Published Paper
Inserted: 8 jan 2002
Last Updated: 18 dec 2002
Journal: Communications in PDE
Volume: 27
Number: 5\&6
Pages: 945-952
Year: 2002
Abstract:
\magnification=\magstep1 \hsize 5.5truein \vsize 8.5truein \hoffset .7truecm
\newfam\numsets \font\textnumsets msbm10 \textfont\numsets \textnumsets \font\scriptnumsets msbm10 at 7pt \scriptfont\numsets \scriptnumsets \font\scriptscriptnumsets msbm10 at 5pt \scriptscriptfont\numsets \scriptscriptnumsets \def\Bbb{\fam \numsets} \mathchardef \subsetneq "2\number\numsets 28
\def\realnumber{{\Bbb R}} \def\complexnumber{{\Bbb C}} \def\integernumber{{\Bbb N}} \def\section#1{\bigbreak\centerline{#1}\nobreak\smallskip\nobreak \noindent \ignorespaces}
\def\diverg{\mathop{\fam 0 div}}
\vglue 2truein {\parindent 0pt \parfillskip 0pt plus1fil \obeylines \everypar{\hfil}{\baselineskip 1.2 \baselineskip % \bf SHARP UPPER BOUNDS FOR THE DEGREE OF REHULARITY OF THE SOLUTIONS TO AN ELLIPTIC EQUATION} \noindent Luigi Greco\qquad Carlo Sbordone \noindent Dipartimento di Matematica e Applicazioni ``R. Caccioppoli" Università degli Studi, Via Cintia -- 80126 NAPOLI, ITALY {\sl email:}\ luigreco@unina.it\qquad sbordone@unina.it \noindent \noindent }
\section{ABSTRACT}
It is well known that local solutions to the linear equation
$$\diverg A(x)\nabla u=0\eqno(1.1)$$
in an open set $\Omega\subseteq \realnumber^n$, are locally Hölder continuous. Here we assume
$\vphantom{A}^tA=A$ satisfying the uniform ellipticity condition
$${
\xi
2\over K}\le \langle A(x)\xi\,,\,\xi\rangle\le K\,
\xi
2\,,\eqno(1.2)$$
with $K\ge 1$, for a.e.\ $x\in \Omega$ and all $\xi\in \realnumber^n$.\par
In dimension $n=2$ the best Hölder exponent is known to be $1/K$.
The optimality of such an exponent is seen by considering the functions
$$uj(x)=xj\,
x
{{1\over K}-1},\qquad j=1,2\eqno(1.3)$$
and the coefficient matrix
$$A(x)={1\over K}\,I+\left(K-{1\over K}\right)\,{x\otimes x\over
x
2}\,.\eqno(1.4)$$
If $K>1$, the functions $u_j$ defined in (1.3) belong to $C^{0,{1\over K}}_{\fam 0 loc}$, but not to
$C^{0,\beta}_{\fam 0 loc}$ for any $\beta>1/K$.\par
In any dimension, equation (1.1) with the coefficient matrix (1.4) has the solutions
$$uj(x)=xj\,
x
{\alpha-1},\qquad j=1,2,\ldots,n\,,\eqno(1.5)$$
where
$$\alpha={1\over 2}\sqrt{(n-2)2+4\,{n-1\over K2}}-{n-2\over 2}\eqno(1.6)$$
(see Lemma 1 below). Notice that for $n=2$ formula (1.6) reduces to the equality $\alpha=1/K$ and
(1.5) reduce to (1.3). As above, the functions defined in (1.5) belong to $C^{0,\alpha}_{\fam 0 loc}
(\Omega)$, but not to $C^{0,\beta}_{\fam 0 loc}(\Omega)$ for any $\beta>\alpha$.\par
Our aim here is to show that, modulo a nonzero moment condition, all solutions of
(1.1), (1.4) enjoy this same property. That is why we call the functions in
(1.3)-(1.5) {\sl principal solutions} of (1.1). In other words, the principal solutions govern the
regularity properties of other solutions. We also study higher order derivatives of the solutions,
by means of higher order moments.\end