Description:

S_{ij} a_{ij}(x) D_{
x}_{ i} D_{
xj} u +
S_{ k} b_{k}(x)D_{x}_{
k} u + c(x)u =
f(x) ,

where the given functions a_{ij}(x) , b_{
k}(x) , c(x) , f(x) are assumed to be measurable (and usually
bounded) and the ellipticity condition is that a_{
ij}(x) is a positive definite symmetric matrix. A nonnegative
solution u(x) may be interpretated as the equilibrium concentration
of a fluid with the first sum describing the (anisotropic, heterogeneous)
diffusion, the second sum the transport, the third term giving the creation
or depletion (due to reaction), and the right hand side the applied force.
As with O.D.E. theory, there are basic results on existence, uniqueness,
regularity, and stability of solutions (with given boundary values).
One may often get much information by specializing to the simplest classical
example of Poisson's equation Du =
f , or one may generalize many results to *quasi-linear* equations
in which the lower order terms b_{k}(x) , c(x) are replaced
by a general possibly nonlinear function b(x,u,Du) (satisfying
some growth condition). A particular important type of elliptic equation
is a divergence structure equation

S_{
ij} D_{xi}(a_{ij}(x) D_{x}_{
j} u) +
S_{k} b_{k}(x)D_{x}_{
k} u + c(x)u =
0

which describes local minimizers of a suitable functional. For example, a solution of the minimal surface equation

S_{
i} D_{xi}(
D_{ xi} u / (1+|Du|^{
2} )^{1/2})
= 0

minimizes the area integrand, and its graph is a minimal surface.

Basic to the regularity theory (and also much existence,
uniqueness, and stability theory) are "a priori" bounds on solutions and
their derivatives in various norms. Some very strong results are
in papers by Di Giorgi, Nash, and Moser from the late 50's.
There were also important new arguments and simplifiications by Krylov-Safanov
(80's) and Caffarelli (90's) which apply to other classes of (fully non-linear)
equations. Most of the material for this course can be found in
the important book "Elliptic Partial Differential Equations of Second
Order" by Gilbarg and Trudinger and papers by Caffarelli.
Fortunately many of the key constructions are in the Amer. Math. Soc. (CIMS)
notes "Elliptic Partial Differential Equations" by Han and Lin. We will
follow these notes for much of the course, and they are available at the
bookstore or online from the AMS.

A brief outline includes the following:

- Harmonic function estimates
- Maximum and comparison principles
- Existence of weak solutions
- Regularity of weak solutions with smooth coefficients
- Regularity of viscosity solutions with bounded measurable coefficients
- Some Harnack inequalities

and some previous exposure to harmonic functions. Although most of the proofs will be elementary maximum principle type arguments, we will also quickly cover the necessary Sobolev theory and functional analysis, so some familiarity with these topics would also be useful.

Sec.6.1 of F. Morgan,

Harmonic Functions and the Mean Value Property: Sec. 1.2 of Han-Lin.

Green's identities, fundamental solutions and Green's function: Sec. 1.3 of Han-Lin.

Compactness of uniformly bounded harmonic functions.

Dirichlet problem for harmonic functions:

See the handout.

The Sobolev and Poincare inequalities:

Sec. 4.5 of L.C.Evans and R.Gariepy,

Energy and decay estimates, Cacciopolli Inequality for A-harmonic functions: Sec. 1.5 of Han-Lin.

Growth of local integrals, Campanato and Morrey criteria for Holder continuity: Sec. 3.2 of Han-Lin.

Holder continuity via harmonic approximation: Sec. 3.3 of Han-Lin.

Holder continuity of the gradient via harmonic approximation: Sec. 3.4 of Han-Lin.

Local boundedness of solutions: Sec. 4.2 (Method 1) of Han-Lin.

See the handout.

Holder continuity via De Giorgi: Sec. 4.3 of Han-Lin

Nonlinear Equations: Sec. 4.5 of Han-Lin..

In the third example, one should assume the existence of an exterior (not interior) touching ball. The

renormalized fundamental solution G(a,x)-G (a,b) then gives the desired barrier.

- Robert Hardt Office: Herman Brown 430; Office hours: 11-12 MWF (and others by appt.),
- Email: hardt@rice.edu, Telephone: ext 3280
- http://math.rice.edu/~hardt/521F02/

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