\magnification=\magstep1
\parindent=0pt
\def \csi {\xi}
\def \l {\lambda}
\def \e {\varepsilon}
\def \Cup {\bigcup}
\def \Cap {\bigcap}
\def \Nabla {\nabla}
\def \Vs {\cal V}
\def \circle {\circ}
\def \Beta {\beta}
\def \Tau {\tau}
\def \hh {{\cal H}}
\def \MD {{\cal MD}}
\def\cer#1{\mathrel{\smash{\mathop{#1} \limits^{\circ} }}}
\def\finedim{\quad \vrule height .9ex width .8ex depth -.1ex}
\def\rr{{\ \cal R \ }}
\def\bar{\overline}
\def\nont{ \rlap {\ /} \Theta }
\def\real{{\bf R}}
\def\nat{{\bf N}}
\def\interi{{\bf Z}}
\def\raz{{\bf Q}}
\newcount\refno
\global\refno=0
\def\refer#1#2#3#4#5{\hangindent=20 pt\hangafter=1
{\global\advance\refno by 1}
$\hbox to 20 pt{[\the\refno]\hfill}${\bf {#1}:}{\sl\ {#2},}{\ #3}{\bf\
#4}{\ #5}\medskip}
\centerline{ \bf Congetture riguardanti alcuni problemi di
evoluzione.}
\bigskip
\centerline{ A Paper in Honor of John Nash }
\bigskip
\centerline{ Ennio De Giorgi }
\bigskip
{\bf Summary.} I have greatly appreciated the invitation to write a paper in
honor of John Nash, a scientist who has been a major source of new ideas for
mathematicians. His work has been a clear example of how the most original
mathematical ideas are often close to the fundamental problems of other
disciplines. I believe that his example is very important for any
student who is still motivated by the drive that the ancients called
{\sl Philosophy}, that is love for Wisdom.
\medskip
The character of my paper is exploratory.
Some conjectures concerning ``evolution problems'' are
presented. They are related to the ``steepest descent'', to
the approximations of Newton's gravitation law, to hyperbolic non linear
equations and to ``descent movements" of manifolds.
The article identifies several questions, of which I do not know the answer,
and points out a number of analogies between problems that are apparently far
from each other. I believe that the study of these conjectures might provide an
opportunity for scientists that are expert in different fields within pure and
applied mathematics to get together and investigate on the connections existing
among various mathematical concepts, such as linear vs. non-linear behaviour,
stability vs. instability, or convergence of different approximation methods,
as well as certain ideas well developed in physics, such as deterministic vs.
non-deterministic behaviour, predictability vs. non-predictability, order and
chaos, etc.
\bigskip
\beginsection Problem 1. Steepest Descent Problems.
We begin with the analysis of some problems related to the ``steepest descent
method''.
We deal with functionals of the following kind
$$F(w,A)=\int_A g(w,\nabla w,\,\ldots\, ,\nabla^k w).
\leqno(1)$$
Here $A$ is a bounded open subset of $\real^n$, $w$ is a smooth real
function, $g$ is a smooth real
function defined on the space of the real polynomials of $n$ variables
with degree not greater than $k$.
We say that a function $\psi\in C^\infty(\real^n)$ is
the ``Euler function'' associated to $F$ and to $w$, and we write
$$\psi={ \cal E }(F,w),$$
if for every bounded open set $A\subset\real^n$ and for every test function
$\tau\in C_0^\infty(A)$ the following condition holds
$$\lim_{\lambda\to0}
{1\over\lambda}\left(F(w+\lambda\tau,A)-F(w,A)\right)=\int_A\tau\,\psi.
\leqno(2)$$
Sometimes the function $g$ may be neither convex nor enjoy
any other condition close to convexity, see [4]. In these cases to
study the differential equation
$${\partial u\over\partial t}=-{ \cal E }(F,u(\cdot,t))\leqno(3)$$
with the initial condition
$$u(x,0)=\varphi(x)\leqno(4)$$
in the cone $\real^n\times[0,+\infty)$ may be very difficult.
Nevertheless a vanishing--viscosity type method seems to be suitable
(see for instance [5]). It reads as follows: one associates $F$ the
functionals $F_\e$ defined for every $\e\not= 0$, by the formula
$$
F_\e(w,A)=F(w,A)+\e^2\int_A{\vert\nabla^{k+1}w\vert}^2
$$
where as usual
$$
|\nabla^s w|^2=\sum_{j_1=1}^n \dots \sum_{j_s=1}^n \Bigl({{\partial^s
u}\over{\partial x_{j_1}\dots \partial x_{j_s}}}\Bigr)^2.
$$
Hence the task is the approximation of the differential equation
$(3)$ with the equation
$${\partial u\over\partial t}=-{ \cal E }(F_\e,u(\cdot,t)) .
\leqno(5)$$
A simple but really not trivial example of this situation is the
following: let $k=1$, $n=1$ (see [10,11]),
$$g\left(w,{\partial w\over\partial
x}\right)={1\over 2}\log\left(1+\left\vert{\partial w\over\partial
x}\right\vert^2\right).$$
In this case the equation $(3)$ can be written as
$${\partial u\over\partial t}=
{\partial\over\partial x}
\left({{\partial u\over\partial x}\over 1+\left\vert{\partial u\over\partial x}\right\vert^2}\right)
={\partial^2 u\over \partial x^2}{{1-\Bigl|{\partial u\over\partial x}
\Bigr|^2}\over\left(1 +\Bigl|{\partial u\over\partial x}\Bigr|^2\right)^2}$$
The study of this equation may be very hard when the condition
$\left\vert{\partial\varphi\over\partial x}\right\vert>1$
is fulfilled by the initial value
$\varphi$ in some subset of
$\real$. Then the functional $F_\e$ can be considered. This one leads to the
``stabilized'' equation
$${\partial u\over\partial t}=
{\partial^2 u\over \partial x^2}{{1-\Bigl|{\partial u\over\partial x}\Bigr|^2}
\over\left(1 +\Bigl|{\partial u\over\partial x}\Bigr|^2\right)^2}
-\e^2{\partial^4u\over\partial x^4},
\leqno (6) $$
and leads to state the first conjecture.
\bigskip
{\bf Conjecture 1.} If $\varphi\in C_0^\infty(\real)$ and for every
$\e\not=0$ $u_\e$ is a solution of the equation $(6)$
with the initial condition $(4)$ and it is bounded in $\real\times
[0,+\infty )$, then for almost all $(x,t )\in \real\times [0,+\infty)$
the limit of $u_\e(x,t)$ there exists as $\e\to 0$.
\bigskip
{\bf Remark 1.} If Conjecture 1 is true, it could be
strengthened by requiring the convergence {\it for all} $(x,t)$, and
then by requiring much stronger convergences
leading to further regularity properties of the limit function.
The convergence may be requested to have a certain degree of uniformity
with respect to the initial value $\varphi$, and then the
continuous dependence of the limit on the initial datum may be also requested.
On the other hand, if any counterexample to Conjecture 1 is found, the
conjecture itself should be weakened. This can be done
in several ways. E.g., the convergence of the solutions of equations
(6) would already be satisfactory for initial data simply belonging to
a dense set in some typical metric or topological space. For instance
analytical functions spaces, classes of trigonometric polynomials or
of functions built up by Gaussian, could be considered to this aim.
Moreover these topological spaces could be endowed with some natural
measure $\mu$, hence the convergence of the $u_\e$ might simply be
required for $\mu$--almost all initial data.
\bigskip
{\bf Remark 2.} A further weakening of Conjecture 1 could be obtained
by relaxing the global condition of convergence for almost all the points
of $\real\times[0,+\infty)$ to the less severe request of a local almost
sure convergence. Namely: there exists
$T>0$ such that $u_\e$ converge for almost all the points of
$\real\times[0,T)$. A first step towards the evidence of Conjecture 1
or the discovery of any counterexample can be the study of sequences of
positive real numbers $(\e_i)$ such that $\e_i\to0$ and the
sequence $(u_{\e_i})$ converges in some sense.
\bigskip
{\bf Remark 3.} Several versions of Conjecture 1 may be
considered: functions defined in $\real$ can be replaced by functions
defined in some interval of $\real$ with the right boundary conditions,
initial data with compact support and bounded solutions may be
replaced respectively by periodic initial data and periodic solutions
in the space variable $x$, bounded solutions can be replaced by
solutions of (6) verifying some decay condition at infinity.
More generally we can consider
$$F(w,A)={1\over2}\int_A\log(1+\vert\nabla w\vert^2)\ .$$
Similarly a new functional, in the case of only one space-variable, such as
$$F(w,A)=\int_A\left(1-\left\vert{\partial w\over\partial
x}\right\vert^2\right)^2 \ , $$
leads to consider the correspondent functional in $\real^n$
$$F(w,A)=\int_A\left(1-\left\vert\nabla w\right\vert^2\right)^2 .$$
Interesting results could be obtained by studying other cases, for
instance setting
$$F_\e(w,A)=\int_A
\e^2w^2+\left(1-\left\vert{\nabla w}\right\vert^2\right)^2+\e^2
{|\nabla^2 w|^2 } . $$
Eventually non-stationary and nonhomogeneous cases can be considered:
the function
$g$ appearing in (1) should depend not only on $w$ and on its derivatives
but also on the space variables $(x_1,\ldots,x_n)$
and possibly on a parameter $t$, that in (3), (5) is a time variable.
\bigskip
\beginsection Problem 2. Approximate Gravitational Problem.
We now formulate a problem related to a reasonable approximation
of Newton's law (see [3,7]); this problem will be called
``approximate gravitational problem'' and it can be
formulated as follows:
\bigskip
{\bf Problem 2.1.} Let $\mu$ be a finite positive $\sigma$-additive measure
defined on the $\sigma$-algebra of
Borel subsets of $\real^6$.
The approximate gravitational problem associated to $\mu$ is a
triplet of functions $(s,v,u)$ satisfying the qualitative conditions
{\bf (A)} and the integral--differential equations {\bf (B)} below:
{\bf (A)}
$$s=s(\e,\csi,\eta,t)\in \bigl[C^1( (\real\setminus\{0\})\times
\real^7)\bigr]^3,$$
$$v=v(\e,\csi,\eta,t)\in \bigl[C^1((\real\setminus\{0\})\times
\real^7)\bigr]^3,$$
$$u=u(\e,x,t) \in C^0((\real \setminus \{0\}) \times \real^4 ).$$
The scalar function $u$ is differentiable with respect to the space variable
$x=(x_1,\,x_2,\,x_3)$ and the spatial gradient $\nabla_x u$ belongs to
$C^0((\real\setminus\{0\})\times\real^4)^3$.
\bigskip
{\bf (B)}
The vector functions $s,\,v$ satisfy
$${\partial s \over \partial t } = v,
\quad {\partial v \over \partial t }=
\alpha(\e,s(\e,\xi,\eta,t),t ), \quad \alpha(\e,x,t )=\nabla_x
u(\e,x,t)\qquad \leqno (1) $$
with the initial conditions
$$s(\e,\csi,\eta,0)=\csi,
\qquad v(\e,\csi,\eta,0)=\eta \leqno (2)$$
whereas $u$ satisfies the integral equation
$$
u(\e,x,t) =
\int_{\real^6} (|x-s(\e,\csi,\eta,t)|^2 +\e ^2 )^{-1/2}
d\mu (\csi,\eta).\leqno (3)
$$
{\bf Remark 1.} Intuitively, $s(\e,\csi,\eta,t)$ represents the
position at time $t$ of a particle located at $t=0$
in the point $\csi$ with speed $\eta$; $v(\e,\csi,\eta,t)$
represents the velocity at time $t$ of the same particle and
$\alpha(\e,x,t)$ represents the acceleration of a particle that
at time $t$ is located in $x$. Finally, $u$ should give, when $\e$ is
small, a reasonable approximation of Newton's potential.
The measure $\mu$ represents an initial distribution of masses
(possibly endowed with some velocity) at time $t=0$.
In order to understand more clearly the meaning of (1), (2), (3), it is
convenient to consider the case of a measure $\mu$ which can be written
as the sum of $n$ Dirac measures, i.e., there exist
$(\csi_h,\eta_h)$ such that
$$\mu(B)=\sum_{h=1}^n m_h \delta(\csi_h,\eta_h)(B)$$
for any Borel set $B$, where
$$\cases { \delta(\csi_h,\eta_h)(B) = 1 & if $(\csi_h,\eta_h)\in B$ \cr\cr
\delta(\csi_h,\eta_h)(B) = 0 & if $(\csi_h,\eta_h) \notin B .$ \cr }$$
In this case the potential $u$ appearing in $(3)$ becomes
$$u(\e,x,t)= \sum_{h=1}^n m_h (|x-s(\e
,\csi_h,\eta_h,t)|^2+ \e^2 ) ^{-1/2}. $$
Hence, the system of equations $(1),\,(2),\,(3)$ appears to be a
reasonable approximation of the classical $n$-body problem.
\bigskip
{\bf Remark 2.} The case in which $\mu$ is not supported in a finite number
of points is suggested by the universe, in which
stars, planets, satellites, gas clouds and other forms of ``diffuse mass''
interact. In this case, (3) can be written in the form
$$
u(\e,x,t) = \int_{\real^6} (|x-\csi|^2 +\e ^2 )^{-1/2}
d\mu^*_t (\csi,\eta),\leqno (3^*)
$$
where the measure $\mu^*_t$ is given for any $t\in\real$ by the
formula
$$
\mu^*_t(B)=\mu\left(\bigl\{(\xi,\eta)\in\real^6:\
\bigl(\sigma(\e,\xi,\eta,t),
v(\e,\xi,\eta,t)\bigr)\in B \bigr\}\right).
$$
Intuitively, $\mu^*_t$ describes for any $t\in\real$ the distribution
and the velocities at time $t$ of the mass in the universe;
the evolution of $\mu_t^*$ essentially describes what would be the
evolution of the universe under an
``approximate gravitational field'' computable from the
``elementary potential''
$$
V_{\e}(x)=(|x|^2+\e^2)^{-1/2} \ ,
\leqno(4)$$
which, for $\e$ small, seems to be a reasonable approximation of
the classical Newton's potential, proportional to
$$
V(x)={1\over |x|}.
$$
To my knowledge, in the literature there is not an exhaustive answer
to the questions about the existence of limits, as $\e\to 0$, of the
functions considered in the approximate gravitational problem. In
particular, I am very interested in the proof or in a counterexample
of the following conjecture.
\bigskip
{\bf Conjecture 1.} If $(s,v,u)$ is a solution of the approximate
gravitational problem associated to $\mu$, then for almost every
point $(x,t)\in{\real}^4$ there exists
$$\lim_{\e\to0} u(\e,x,t) \in\real;$$
moreover, for almost every $(\csi,\eta,t)\in{\real}^7$, there exist
the limits
$$\lim_{\e\to0} s(\e,\csi,\eta,t),$$
$$\lim_{\e\to0} v(\e,\csi,\eta,t).$$
\bigskip
{\bf Remark 3.} It is very likely that Conjecture 1 holds for large classes of
measures $\mu$ but not for all measures $\mu$ considered in Problem 2.1.
Up to now, I didn't find in the literature any counterexample; I think
that a counterexample may be looked for in the class of measures $\mu$
satisfying the conditions
$$\mu=\sum\limits_{h=1}^{+\infty}m_h\delta(\csi_h,\eta_h),\qquad
\sum\limits_{h=1}^{+\infty} m_h<+\infty,\qquad
\sum\limits_{h=1}^{+\infty} \vert\eta_h\vert=+\infty.$$
More difficult but also more interesting is the case
$$\mu=\sum\limits_{h=1}^{+\infty}m_h\delta(\csi_h,\eta_h),\qquad
\sum\limits_{h=1}^{+\infty} m_h<+\infty,\qquad
\sup_h\,\left(\vert\csi_h\vert+\vert\eta_h\vert\right)\,<+\infty$$
corresponding to a situation in which initially
all the masses are confined in a bounded region in space and the set
of their initial velocities is bounded as well.
\bigskip
We can also state a weaker conjecture.
\bigskip
{\bf Conjecture 2.} Let ${\cal M}(\real^6)$ be the set of positive
$\sigma$-additive measures $\mu$ defined on Borel subsets of
$\real^6$ and satisfying the condition
$\mu(\real^6)<+\infty$. Endowing ${\cal M}(\real^6)$ with
the distance
$$\delta(\mu,\nu)=\sup_B\left(\mu(B)-\nu(B)\right)+\sup_B\left(\nu(B)-
\mu(B)\right),$$
the set of measures $\mu$ satisfying Conjecture 1 is dense.
\bigskip
Moreover, it would be interesting to evaluate the stability with respect to
$\mu$ of the limits as $\e\to 0$ of the solutions of the approximate
gravitational problems, in the cases where the limit exist.
Very likely there is not always an high stability of the limits, hence
it might be convenient to introduce a weaker condition, called
$\alpha$--stability, with the following definition.
\bigskip
{\bf Definition 1.} Given a measure $\mu$, the functions $(s,v,u)$
solving the approximate gravitational problem associated to $\mu$ and a
real number $\alpha\geq0$, we say that $(s,v,u)$ is
an $\alpha\,$--$\,$stable solution of
the approximate gravitational problem associated with $\mu$ if the limits
$$\lim_{\e\to0}u(\e,x,t),$$
$$\lim_{\e\to0}s(\e,\csi,\eta,t),$$
$$\lim_{\e\to0}v(\e,\csi,\eta,t),$$
exist for almost every $(x,t)\in\real^4$ and almost every
$(\csi,\eta,t)\in\real^7$. Moreover, we ask that
for any infinitesimal sequence $(\e_i)$, and any sequence of measures
$(\mu_i)$ satisfying the condition
$$\lim_{i\to+\infty}{\e_i}^{-\alpha}\delta(\mu_i,\mu)=0,$$
denoting by $(s_i,v_i,u_i)$ the solutions of the approximate
gravitational problem corresponding to $\mu_i$, we have
$$\lim_{i\to+\infty}
u_i(\e_i,x,t)=\lim_{\e\to0}u(\e,x,t),$$
for almost every $(x,t)\in\real^4$, and, for almost every
$(\csi,\eta,t)\in\real^7$ we ask that
$$\lim_{i \to + \infty }
s_i(\e_i,\csi,\eta,t)=\lim_{\e\to0}
s(\e,\csi,\eta,t),$$
$$\lim_{i \to + \infty }
v_i(\e_i,\csi,\eta,t)=\lim_{\e\to0}
v(\e,\csi,\eta,t).$$
We can now formulate a meaningful reinforcement of Conjecture 2
with the following
\bigskip
{\bf Conjecture 3.} There exist positive numbers $\alpha$ such that
the set of all measures $\mu$ corresponding to $\alpha\,$--$\,$stable
solutions of the approximate gravitational problem is dense in
${\cal M}(\real^6)$.
\bigskip
{\bf Remark 4.} If Conjecture 3 were true, it would be interesting to
compute the infimum of the numbers $\alpha$ for which the density property
is true; this infimum is very likely strictly positive. Taking into account
that $\alpha\,$--$\,$stability becomes weaker as $\alpha$ increases,
the infimum would give an indication of the global degree of stability
of the approximate gravitational problem.
\bigskip
\beginsection 3. Some Variants of the Approximate Gravitational Problem.
We can consider many variants of the gravitational problem shown in \S 2. We
could imagine that the force acting between 2 bodies is a sum of two parts,
one attractive and the other repulsive, for instance replacing the
equation (3) of \S 1 with
$$u(\e,x,t) =\int_{\real^6} (|x-s|^2 + \e ^2 )^{-1/2} -
c^2 ( |x-s|^2 +\e^2)^{-1} d\mu(\csi,\eta)$$
where $c$ is a positive constant.
After the limits for $\e \to 0$, the convergence of such limits when $c\to 0$
should also be considered; this means, in some sense, that the repulsive
forces act only when the distance is very small. Finally, we could imagine
even more complex expressions, corresponding to forces depending not only on
the position but also on the velocity of the particles, for instance
studying integrals like
$$ \alpha(\e,x,y,t)=\int_{\real^6}
\phi(\e,x,y,s(\e,\csi,\eta,t) ,v(\e,\csi,\eta,t) )
d\mu(\csi,\eta)$$
hence differential equations,
$${\partial v \over \partial t} = \alpha(\e,s,v,t) .$$
Clearly, in modelling very complex physical theories, many further
complications may arise and these are beyond the aim of this note,
which is only directed to underline some kinds of mathematical problems
that subsequently can also produce interesting physical models.
\bigskip
Coming back to the approximate gravitational problem, particularly to
Remark 2 of \S 2, it can be studied the case in which the mass
$\mu^*$ is continuously distributed, that is, there exists a function
$\varphi=\varphi(\e,x,y,t) \in C^\infty\bigl(\real\setminus\{0\}
\times\real^7\bigr)$ such that
$$
\mu^*_t(B)=\int_B \varphi(\e,x,y,t) dxdy;
\leqno(1)$$
then the function $u(\e,x,t)$ will be given by the formula
$$u(\e,x,t)=\int_{\real^6} \left(\vert
x-\csi\vert^2+\e^2\right)^{-1/2}\,\varphi(\e,\csi,\eta,
t)\,d\csi\,d\eta,$$
moreover $\varphi$ will satisfies the first order linear differential
equation
$${\partial \varphi\over\partial t}=-\sum_{h=1}^3\left({\partial
\varphi\over \partial x_h}y_h+{\partial\varphi\over\partial y_h}{\partial
u\over\partial x_h} \right)$$
whose characteristics are determined by equations (1), (2) of Problem 2.1.
Hence, we can state the problem of existence of the limits as
$\e\to 0$ of the function $\varphi(\e, x, y,t)$ and we can study the possible
relations with the limits considered in the Conjectures 1, 2, 3 of \S 2.
Probably in this analysis, interesting phenomena may happen,
concerning the solutions (generalized or not) of the first order partial
differential equations.
\bigskip
\beginsection 4. Limits of Variational Problems Related to Hyperbolic
Equations.
In the research of suitable approximations of difficult and unstable
problems with easier and more stable ones, we can include the idea of
finding solutions of evolution problems as a limit of solutions of
minimizing problems. An interesting example where this approach was
successfully applied is in the paper [8]. A possible variant of
Ilmanen's result is given by the following conjecture.
\bigskip
{\bf Conjecture 1.} Let be given $\varphi,\psi\in C_0^\infty(\real^n)$, $k> 1$
integer. For every $\lambda>0$ let $w_\l=w_\l(x_1,\ldots,x_n,t)$ be
a minimizer of the functional
$$
F_\l(u)=\int_{\real^n\times [0,+\infty[} e^{-\l t}\left[ \Bigl|
{{\partial^2 u}\over{\partial t^2}}\Bigr|^2 + \l^2 |\nabla_x u|^2 +\l^2
u^{2k} \right] dx\,dt
\leqno(1)$$
in the class of all functions $u$ satisfying the initial conditions
$$
u(x,0)=\varphi(x),\qquad\qquad {{\partial u}\over{\partial
t}}(x,0)=\psi(x). $$
Then, there exists $\displaystyle\lim_{\l\to +\infty} w_\l (x,t)= w_0(x,t)$,
and satisfies the equation
$$
{{\partial^2 w_0}\over{\partial t^2}}=\Delta_xw_0-kw_0^{2k-1}.
\leqno(2)$$
\bigskip
{\bf Remark 1.} If this conjecture is false, then we can imagine
various modifications, for instance, asking whether it holds for a dense set
of initial data, replacing $C^\infty$ with the analytic functions
rapidly decreasing, considering only periodic functions, in particular
trigonometric polynomials, or even considering functionals $F_\l$
different from the one defined in (1) which share minimizers converging to
solutions of equation (2). Otherwise, if the conjecture can be
proved, then we can think to a large number of generalizations, starting
from a functional $F$ of the calculus of variations and associating
to it the functional
$$
F_\l(u)=\int_{\real^n\times [0,+\infty[} e^{-\l t} \Bigl|
{{\partial^2 u}\over{\partial t^2}}\Bigr|^2\, dxdt + \l^2 \int_0^{+\infty}
e^{-\l t} F\bigl(u(\cdot,t)\bigr)\,dt.
$$
\bigskip
\beginsection 5. Motion of Manifolds According to Euler Functions.
In order to define the motion of embedded manifolds of any dimension
and codimension, we establish some notations.
For every open set $A\subset\real^n$ we denote as usual by $C^\infty (A)$
the space of real smooth functions on $A$. If $E$ is not open, we say that
$f:E\to\real$ belongs to $C^\infty(E)$ if $f$ admits a smooth extension
to some open set $A$ which includes $E$.
Given open sets $A,\Omega$, we write $A\subset\subset\Omega$
when $A$ is relatively compact in $\Omega$. Finally, for every $E\subset\real^n
$
we define the function $\eta_E:\real^n\to [0,\infty)$ as follows:
$$
\eta_E(x)={1\over 2} [{\rm dist}(x,E)]^2=
\inf\left\{{{|x-\xi|^2}\over{2}};\ \xi\in E\right\} .
\leqno(1)
$$
Now we can give the following definition of $h$-dimensional manifolds of
class $C^\infty$ in an open set $\Omega$.
\medskip
{\bf Definition 1.}~ Let $\Omega$ be an open subset of $\real^n$,
and let $h$ be an integer with $0\le h\le n$.
We say that a set $E$ is an $h$-dimensional manifold
of class $C^\infty$ in $\Omega$, and we write $E\in V_hC^\infty(\Omega)$,
when the following three conditions hold:
\smallskip\line{(a)\hfil
$\Omega\cap E=\left\{ x\in\Omega :\eta_E(x)=0\right\}$; \hfil}
\smallskip\line{(b)\hfil
there exists an open set $A\supset\Omega\cap E$ such that
$\eta_E\in C^\infty(A)$; \hfil}
\smallskip\line{(c)\hfil
for every $x\in E\cap\Omega$ the Hessian matrix
$\nabla^2\eta_E(x)$ has rank $n-h$. \hfil}
\medskip
When $g$ is a smooth function defined on the class of all real polynomials
of
$n$-variables with degree $\le k$, and $E\in V_hC^\infty(\Omega)$, then for
every open set $A\subset\subset\Omega$ we define the functional
$$
F(E,A)=\int_{A\cap E}g\bigl(\eta_E(x),\nabla\eta_E(x),\ldots,
\nabla^k\eta_E(x)\bigr)\,d\hh^h(x),
\leqno(2)
$$
where $\hh^h$ is the $h$-dimensional Hausdorff measure. Since both $\eta_E$
and $\nabla\eta_E$ vanish on $E$, this functional is non-trivial when
$k\ge 2$.
\medskip
We give now the definition of Euler function associated with $E$ and $F$.
\medskip
{\bf Definition 2.}~ Let $\Omega$ be an open subset of $\real^n$,
$E\in V_hC^\infty (\Omega)$, $\psi\in (C^\infty(E))^n$, and let $F$
be given as in (2). We say that the vector function $\psi$ is the
{\it Euler function} of $F$ on $E$, and we write $\psi\in {\cal E}(F,E)$,
if for every open set $A\subset\subset\Omega$ and every
$\tau\in (C^\infty_0(A))^n$ there holds
$$
\lim_{\l\to 0}{1\over\l} \big( F(E_\l,A) -F(E,A) \big)=
\int_{E\cap A}\sum_{j=1}^n\tau_j(x)\psi_j(x)\,d\hh^h(x),
$$
where we have set $E_\l=\left\{x+\l\tau(x);\ x\in A\cap E\right\}$.
\medskip
Now we can give the definition of gradient flow according to the Euler
function.
\medskip
{\bf Definition 3.}~ Let be given $E\subset\real^n$, $T\in ]0,+\infty]$,
$u=u(x_1,\ldots,x_n,t)\in C^\infty(E\times [0,T[)$, and let $F$ be defined
as
in (2). We say that $u$ is a {\it $F$-gradient flow} starting from $E$,
and we write $u\in\MD(F,E,T)$, if the following conditions are fulfilled:
\smallskip\line{{\bf (A)}\hfil
$u(x,0)=x$ for every $x\in E$; \hfil }
\smallskip
{\bf (B)} for every $(x,t)\in E\times[0,T[$
there exists an open set $A(x,t)\subset\real^n$
such that $x\in A(x,t)$, and if we set
$L(x,t)=\{u(x,t);\ x\in E\cap A(x,t)\}$,
$$
{\partial u\over\partial t}(x,t)={\cal E}\bigl(F,L(x,t)\bigr)(u(x,t))
\qquad\forall\,(x,t)\in E \times [0,T[ \, .
$$
\medskip
{\bf Remark 1.}~ When $g$ is a positive constant, $F(E,A)$ is proportional
to
$\hh^h(E\cap A)$, and then the previous definition gives the usual
motion by mean curvature (see for instance [1]). In general we do not
expect
that the gradient flow exists for every $F$ and every
initial data $E$, but we can still try to approximate the functional $F$
as we did in \S 1. More precisely, if $F$ is defined by (2), we can set for
every $\e>0$
$$
F_\e (E,A)=F(E,A)+\e^2\int_{E\cap A}|\nabla^{k+1}\eta_E|^2\,d\hh^h \ .
\leqno(3)
$$
Then we have the following
\medskip
{\bf Conjecture 1.}~ If $E$ is a compact set in $V_hC^\infty(\real^n)$, and
$F$ is given as in (2), for every $\e>0$ there exist $T_\e>0$ and $u_\e\in
\big( C^\infty(E\times [0,T_\e[) \big)^n$ such that
$u_\e\in \MD(F_\e,E,T_\e)$.
\medskip
When $g$ is a positive constant, we think that $T_\e$ does not depends on
$\e$, as stated in the following
\medskip
{\bf Conjecture 2.}~ In the hypothesis of Conjecture 1, if we assume
moreover
that $k>h$ and $g$ is a positive constant, then $T_\e\equiv +\infty$ and
then
$u_\e\in \big( C^\infty(E\times [0,+\infty[)\big)^n$.
\medskip
{\bf Remark 2.}~
Conjecture 2 basically gives a standard procedure to approximate the
curvature flow. Regarding the limiting behaviour (as $\e\to 0$) of these
approximating flows, the problems should be similar to those described in
the previous sections. To begin, we suggest the following result, which is
more or less a uniqueness result for the curvature flow defined by the
``viscosity method''.
\medskip
{\bf Conjecture 3.}~ In the hypothesis of Conjecture 2, for
${\cal H}^{h+1}$-almost every $(x,t)\in E\times [0,+\infty)$
there exists the limit $\lim_{\varepsilon \to 0} u_\varepsilon (x,t)$.
\medskip
{\bf Remark 3.}~ Conjecture 3 can be weakened in many different ways, e.g.
by assuming that $E$ belongs to $V_h^\omega (\real^n)$ (and not only
$V_h^\infty(\real^n)$), that is, assuming that the function $\eta_E$
in Definition 1 is analytic on the open set $A$. Also we could try to
verify
Conjecture 3 for all $E$ in a some dense subset of the class of
all compact elements of $V_hC^\omega(\real^n) $, endowed with a suitable
topology.
If Conjecture 3 is true, examples of non-constant $g$ should be
examined. From a physical viewpoint, it would be interesting to
consider equations of the form
$$
{\partial^2 u \over \partial t^2} = {\cal E}\bigl(F,L(x,t)\bigr)(u(x,t)) .
$$
\bigskip
{\bf Acknowledgements.}~This note is the result of several conversations
I had with many friends and colleagues. In particular I want to thank Luigi
Ambrosio, Giovanni Bellettini, Giuseppe Bertin, Ivar Ekeland, Antonio Leaci,
Carlo Mantegazza, Antonio Marino, Stefano Mortola, Diego Pallara, Sergio
Spagnolo and Piero Villaggio.
\bigskip
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\bye