[(ii)]
One can easily verify \ref{hp:psidomcar1bis} when the operator
$\A$ belongs to a family of operators arising in the pricing of a
contingent claim in the multidimensional case (e.g., the
d-dimensional Black and Scholes operator for every $d\geq 1$; see
Section \ref{APPLICATIONS}).
\end{enumerate}
 \end{Remark}



Lemma \ref{psi-estimates-lemma} below shows that \ref
{hp:psidomcar1bis} implies \ref{hp:psidomcar1} and, actually,
something more.
\begin{Lemma}
\label{psi-estimates-lemma} Under Assumption
\ref{hp:psidomcar1bis}, there exist real numbers $r_{1}>0$ and
$1<L<2$ such that
\begin{enumerate}
\item[(i)]
for every $0<r\leq r_1$ and $x_0\in \{x\in\reals^d-Z :
dist(x,Z)<r_0/2\;\hbox{\rm or}\;dist(x,0)>2R_0 \} \equiv{\mathcal
D}\left(r_0/2,2R_0\right)$ we have
\[
\frac{1}{L}\leq \min_{i=1,\dots,d}\inf_{x\in R(x_0,10r\psi )}
\frac{\vert\psi _i(x)\vert}{\vert\psi_i(x_{0})\vert}
\leq\max_{i=1,\dots,d}\sup_{x\in R(x_0,10r\psi)}
\frac{\vert\psi_i(x)\vert}{\vert\psi_i(x_0)\vert}\leq L;
\]
\item[(ii)]
for every $r<r_1$ there exists a countable set
$N_r\subset{\mathcal  D}\left(r_0/2,2R_0\right)$ such that
\begin{enumerate}
\item[(a)]
the family of rectangles ${\mathcal
F}_1\equiv\left\{R(x_0,r\psi)\right\}_{x_0\in N_r}$ is a covering
of ${\mathcal  D}\left(r_0/2,2R_0\right)$,
\item[(b)]
every rectangle of the family ${\mathcal
F}_2\equiv\left\{R(x_0,2r\psi)\right\}_{x_0\in N_r}$ does not
contain any point of $Z$, and has a nonempty intersection with at
most a fixed number $n_r$ of other rectangles of ${\mathcal F}_2$
itself.
\end{enumerate}
\end{enumerate}
\end{Lemma}
\proof


{\bf Proof of (i).}


{\bf Step (i)--1.} We start by observing that, for $r_1>0$
sufficiently small, for every $0<r<r_1$, and for each
$x_0\in{\mathcal  D}(r_0/2,2R_0)$, we have
\[
R(x_0,10r\psi)\subset{\mathcal  D}(r_0,R_0).
\]
Therefore, for each $x_0\in{\mathcal D}(r_0/2,2R_0)$ and every
$x\in R(x_0,10r\psi)$, we have
\[
\vert D_j\psi_i(x)\vert\leq\alpha,
\]
for all $i,j=1,\dots,d$.


Indeed, if $x_0\in{\mathcal D}(r_0/2,2R_0)$, we have either $\Vert
x_0\Vert <2R_0$, or $\Vert x_0\Vert\geq 2R_0$. Let $\Vert x_0\Vert
<2R_0$. We have then $d\left(x_0,Z\right)<r_0/2$, and taking $x\in
R(x_0,10r\psi)$ we can write
\begin{align}
\label{innerestimate} d\left(x,Z\right) &\leq
d\left(x_0,Z\right)+\Vert x-x_0\Vert \leq r_0/2+ \Big
(\sum_{i=1}^{d}\vert x_i-x_i^{(0)}\vert^{2} \Big )^{1/2}
\\
&\leq  r_0/2+ \Big (\sum_{i=1}^d\vert 10r\psi_i (x_0 )\vert^2 \Big
)^{1/2} \leq
r_0/2+10d^{1/2}r_1\Vert\psi\Vert_{L^{\infty}\left(B\left(0,2R_0\right)\right)},
\nonumber
\end{align}
where
$\Vert\psi\Vert_{L^{\infty}\left(B\left(0,2R_0\right)\right)}
\equiv \max_{i=1,\dots,d}\sup_{x\in
B\left(0,2R_0\right)}\left\{\vert\psi_i(x)\vert\right\}$. Now,
(\ref{innerestimate}) clearly implies that choosing $r_1$ small
enough we obtain
\[
d\left(x,Z\right)<r_0.
\]


Suppose now $x_0\in{\mathcal  D}(r_0/2,2R_0)$ and $\Vert
x_0\Vert>2R_0$. Taking $x\in R(x_0,10r\psi)$ we have then $\Vert
x\Vert>R_0$. Indeed,
\begin{align}
\Vert x\Vert &\geq \Vert x_0\Vert -\Vert x-x_0\Vert =\Vert
x_0\Vert - \Big (\sum_{i=1}^{d}\vert x_i-x_i^{(0)}\vert^{2} \Big
)^{1/2} \nonumber\\ &\geq \Vert x_0\Vert -10r \Big
(\sum_{i=1}^d\vert\psi_i\left(x_0\right)\vert^2 \Big )^{1/2}
\label{outerestimate1}.
\end{align}
Now, the point $\tilde{x}_0\equiv\frac{R_0}{\Vert x_0\Vert}x_0$
belongs to the boundary of $B\left(0,R_0\right)$, and the line
segment between $\tilde{x}_0$ and $x_{0}$ is contained entirely in
${\mathcal D}(r_0,R_0)$. Then, by virtue of the differentiability
of $\psi_i$ and of \ref{hp:psidomcar1bis} (iii), we get, for
$i=1,\dots,d$,
\[
\psi_i\left(x_0\right)=\psi_i\left(\tilde{x}_0\right)
+D_i\psi_i\left(\xi\right)\left(x_i^{(0)}-\tilde{x}_i^{(0)}\right),
\]
for a suitable $\xi\in[\tilde{x}_0,x_{0}]\subseteq{\mathcal
D}(r_0,R_0)$, and it follows that
\begin{equation}
\label{outerestimate2} \vert\psi_i\left(x_0\right)\vert \leq
\vert\psi_i\left(\tilde{x}_0\right)\vert+\alpha\Vert\tilde{x}_0-x_0\Vert
\leq \Vert\psi\Vert_{L^{\infty}\left(B\left(0,R_0\right)\right)}
+\alpha\left(\Vert x_0\Vert-R_0\right).
\end{equation}
Combining (\ref{outerestimate1}) and (\ref{outerestimate2}), we
have then
\[
\Vert x\Vert\geq\Vert x_0\Vert
-10r\left(\Vert\psi\Vert_{L^{\infty}\left(B\left(0,R_0\right)\right)}
+\alpha\left(\Vert x_0\Vert-R_0\right)\right)\sqrt{d},
\]
namely
\[
\Vert x\Vert\geq \Vert x_0\Vert\left(1-10r\alpha\sqrt{d}\right)
-10r\left(\Vert\psi\Vert_{L^{\infty}\left(B\left(0,R_0\right)\right)}-\alpha
R_0\right)\sqrt{d}.
\]
Finally, wiping off the term $\alpha R_{0}$ and recalling that
$\Vert x_0\Vert>2R_0$, we have
\[
\Vert x\Vert\leq 2R_0\left(1-10r\alpha\sqrt{d}\right)
-10r\sqrt{d}\Vert\psi\Vert_{L^{\infty}\left(B\left(0,R_0\right)\right)}.
\]
>>>From the latter it easily follows that choosing $r_1$ small enough
we obtain the desired $\Vert x\Vert\geq R_0$.


{\bf Step (i)--2: Conclusion.}


We are now in a position to prove (i) of the lemma. Let
$x_0\in{\mathcal  D}(r_0/2,2R_0)$, and choose $r_1>0$ sufficiently
small. Thanks to (iii) of Assumption \ref{hp:psidomcar1bis}, for
every $i=1,\dots,d$ and every $x\in R(x_0,10r\psi)$, we can write
\begin{equation}
\label{ass.-(ii)-1}
\psi_i(x)=\psi_i(x_0)+D_i\psi_i(\xi)(x_i-x_i^{(0)}),
\end{equation}
for a suitable $\xi\in[x,x_0]$. It follows that
\[
\vert\psi_i(x)\vert\leq\vert\psi_i(x_0)\vert+10\alpha
r\vert\psi_i(x_0)\vert,
\]
which implies
\begin{equation}
\label{ass.-(ii)-2}
\frac{\vert\psi_i(x)\vert}{\vert\psi_i(x_0)\vert}\leq 1+10\alpha
r.
\end{equation}
Similarly, from (\ref{ass.-(ii)-1}), we have also
\[
\vert\psi_i(x)\vert \geq\vert\psi_i(x_0)\vert -\vert
D_i\psi_i(\xi)\vert\vert x_i-x_i^{(0)}\vert
\geq\vert\psi_i(x_0)\vert-10\alpha r\vert\psi_i(x_0)\vert,
\]
and the latter yields
\begin{equation}
\label{ass.-(ii)-3}
\frac{\vert\psi_i(x)\vert}{\vert\psi_i(x_0)\vert}\geq 1-10\alpha
r.
\end{equation}
Therefore the claim (i) follows by choosing again $r_1$ small
enough.


\smallskip


\noindent {\bf Proof of (ii).} Now we prove (ii). This proof is
more technical and adapts some arguments used in the proof of the
so-called Besicovitch lemma (see \cite{Be-1928, Be-1938,
Be-1939}). However, we remark that we cannot apply here the
Besicovitch lemma straightforwardly, since
\begin{itemize}
\item
the set $\reals^d-Z$ to cover is not bounded;
\item
the family of sets ${\mathcal
F}=\left\{R(x,r\psi)\right\}_{x\in\reals^d-Z}$ is not uniformly
regular in the sense used by Besicovitch (see, e.g.,
\cite{Be-1939}).
\end{itemize}
We divide the proof into three steps:


{\bf Step (ii)--1: Extracting the \lq\lq good\rq\rq  subfamily.}


To extract the desired covering of ${\mathcal
D}\left(r_0/2,2R_0\right)$ we use a recursive argument. First, we
consider the family ${\mathcal
F}\equiv\left\{R(x,r\psi)\right\}_{x\in{\mathcal
D}\left(r_0/2,2R_0\right)}$ of the rectangles having centers in
all points of ${\mathcal D}\left(r_0/2,2R_0\right)$, and, by
slicing each side into three equal parts, we divide every
rectangle $R(x,r\psi)$ of ${\mathcal F}$ into $3^{d}$ rectangles,
each of which has the $i$-th side of length
$\frac{1}{3}r\left|\psi_{i}\left(x\right)\right|$ for $i=1,...,d$.
We write then $R(x,r\psi/3)$ for the internal rectangle. Next,
among all rectangles of ${\mathcal  F}$ we choose a subfamily
${\mathcal  R}_{0}$ such that


\begin{enumerate}
\item[(a)]
the smallest semiaxis is $\geq r\cdot 2^{0}=r$;


\item[(b)]
the internal rectangles $R(x,r\psi/3)$ are pairwise disjoint;


\item[(c)]
the family is maximal with respect to the inclusion, namely, there
is no subfamily of ${\mathcal F}$ enjoying (a) and (b) above and
containing ${\mathcal R}_{0}$ strictly).
\end{enumerate}
Clearly, if all $\psi(x)$'s in the set ${\mathcal
D}\left(r_0/2,2R_0\right)$ take values $<1$ such a family is
empty. Anycase ${\mathcal R}_{0}$ is at most countable, because of
the two properties (a) and (b) above. If ${\mathcal R}_{0}$ covers
${\mathcal  D}\left(r_0/2,2R_0\right)$ we stop here. If not we
choose another subfamily ${\mathcal R}_{1}$ of ${\mathcal F}$
satisfying


\begin{enumerate}
\item
the centers belong to ${\mathcal
D}\left(r_0/2,2R_0\right)-\cup{\mathcal R}_0$, where
$\cup{\mathcal R}_{0}$ stands for the union of all rectangles
belonging to ${\mathcal  R}_{0}$;


\item
the smallest semiaxis is $\geq r\cdot 2^{-1}=r/2$;


\item
the internal rectangles $R(x,r\psi/3)$ are pairwise disjoint, and
have also empty intersection with all internal rectangles of the
family ${\mathcal R}_{0}$;


\item
the family is maximal with respect to inclusion; namely, there is
no subfamily of ${\mathcal F}$ enjoying the properties above and
containing ${\mathcal R}_{1}$ strictly).
\end{enumerate}


As above, if ${\mathcal R}_{0}\cup{\mathcal R}_{1}$ covers
${\mathcal D}\left(r_0/2,2R_0\right)$ we stop here. If not we
iterate the procedure. In any case we generate a sequence
$\left\{{\mathcal R}_{n}\right\}_{n\geq 0}$ of subfamilies of
${\mathcal  F}$ such that every ${\mathcal  R}_{n}$ satisfies


\begin{enumerate}
\item
the centers belong to ${\mathcal
D}\left(r_0/2,2R_0\right)-\cup_{k=0}^{n-1}\left(\cup{\mathcal
R}_k\right)$, where $\cup{\mathcal  R}_{k}$ stands for the union
of all sets belonging to ${\mathcal  R}_{k}$);


\item
the smallest semiaxis is $\geq r\cdot 2^{-n}$;


\item
the internal rectangles $R(x,r\psi/3)$ are pairwise disjoint and
have also empty intersection with all internal rectangles of the
families ${\mathcal R}_{k}$ for $k=1,\dots,n-1$;


\item
the family is maximal with respect to inclusion.
\end{enumerate}


If this procedure stops after a finite number of steps we have
found a countable covering. If not, we consider the generated
countable family $\cup_{n=0}^{\infty}{\mathcal R}_n$ and we prove
below that it is a covering for ${\mathcal
D}\left(r_0/2,2R_0\right)$.


\textbf{Step (ii)--2: The \lq\lq good\rq\rq  subfamily is a
covering.}


We argue by contradiction. Let $x_0\in{\mathcal
D}\left(r_0/2,2R_0\right) -\cup_{n=0}^{\infty}\left(\cup{\mathcal
R}_{n}\right)$. Then consider the rectangle $R(x_0,r\psi/3)$, and
let $n$ be the first integer such that
$\min_{i=1,\dots,d}\psi_{i}\left(x_0\right)\geq 2^{-n}$. By the
maximality of the families ${\mathcal  R}_{k}$ for $0\leq k\leq
n$, there exist $k\in\left\{0,\dots,n\right\}$ and
$x_k\in{\mathcal  D}\left(r_0/2,2R_0\right)$ such that
$R(x_k,r\psi)\in{\mathcal  R}_{k}$ and $R(x_0,r\psi/3)\cap
R(x_k,r\psi/3)\neq\emptyset$. This implies that we have
\begin{equation}
\label{eq:contradictionricoprimento1}
\left|x_{i}^{(0)}-x_{i}^{(k)}\right|
\leq\frac{r}{3}\left(\vert\psi_{i}\left(x_0\right)\vert
+\vert\psi_{i}\left(x_k\right)\vert\right).
\end{equation}
for every $i=1,\dots,d$. On the other hand, since $x_0\notin
R(x_k,r\psi)$, there exists at least an index
$i_0\in\left\{1,\dots,d\right\}$ such that
\begin{equation}
\label{eq:contradictionricoprimento2} \vert
x_{i_0}^{(0)}-x_{i_0}^{(k)}\vert
>r\vert\psi_{i_0}\left(x_k\right)\vert.
\end{equation}
Combining (\ref{eq:contradictionricoprimento1}) and
(\ref{eq:contradictionricoprimento2}), we have then
\[
\vert\psi_{i_0}\left(x_k\right)\vert
<\frac{1}{3}\left(\vert\psi_{i_0}\left(x_k\right)\vert
+\vert\psi_{i_0}\left(x_0\right)\vert\right),
\]
or equivalently
\begin{equation}
\label{eq:contradictionricoprimento3}
\frac{\vert\psi_{i_0}\left(x_0\right)\vert}{\vert\psi_{i_0}\left(x_k\right)\vert}>2.
\end{equation}
But this is impossible. In fact, by Assumption
(\ref{hp:psidomcar1bis})--(iii), we have also
\begin{equation}
\label{eq:contradictionricoprimento4}
\vert\psi_{i_0}\left(x_0\right)-\psi_{i_0}\left(x_k\right)\vert
=\vert D_{i_0}\psi_{i_0}\left(\xi\right)\vert \vert
x_{i_0}^{(0)}-x_{i_0}^{(k)}\vert,
\end{equation}
for suitable $\xi\in[x_0,x_k]$. Hence, combining
(\ref{eq:contradictionricoprimento1}) and
(\ref{eq:contradictionricoprimento3}), and taking into account
Assumption \ref{hp:psidomcar1bis}--(ii), we obtain
\[
\vert\vert\psi_{i_0} (x_0 )\vert -\vert\psi_{i_0} (x_k )\vert\vert
\leq \vert D_{i_0}\psi_{i_0} (x_k )\vert \vert
x_{i_0}^{(0)}-x_{i_0}^{(k)}\vert \leq\alpha\frac{r}{3}
(\vert\psi_{i_0} (x_0 )\vert +\vert\psi_{i_0} (x_k )\vert ),
\]
or equivalently
\[
\Big |\frac{\vert\psi_{i_0}\left(x_0\right)\vert}
{\vert\psi_{i_0}\left(x_k\right)\vert}-1\Big | \leq
\alpha\frac{r}{3}\Big (\frac{\vert\psi_{i_0}\left(x_0\right)\vert}
{\vert\psi_{i_0}\left(x_k\right)\vert}+1\Big ).
\]
The latter yields
\[
\frac{\vert\psi_{i_0}\left(x_0\right)\vert}
{\vert\psi_{i_0}\left(x_k\right)\vert}
\leq\frac{1+\frac{\alpha}{3}r}{1-\frac{\alpha}{3}r},
\]
and choosing $r_1$ small enough, for every $0<r<r_1$, we can make
\[
\frac{1+\frac{\alpha}{3}r}{1-\frac{\alpha}{3}r}<2,
\]
which contradicts (\ref{eq:contradictionricoprimento3}).


{\bf Step (ii)--3. The \lq\lq good\rq\rq  subfamily enjoys the
finite intersection property}. We write ${\mathcal F}_1$ for the
countable covering of rectangles defined above, and we write $N_r$
for the set of all centers of the rectangles of ${\mathcal F}_1$.
Hence, we are left to prove that the family ${\mathcal
F}_2\equiv\left\{R(x,2r\psi)\right\}_{x\in N_r}$ enjoys the
announced properties.


Notice first that the first two steps of our proof assure that the
family ${\mathcal F}_2$ covers ${\mathcal D}\left(r_0,R_0\right)$
and has empty intersection with $Z$. Therefore, we need only to
show that every rectangle of ${\mathcal  F}_2$ has a nonempty
intersection with at most a fixed number $n_r$ of the others. To
this end, we fix any $x_0\in N_r$ and we claim that


- if $x\in N_r$ is such that $R\left(x,2r\psi\right)\cap
R(x_0,2r\psi)\neq\emptyset$, then $x\in R(x_0,10r\psi)$;


- there are at most $60^d$ points in $N_r\cap R(x_0,10r\psi)$.


To prove the first claim, take $x\in N_r$ such that
$R(x,2r\psi)\cap R(x_0,2r\psi)\neq\emptyset$. Then we have
\begin{equation}
\label{eq:distcentri1} \vert x_i-x_{i}^{(0)}\vert \leq
2r\left(\vert\psi_i(x)\vert+\vert\psi_i\left(x_0\right)\vert\right).
\end{equation}
for every $i=1,\dots,d$. In addition, by virtue of the part (i) of
the lemma, for any $y\in R(x,2r\psi)\cap R(x_0,2r\psi)$, we have
\[
\frac{\vert\psi_i\left(y\right)\vert}{\vert\psi_i\left(x_0\right)\vert}<2
\quad\hbox{and}\quad
\frac{1}{2}<\frac{\vert\psi_i\left(y\right)\vert}{\vert\psi_i\left(x\right)\vert},
\]
so that
\begin{equation}
\label{eq:distcentri2} \vert\psi_i\left(x\right)\vert
<2\vert\psi_i\left(y\right)\vert
<4\vert\psi_i\left(x_0\right)\vert.
\end{equation}
Hence, combining (\ref{eq:distcentri1}) and
(\ref{eq:distcentri2}), the desired claim immediately follows.


To prove the second claim, we recall that, by construction, we
have
\[
\vert x_i-y_i\vert
>\frac{r}{3}\left(\vert\psi_i(x)\vert+\vert\psi_i(y)\vert\right).
\]
for all $x,y\in N_r$ and every $i=1,\dots,d$. On the other hand,
we have also
\[
\frac{\vert\psi_i(z)\vert}{\vert\psi_i\left(x_0\right)\vert}>\frac{1}{2}.
\]
for every $z\in R(x_0,10r\psi)$. Thus, we can write
\[
\vert x_i-y_i\vert >\frac{r}{3}\vert\psi_i\left(x_0\right)\vert,
\]
for all $x,y\in N_r\cap R(x_0,10r\psi)$ and every $i=1,\dots,d$.
The latter implies, by suitably slicing the rectangle
$R(x_0,10r\psi)$, that the points $x\in N_r\cap R(x_0,10r\psi)$
can be $60^{d}$ at most. \hfill\qed


The following lemma is also needed.


\begin{Lemma}
\label{psi-covering-first-cons.} Suppose that Assumption
\ref{hp:psidomcar1} holds true, and let $f$ and $g$ be measurable
functions satisfying
\[
\int_{B(x,r\psi )}\left|g(y)\right|^2dy
\leq\int_{R(x,2r\psi)}\left|f(y)\right|^2dy
\]
for every $x\in N_r$. Then $g$ belongs to $L^2$ whenever $f$ does.
\end{Lemma}


\proof Condition \ref{psi-estimates-lemma} (ii) clearly implies
that for every $x\in N_r$ the corresponding rectangle
$R\left(x,2r\psi\right)$ of ${\mathcal  F}_2$ can be covered by
not more than $n_r$ rectangles of the family ${\mathcal  F}_1$,
and that each rectangle of ${\mathcal  F}_1$ can appear in not
more than $n_r$ coverings of different rectangles of the family
${\mathcal  F}_2$. Therefore we can write
\begin{equation}
\label{psi-covering-first-cons.-1} \sum_{x\in
N_r}\int_{R\left(x,2r\psi\right)}\left|f\left(y\right)\right|^2dy
\leq n_r^2\sum_{x\in
N_r}\int_{R\left(x,r\psi\right)}\left|f\left(y\right)\right|^2dy.
\end{equation}
Moreover, since Condition \ref{psi-estimates-lemma} (ii) implies
also that each rectangle of the family ${\mathcal  F}_1$ has
nonempty intersection with not more than $n_r$ different
rectangles of  ${\mathcal F}_1$ itself, we have
\begin{equation}
\label{psi-covering-first-cons.-2} \sum_{x\in
N_r}\int_{R\left(x,r\psi\right)}\left|f\left(y\right)\right|^2dy
\leq n_r\int_{{\reals}^d}\left|f\left(y\right)\right|^2dy.
\end{equation}
Combining (\ref{psi-covering-first-cons.-1}) and (\ref
{psi-covering-first-cons.-2}), we obtain
\begin{align*}
\int_{{\reals}^d}\left|g\left(y\right)\right|^2dy & \leq\sum_{x\in
N_r}\int_{R\left(x,r\psi\right)}\left|g\left(y\right)\right|^2dy
\leq\sum_{x\in N_r}\int_{B\left( x,2r\psi
\right)}\left|f\left(y\right)\right|^2dy
\\
& \leq n_r^3\int_{{\reals}^d}\left|f\left(y\right)\right|^2dy,
\end{align*}
which yields the lemma. \qed


We are now in a position to carry on our localization procedure,
to obtain a full characterization of the domain.


For each $x_0\equiv(x_1^{(0)},\dots,x_d^{(0)})\in\reals^d-Z$,
consider the change of variables
$T_{x_0,\psi}:\reals^d\to\reals^d$ given by
\[
T_{x_0,\psi}(x) \! \bydef \! \big ((x_1-x_1^{(0)})/ |\psi_1(x_0)
|,\dots, (x_d-x_d^{(0)})/ |\psi_d(x_0) |\big )
\equiv\left(\widetilde{x}_1,\dots,\widetilde{x}_d\right),
\equiv\widetilde{x}
\]
with inverse
\[
T_{x_0,\psi}^{-1}(\widetilde{x})
=(x_1^{(0)}+\left|\psi_1(x_0)\right|\widetilde{x}_1,\dots,
x_d^{(0)}+\left|\psi_d(x_0)\right|\widetilde{x}_d)
=(x_1,\dots,x_d)=x.
\]
Clearly $ T_{x_0,\psi}(R(x_0,r\psi))=R(0,r)$ and $
T_{x_0,\psi}^{-1}(R(0,r))=R(x_0,r\psi), $ where, for every $r>0$,
we write $R(0,r)$ for the $d$-dimensional cube of $\reals^d$
centered at $0$ with semiaxes of length $r$. Exploiting Lemmas
\ref{psi-covering-first-cons.} and \ref{psi-estimates-lemma}, we
obtain
\begin{Proposition}
\label{L^2-domain-char} Under Assumption \ref{hp:psidomcar1}, the
functions $$ \psi_i(x)\psi_j(x)D_{i,j}u(x) $$
 belong to $L^2$ for all $i,j=1,\dots,d$.
More precisely, we have $u\in H_{(\gamma^2,\gamma\psi,\psi^2)}^2$
and the estimate
\begin{equation}
\label{eq:StewartL2} |\lambda|\Vert u\Vert_{L^2}
+|\lambda|^{1/2}\Vert u\Vert_{H_{(\gamma,\psi)}^1} +\Vert
u\Vert_{H_{(\gamma^2,\gamma\psi,\psi^2)}^2} \le K\Vert
f\Vert_{L^2}
\end{equation}
holds true for a suitable $K>0$.
\end{Proposition}
\proof Writing
\begin{equation}
p(x)\bydef\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_{i,j}u(x),
\label{sum-prop.}
\end{equation}
we know that $p$ belongs to $L^2$, and it is easily seen that, for
each $x_{0}\in\reals^{d}-Z$, the change of variables
\[
x\bydef T_{x_0,\psi}^{-1}(\widetilde{x})
\]
allows us to rewrite (\ref{sum-prop.}) in the form
\begin{equation}
\widetilde{p}(\widetilde{x}) \bydef
p\left(T_{x_0,\psi}^{-1}(\widetilde{x})\right)
=\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(\widetilde{x})\widetilde{\psi}_j(\widetilde{x})}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(\widetilde{x})\widetilde{D}_{i,j}\widetilde{u}(\widetilde{x}),
\label{utilde-equ.}
\end{equation}
with obvious meaning of the \lq\lq tilda\rq\rq-labeled functions,
and with $\widetilde{p}$ still belonging to $L^2$. Now, given any
smooth cut-off function on $\reals^d$ such that
\[
\theta_r(\widetilde{x})= \left\{
\begin{array}{ll}
1 & \hbox{if}\;\widetilde{x}\in R(0,r)\\ 0 &
\hbox{if}\;\widetilde{x}\notin R(0,2r)
\end{array},
\right.
\]
we can consider the function
\[
v(\widetilde{x})\bydef\theta_r(\widetilde{x})\widetilde{u}(\widetilde{x}).
\]
Clearly $v\in L^2$ and, as a distribution, it satisfies
\begin{align}
\label{eq:locv} &
\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)D_{i,j}v
 = \theta_r(x)\widetilde{p}
 \\
& +\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x) \left(
\widetilde{u}D_{i,j}\theta_r(x)+D_i\theta_r(x)D_j\widetilde{u}+D_j\theta_r(x)D_i\widetilde{u}
\right),^1 \nonumber
\end{align}
\footnote{Here we dropped the cumbersome \lq\lq tilda\rq\rq label
for the independent variables since no confusion can arise.}



whose right hand side, say $\widetilde{q}$, fulfills
$\widetilde{q}\in L^2(R(0,2r))$, and
$\widetilde{q}=0\;\hbox{on}\;\partial R(0,2r)$. On the other hand,
under our hypotheses, the second-order differential operator
\[
v\to\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)D_{i,j}v
\]
is strongly elliptic in $R(0,2r)$, and, thanks to well known
regularity results (see \cite[Theorem 17.2, p. 67]{Fried},
\cite[8.3, p. 173]{GT} and \cite{CH}), it follows that $v\in
H^2(R(0,2r))$. In addition, using Assumption
\ref{hp:nonvariational}, a similar argument to that in the proof
of \cite[Theorem 8.8, p.173]{GT} assures us that there exists a
suitable constant $K>0$, independent of the particular rectangle
$R(x,2r\psi)$ considered, such that the estimate
\begin{equation}
\label{v-L2-estimate} \Vert v\Vert_{H^2(R(0,r))}\leq K \Big (
\Vert\widetilde{q}\Vert_{L^2(R(0,2r))}
+\sum_{i=1}^d\frac{1}{\left|\psi_i (x_0)\right|}
\Vert\widetilde{\gamma}\widetilde{\psi}_iD_i v\Vert_{L^2(R(0,2r))}
\Big )
\end{equation}
holds true.


Indeed, to prove (\ref{v-L2-estimate}), observe first that, for
every $\varphi\in C_c^\infty(R(0,2r))$, we have
\begin{equation}
\label{v-L2-estimate-prel.-1}
\int_{R(0,2r)}\widetilde{q}(x)\varphi(x)dx
=\int_{R(0,2r)}\sum_{i,j=1}^d
\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)D_{i,j}v(x)\varphi(x)dx.
\end{equation}
In particular, choosing any $h \! \in \! \reals$ such that $0 \! <
\! |2h | \! <\hbox{\rm dist}(\hbox{\rm supp}(\varphi),\partial
R(0,2r))$, and replacing $\varphi$ in
(\ref{v-L2-estimate-prel.-1}) with the difference quotient
\[
\triangle^h\varphi\bydef \frac{\varphi(x+he_k)-\varphi(x)}{h},
\]
for some $k=1,\dots,d$, we obtain
\begin{align}
\label{v-L2-estimate-prel.-2} & \int_{R(0,2r)}\sum_{i,j=1}^d
\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)D_{i,j}v(x)\triangle^{-h}\varphi(x)dx
\nonumber
 \\
&= -\int_{R(0,2r)}\sum_{i,j=1}^dD_j \Big (
\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x) \Big ) D_iv(x)\triangle^{-h}\varphi(x)dx
\nonumber\\ & +\int_{R(0,2r)}\sum_{i,j=1}^d\triangle^h \Big (
\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)D_iv(x) \Big ) D_j\varphi(x)dx.
\end{align}
On the other hand, we have
\begin{align}
\label{v-L2-estimate-prel.-3} & \triangle^h \Big (
\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)D_jv(x) \Big ) \nonumber
 \\
&= \frac{\widetilde{\psi}_i(x+he_k)\widetilde{\psi}_j(x+he_k)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x+he_k)D_j\triangle^hv(x) \nonumber\\ &
+\triangle^h\Big
(\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)\Big )D_jv(x).
\end{align}
Therefore, combining (\ref{v-L2-estimate-prel.-1}),
(\ref{v-L2-estimate-prel.-2}) and (\ref{v-L2-estimate-prel.-3}),
it follows that
\begin{align}
\label{v-L2-estimate-prel.-4} & \int_{R(0,2r)}\sum_{i,j=1}^d
\frac{\widetilde{\psi}_i(x+he_k)\widetilde{\psi}_j(x+he_k)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x+he_k)D_i\triangle^hv(x)D_j\varphi(x)dx
\nonumber
\\
&= \int_{R(0,2r)}\widetilde{q}(x)\triangle^{-h}\varphi(x)dx
\nonumber\\ & +\int_{R(0,2r)}\sum_{i,j=1}^dD_j \Big
(\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)\Big )D_iv(x)\triangle^{-h}\varphi(x)dx
\nonumber\\ & -\int_{R(0,2r)}\sum_{i,j=1}^d\triangle^h \Big
(\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)\Big )D_iv(x)D_j\varphi(x)dx.
\end{align}
Now, writing $\Vert D\varphi\Vert_{L^2(R(0,2r))}$ as a shorthand
for $\sum_{i=1}^d\int_{R(0,2r)}\left|D_i\varphi(x)\right|^2dx$,
and applying \cite[Lemma 7.23, p. 161]{GT}, we have
\begin{equation}
\label{v-L2-estimate-prel.-5}
\int_{R(0,2r)}\left|\widetilde{q}(x)\triangle^{-h}\varphi(x)\right|dx
\leq\Vert \widetilde{q}\Vert_{L^2(R(0,2r))}\Vert
D\varphi\Vert_{L^2(R(0,2r))}.
\end{equation}
Moreover, since (\ref{Non-Var.-Case-I-Golden-Cond.}) of Assumption
\ref{hp:nonvariational} implies
\[
\left|D_j(\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)\widetilde{a}_{i,j}(x))\right|
\leq
B_2E^{1/2}\left|\psi_j(x_0)\right|\vert\widetilde{\psi}_i(x)\vert\widetilde{\gamma}(x),
\]
we can write
\begin{align}
\label{v-L2-estimate-prel.-6} & \int_{R(0,2r)}\sum_{i,j=1}^d\Big
|D_j \Big (\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x)\Big )D_iv(x)\triangle^{-h}\varphi(x)\Big
|dx \nonumber
\\
&\leq  B_2E^{1/2}d\int_{R(0,2r)}
\sum_{i=1}^d\frac{\vert\widetilde{\psi}_i(x)\vert\widetilde{\gamma}(x)}
{\left|\psi_i(x_0)\right|} D_iv(x)\triangle^{-h}\varphi(x)dx
\nonumber\\ &\leq  B_2E^{1/2}d\Vert D\varphi\Vert_{L^2(R(0,2r))}
\sum_{i=1}^d\frac{1}{\vert\psi_i(x_0)\vert}
\Vert\widetilde{\psi}_i\widetilde{\gamma}D_iv\Vert_{L^2(R(0,2r))},
\end{align}
and similarly
\begin{align}
\label{v-L2-estimate-prel.-7} &
\int_{R(0,2r)}\sum_{i,j=1}^d\triangle^h \Big (
\frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x) \Big ) D_iv(x)D_j\varphi(x)dx \nonumber
\\
&\leq  B_2E^{1/2}d\Vert D\varphi\Vert_{L^2(R(0,2r))}
\sum_{i=1}^d\frac{1}{\left|\psi_i(x_0)\right|}
\Vert\widetilde{\psi}_i\widetilde{\gamma}D_iv\Vert_{L^2(R(0,2r)} .
\end{align}
Hence, combining (\ref{v-L2-estimate-prel.-4}) with
(\ref{v-L2-estimate-prel.-5}), (\ref{v-L2-estimate-prel.-6}) and
(\ref{v-L2-estimate-prel.-7}), we obtain
\begin{align}
\label{v-L2-estimate-prel.-8} & \int_{R(0,2r)}
\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(x+he_k)\widetilde{\psi}_j(x+he_k)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(x+he_k)D_i\triangle^hv(x)D_j\varphi(x)dx
\nonumber
\\
&\leq \! \Vert D\varphi\Vert_{L^2(R(0,2r))} \Big
(\Vert\widetilde{q}\Vert_{L^2(R(0,2r))} \!+\!2B_2E^{\frac
12}d\sum_{i=1}^d \frac{1}{ |\psi_i(x_0) |} \Vert
\widetilde{\psi}_i\widetilde{\gamma}D_iv\Vert_{L^2(R(0,2r))}\Big).
\end{align}
>>>From the ellipticity condition, we obtain that
\begin{equation}
\label{v-L2-estimate-prel.-9}
E\int_{R(0,2r)}\sum_{i=1}^d\left|D_i\varphi(x)\right|^2dx
\leq\int_{R(0,2r)}\sum_{i,j=1}^d\widetilde{a}_{i,j}(x+he_k)D_i\varphi(x)D_j\varphi(x)dx.
\end{equation}
On the other hand, since $v$ has compact support, by the usual
density argument, we can replace $\varphi(x)$ with $\triangle^hv$
in (\ref{v-L2-estimate-prel.-9}). Therefore, combining
(\ref{v-L2-estimate-prel.-8}) with (\ref{v-L2-estimate-prel.-9}),
and taking into account (i) of Lemma \ref{psi-estimates-lemma}, we
can write
\begin{align}
\label{v-L2-estimate-prel.-10} &
E\int_{R(0,2r)}\sum_{i=1}^d\left|D_i\triangle^hv(x)\right|^2dx
 \leq  L^2\Vert D\triangle^hv\Vert_{L^2(R(0,2r))}
\nonumber\\ & \times \Big (\Vert\widetilde{q}\Vert_{L^2(R(0,2r))}
+2B_2Ed\sum_{i=1}^d\frac{1}
 {|\psi_i(x_0) | }
\Vert\widetilde{\psi}_i\widetilde{\gamma}D_iv\Vert_{L^2(R(0,2r))}\Big).
\end{align}
Finally, thanks to Young's inequality, the desired
(\ref{v-L2-estimate}) follows.


Now, since
\[
\left|\det
J(T_{x_0,\psi}^{-1})(x)\right|=\prod_{i=1}^d\left|\psi_i(x_0)\right|,
\]
changing the variables back, and recalling the definition of
$\widetilde{q}$, we have
\begin{align}
\label{v-L2-estimate-a} & \sum_{i,j=1}^d
\Vert\psi_i(x)\psi_j(x)D_{i,j}u(x)\Vert_{L^2(B(x_0,r\psi))}
\nonumber
\\
&= \prod_{i=1}^d\left|\psi_i(x_0)\right|^{1/2} \sum_{i,j=1}^d \Big
\| \frac{\widetilde{\psi}_i(x)\widetilde{\psi}_j(x)}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
D_{i,j}\widetilde{u}(x) \Big  \|_{L^2(R(0,r))}\\ &\leq
L^2\prod_{i=1}^d\left|\psi_i(x_0)\right|^{1/2} \sum_{i,j=1}^d\Vert
D_{i,j}\widetilde{u}(x)\Vert_{L^2(R(0,r))} \nonumber\\ &\leq
KL^2\prod_{i=1}^d\left|\psi_i(x_0)\right|^{1/2} \Big
(\Vert\widetilde{q}\Vert_{L^2(R(0,2r))} +\sum_{i=1}^d \Big
\|\widetilde{\gamma} \frac{\widetilde{\psi}_i}{\left|\psi_i
(x_0)\right|} D_i\widetilde{u} \Big \|_{L^2(R(0,2r))}\Big )
\nonumber\\ &\leq  KL^2\Big ( \Vert g\Vert_{L^2(B(x_0,2r\psi))}
+\sum_{i=1}^d\Vert\gamma\psi_iD_i u\Vert_{L^2(B(x_0,2r\psi))}\Big
), \nonumber
\end{align}
where $g(x)\bydef\widetilde{q}\left((T_{x_0,\psi}(x)\right)$ for
every $x\in B(x_0,2r\psi)$. Moreover, since
\[
\max\limits_{i=1,\dots,d}\sup_{x\in B(x_0,2r\psi)} \left|
D_i\theta_r\left(T_{x_0,\psi}(x)\right)\right|
\leq\frac{1}{r}\Vert D\theta\Vert_\infty,
\]
and
\[
\max\limits_{i,j=1,\dots,d}\sup_{x\in B(x_0,2r\psi)}
\left|D_{i,j}\theta_r\left(T_{x_0,\psi}(x)\right)\right|
\leq\frac{1}{r^2}\Vert D^2\theta\Vert_\infty,
\]
it follows that
\begin{align}
\label{v-L2-estimate-b} & \Big |
\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(\widetilde{x})\widetilde{\psi}_j(\widetilde{x})}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(\widetilde{x})
\widetilde{u}(\widetilde{x})D_{i,j}\theta_r(\widetilde{x}) \Big |
\nonumber
\\
&= \Big | \sum_{i,j=1}^d
\frac{\psi_i(x)\psi_j(x)}{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
a_{i,j}(x)u(x)D_{i,j}\theta_r\left(T_{x_0,\psi}(x)\right) \Big |
 \\
&\leq \sum_{i,j=1}^d \Big |
\frac{\psi_i(x)\psi_j(x)}{\psi_i(x_0)\psi_j(x_0)} \Big | \Vert
a_{ij}\Vert_\infty\left|u(x)\right|\frac{1}{r^2}\Vert
D^2\theta\Vert_\infty
 \leq \frac{1}{r^2}d^2AL^2\Vert D^2\theta\Vert_\infty\left|u(x)\right|,
\nonumber
\end{align}
and
\begin{align}
\label{v-L2-estimate-c} & \Big |
\sum_{i,j=1}^d\frac{\widetilde{\psi}_i(\widetilde{x})\widetilde{\psi}_j(\widetilde{x})}
{\left|\psi_i(x_0)\right|\left|\psi_j(x_0)\right|}
\widetilde{a}_{i,j}(\widetilde{x})
D_i\theta_r(\widetilde{x})D_j\widetilde{u}(\widetilde{x}) \Big |
\nonumber
\\
&= \Big |
\sum_{i,j=1}^d\frac{\psi_i(x)\psi_j(x)}{\left|\psi_i(x_0)\right|}
a_{i,j}(x)D_i\theta_r\left(T_{x_0,\psi}(x)\right)D_ju(x) \Big |
\nonumber\\ &\leq \sum_{i,j=1}^d \left|
\frac{\psi_i(x)\psi_j(x)}{\psi_i(x_0)} \right| \Vert
a_{ij}\Vert_\infty\frac{1}{r}\Vert
D\theta\Vert_\infty\left|D_ju(x)\right| \nonumber\\ &\leq
\frac{1}{r}dAL\Vert D\theta\Vert_\infty\sum_{j=1}^d\left|
\psi_j(x)\right|\left|D_ju(x)\right|.
\end{align}
Combining (\ref{eq:locv}) with (\ref{v-L2-estimate-b}) and
(\ref{v-L2-estimate-c}), we can write
\begin{align}
\label{v-L2-estimate-d} \Vert h\Vert_{L^2(B(x_0,2r\psi))} &\leq
\Vert g\Vert_{L^2(B(x_0,2r\psi))} +\frac{1}{r^2}d^2AL^2\Vert
D^2\theta\Vert_\infty \Vert u\Vert_{L^2(B(x_0,2r\psi))}
\nonumber\\ & +\frac{2}{r}dAL\Vert D\theta\Vert_\infty
\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2(B(x_0,2r\psi))},
\end{align}
and, by virtue of (\ref{v-L2-estimate-a}) and
(\ref{v-L2-estimate-d}), it follows that
\begin{align}
\label{pre-global-est.} & \sum_{i,j=1}^d
\Vert\psi_i(x)\psi_j(x)D_{i,j}u(x)\Vert_{L^2(B(x_0,r\psi))}
\nonumber
\\
&\leq KL^2\biggl( (\Vert
g\Vert_{L^2(B(x_0,2r\psi))}+\frac{1}{r^2}d^2AL^2\Vert
D^2\theta\Vert_\infty \Vert u\Vert_{L^2(B(x_0,2r\psi))})
 \\
& +\frac{2}{r}dAL\Vert D\theta\Vert_\infty
\sum_{i=1}^d\Vert\psi_iD_iu\Vert_{L^2(B(x_0,2r\psi))}
+\sum_{i=1}^d\Vert\gamma\psi_iD_iu\Vert_{L^2(B(x_0,2r\psi))}
\biggr).\nonumber
\end{align}
Summing up to the covering, thanks to Lemma
\ref{psi-covering-first-cons.} and (\ref{L^2-mid.-est.}), we can
conclude that $u\in H_{(\gamma^2,\gamma\psi,\psi^2)}^2$.


With regard to the proof of (\ref{eq:StewartL2}), notice that,
rewriting (\ref{non-var.-case-res.-equ.}) as
\[
\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)D_{i,j}u
=\left(\lambda+\gamma^2(x)\right)u-\sum_{i=1}^db_i(x)D_iu-f(x),
\]
we obtain
\[
p(x)=\left(\lambda+\gamma^2(x)\right)u-\sum_{i=1}^db_i(x)D_iu-f(x).
\]
Hence
\[
\Vert p\Vert_{L^2}\le|\lambda|\Vert u\Vert_{L^2}
+\Vert\gamma^2u\Vert_{L^2}+\sum_{i=1}^d\Vert
b_iD_iu\Vert_{L^2}+\Vert f\Vert_{L^2}.
\]
Finally, combining the latter with the global estimate which
arises from (\ref{pre-global-est.}) by summing on the covering,
and taking into account (\ref{L^2-after-mid.-est.}), we can
complete the proof. \hfill\qed


\subsection{Generation in $L_\xi^2(\reals^d)$ }
In this subsection we consider the problem of the generation of
analytic semigroups on a weighted Sobolev space for a suitable
modification of the differential operator considered up to now.
For this task we choose a weight function $\xi \in H_{\loc}^n$,
and we invoke the related weighted Sobolev spaces $L_\xi^2$,
$H_{\xi,(\gamma,\psi)}^1$ and
$H_{\xi,(\gamma^2,\gamma\psi,\psi^2)}^2$ defined in Section 2. Our
result is


\begin{Theorem}
\label{pr:L2weight} Assume that Assumption \ref{hp:domcar} still
holds true when replacing the first-order term of the operator
$\A$ with
\[
\sum_{i=1}^db_iD_i +\sum_{i,j=1}^d\psi_i\psi_ja_{i,j} \Big
(\frac{D_i\xi}{\xi}D_j+\frac{D_j\xi}{\xi}D_i\Big)
\]
and the zero-order term with
\[
-\gamma^2+\sum_{i,j=1}^d\psi_i\psi_ja_{i,j} \Big
(\frac{D_{i,j}\xi}\xi+2\frac{D_i\xi D_j\xi}{\xi^2}\Big )
+\sum_{i=1}^db_i\frac{D_i\xi}\xi ;
\]
then the operator $\A$ has a realization
$\A_{2,\xi}:D(\A_{2,\xi})\to L_\xi^2$ which generates an analytic
semigroup on $L_\xi^2$. Moreover, for each $\lambda\in\complex$
such that ${\rm Re}\;\lambda>0$, the resolvent equation $\lambda
u-\A_{2,\xi }u=f$ has, for every $f\in L^2$, a unique solution
$u\in D(\A_{2,\xi })$, which satisfies the estimate
\begin{equation}
\label{eq:StewartL2weight} |\lambda|\Vert
u\Vert_{L_\xi^2}+|\lambda|^{1/2}\Vert
u\Vert_{H_{\xi,(\gamma,\psi)}^{1,2}} +\Vert
u\Vert_{H_{\xi,(\gamma^2,\gamma\psi,\psi^2)}^{2,2}} \le C\Vert
f\Vert_{L_\xi^2},
\end{equation}
for a suitable constant $C>0$. In particular we have $D(\A_{2,\xi
})=H_{\xi,(\gamma^2,\gamma\psi,\psi^2)}^2$.
\end{Theorem}


\proof Given $f\in L_\xi^2$, let us consider the formal resolvent
equation
\[
\lambda u-\A u=f.
\]
Multiplying both the sides by the weight $\xi$, we can write
\[
\lambda\xi u-\A(\xi u)+(\A(\xi u)-\xi\A u)=\xi f.
\]
On the other hand, we have
\[
\xi\A u-\A(\xi u)
=\sum_{i,j=1}^d\psi_i\psi_ja_{i,j}\left((D_{i,j}\xi)u+D_j\xi
D_iu+D_i\xi D_ju\right) +\sum_{i=1}^db_i(D_i\xi)u,
\]
and since
\[
D_i\xi D_ju=\frac{D_i\xi}{\xi}(\xi
D_ju)=\frac{D_i\xi}{\xi}\left(D_j(\xi u)-uD_j\xi\right),
\]
we can rewrite
\begin{align*}
& \A(\xi u)-\xi\A u = \sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)\\
& \times \Big (\frac{D_{i,j}\xi(x)}{\xi(x)}\xi(x)u
+\frac{D_i\xi}{\xi(x)}D_j(\xi u) +\frac{D_j \xi}{\xi(x)}D_i(\xi
u)+2\frac{D_i\xi D_j\xi}{\xi^2}(\xi u)\Big )\\ &
+\sum_{i=1}^db_i\frac{D_i\xi}{\xi}(\xi u).
\end{align*}
Therefore, if we consider the operator
$\overline{\A}_2:D(\overline{\A}_{2 })\to L^2$ given by
\[
D(\overline{\A}_2)\bydef D(\A_2),
\]
and
\begin{eqnarray*}
\overline{\A}_2 v &\bydef& \A
v+\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x) \Big
(\frac{D_i\xi}{\xi(x)}D_jv+\frac{D_j\xi}{\xi(x)}D_iv\Big )\\
&&+\sum_{i,j=1}^d\psi_i(x)\psi_j(x)a_{i,j}(x)\frac{D_{i,j}\xi(x)}{\xi(x)}v
+\sum_{i=1}^db_i\frac{D_i\xi}\xi v,
\end{eqnarray*}
our hypotheses assure that, for each $\lambda\in\complex$ such
that ${\rm Re}\;\lambda>0$, the resolvent equation
\[
(\lambda-\overline{\A}_2)v=g
\]
has, for every $g\in L^2$, a unique solution $v\in
D(\overline{\A}_2)$, which satisfies the estimate
\begin{equation}
\label{eq:stimapeso1} |\lambda|\Vert
v\Vert_{L^2}+|\lambda|^{1/2}\Vert v\Vert_{H_{(\gamma,\psi)}^1}
+\Vert v\Vert_{H_{(\gamma^2,\gamma \psi,\psi^2)}^2} \le K\Vert
g\Vert_{L^2},
\end{equation}
for a suitable $K>0$. Now, writing
\[
D(\A_{2,\xi})\bydef \left\{u\in L_{\xi}^2 : u\xi\in
D(\overline{\A}_2)\right\}, \quad \A_{2,\xi}u\bydef\A u,
\]
and multiplying by $\xi$, the resolvent equation $\lambda
u-\A_{2,\xi}u=f$ becomes
\[
\lambda(\xi u)-\overline{\A}_2(\xi u)=\xi f.
\]
Therefore, (\ref{eq:StewartL2weight}) follows by a straightforward
application of (\ref{eq:stimapeso1}) to $v=\xi u$. Finally, we
have
\[
D(\A_{2,\xi}) =H_{\xi,(\gamma^2,\gamma\psi,\psi^2)}^2,
\]
as it easily follows from the definition of our weighted spaces
(see Section \ref{PRE}).



\begin{Remark}
\label{rm:weight1} \rm The problem of determining the type of
weights for which the assumptions of the above proposition are
satisfied now arises. For instance, it can be checked that,
choosing $\xi(x)=\left(c+|x|^{2d}\right)^{-1}$, where
$c\in\reals_+$, the contribution of the terms depending on $\xi$
in the operator $\overline{\A}$ in Assumption
\ref{hp:nonvariational-1} goes to zero as $d$ increases. It
follows that one can always find a good weight of the form
described above, for which Proposition \ref{pr:L2weight} holds
true.
\end{Remark}


\section{Examples}
\label{APPLICATIONS}


\begin{Example}
\rm We consider here the PDE for the price of a European
contingent claim (see, e.g., \cite{WIL93}) in the multifactor
case. Assume that the price process of the given underlying $d\geq
1$ assets $\left(X_t\right)_{t\geq 0}$, where
$X_t=\left(X_t^{1},\dots,X_t^{d}\right)$ satisfies the SDE
\[
dX_t=rX_tdt+\sigma\left({\rm diag}X_t\right)dW_t,\quad \forall
t\geq 0
\]
where $\left({\rm diag}X_t\right)$ is the diagonal matrix with the
components of $\left(X_t\right)_{t\geq 0}$ on the main diagonal,
$r$ is the interest rate of a reference riskless asset in the
market, $\sigma$ is a given $d$-order matrix such that, writing
$\sigma^*$ for the transpose of $\sigma$, the matrix
$\sigma^*\sigma$ is positive definite, and
$\left(W_t\right)_{t\geq 0}$, is a $d$-dimensional Wiener process,
$W_t=\left(W_t^{1},\dots,W_t^{d}\right)$.


Write $v\equiv v(x,t)$, where $x\equiv\left(x_1,\dots,x_d\right)$,
for the no-arbitrage price of a contingent claim having payoff
$g\equiv g(x)$ at the expiration time $T$. Then, under the
so-called no-arbitrage assumption, it is well known that $v$
solves the backward Kolmogorov PDE
\[
D_tv+\A v=0,
\]
with the terminal condition $v(x,T)=g(x)$, where
\[
\A v=\frac{1}{2}{\rm Tr}\;\left(\left(\sigma{\rm diag}x\right)
\left(D_{i,j}v\right)(\sigma{\rm diag}x)^*\right)
+r\sum_{i=1}^dx_iD_i v-rv,
\]
and $\left(D_{i,j}v\right)$ denotes the $d$-order matrix having
entries for $D_{i,j}v$, for $i,j=1,\dots,d$.


It can be easily verified that the second-order operator $\A$ fits
all our assumptions. Of course, for financial applications we are
interested in solving the above PDE on the positive orthant, not
in all $\reals^d$, but this does not affect our results. Indeed,
it can be proved that the operator $\A$ generates a
positivity-preserving semigroup (see the second part of this
paper).


Notice that this equation can be also studied (and solved
explicitly) by taking the change of variable (see
\cite{BLA,WIL93}) $y_i =\ln x_i$ for all $i=1,\dots,d$. Our
approach has the advantage of proving a general generation
property and a characterization of the domain which turns out to
be useful when one considers some modifications of the basic model
above. A first example is the case when $\sigma$ depends on $x$,
but preserves differentiability and strong ellipticity (see, e.g.,
\cite{WIL93}). Another example is discussed below.


Assume that the underlying assets pay dividends with a rate
$\rho\equiv\rho(x,t)$, and that they are subjected to a tax rate
$\epsilon\equiv\epsilon(x,t)$. Then we have the following
no-arbitrage PDE
\[
D_tv+\frac{1}{2}{\rm Tr}\;\left(\left(\sigma{\rm diag}x\right)
\left(D_{i,j}v\right) (\sigma{\rm diag}x)^*\right)
+(r-\rho)(1-\epsilon)\sum_{i=1}^dx_iD_iv-rv,
\]
with the terminal condition $v(x,T)=g(x)$. In the case $d=1$,
$\rho=0$, $\epsilon=0$ and $g(x)= (x-E)^+$, where $E$ is the
maturity price, we obtain the well-known Black and Scholes
equation described in \cite{BLA}. Also multifactor models, such as
the ones appearing in \cite{DUFF}, options on futures contracts,
and swaps can be treated in our framework (see \cite{BARGOVE},
\cite{WIL93}), along with the example below.


\end{Example}


\begin{Example} \rm
 We consider here the structure model of interest rate derivatives.
For the so-called affine single-term structure model the interest
rate is modeled by the stochastic process $\left(X_t\right)_{t\geq
0}$ satisfying the differential equation
\begin{equation}
\label{RSTOC}
dX_t=\left(\alpha_1(t)+\alpha_2(t)X_t\right)dt+\left(\beta_1(t)+\beta_2(t)X_t\right)dW_t.
\end{equation}
Suitably choosing the coefficients $\alpha_1(t),$ $\alpha_2(t),$
$\beta_1(t)$ and $\beta_2(t)$, different term-structure models can
be obtained. In particular (see part II of this work), two models
fitting our framework can be obtained by choosing
\begin{enumerate}
\item
\label{DOT} $\alpha_1=\alpha_2=\beta_1=0$ \cite{DO78}
\item
\label{BRE} $\beta_1=0$ \cite{BRE82}.
\end{enumerate}
\noindent The price of a zero-coupon bond maturing at date $T$ is
the solution of the Cauchy problem
\[
D_tv+\A v=0,
\]
with the end terminal condition $v(x,T)=1$, where
\[
\A v =\tfrac{1}{2} (\beta_1+\beta_2x )^2D_{x,x}v +
(\alpha_1+\alpha_2x )D_xv.
\]
We remark in addition that our results allow us to treat also
multifactor models with time-dependent coefficients (see
\cite{DUFF,BARGOVE}), and also semilinear perturbations of the
above equations, as already shown in the previous example.


\end{Example}


\begin{Example} \rm
 The following equation, coming from nonlinear filtering,
is considered in \cite{BaraseC,Sheu}:
\[
D_t=D_{x,x}+xD_xv-x^2v, \quad t >0, x\in\reals
\]
with initial condition $v(0,x)=g(x)$. It can be easily checked
that the second order operator defined by the right-hand side of
the above equation satisfies our assumptions.



\end{Example}


\noindent {\bf Acknowledgments.}
 Thanks to E. Barucci for useful suggestions about financial
applications and G. Da Prato and A. Lunardi for useful
discussions.


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\end{document}
