MATH 521: Topics in Real Analysis, Fall 2002
Elliptic Partial Differential Equations
The general 2nd order linear elliptic partial
differential equation (on a bounded domain in Rn) has
Sij aij(x) D
x i D
xj u +
S k bk(x)Dx
k u + c(x)u =
where the given functions aij(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 bk(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
ij Dxi(aij(x) Dx
j u) +
k u + c(x)u =
which describes local minimizers of a suitable functional.
For example, a solution of the minimal surface equation
D xi u / (1+|Du|
minimizes the area integrand, and its graph is a minimal
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:
Prerequisites for the course include some knowledge of basic analysis
as in Math 425
- 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.
Tues.Thurs 1-2:15 PM in Herman Brown 438, Math. Commons Room
Brief Syllabus and References:
Introduction, Minimal Surface Equation and Area-Minimality:
Sec.6.1 of F. Morgan, Geometric
Measure Theory, Academic Press, 1988.
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. Perron
Method for the Dirichlet Problem (ps file
, pdf file
The Sobolev and Poincare inequalities:
Sec. 4.5 of L.C.Evans
and R.Gariepy, Measure Theory and Fine Properties of Functions, CRC
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. L
p Properties, Distribution Derivatives, and H1 (
, pdf file
Holder continuity via De Giorgi: Sec. 4.3 of Han-Lin
Nonlinear Equations: Sec. 4.5 of Han-Lin..
September 16 Lecture. In the definition of a barrier, only
continuity (and not smoothness) is required.
In the third example, one should assume the existence of an exterior (not
interior) touching ball. The
renormalized fundamental solution
(a,b) then gives the desired barrier.
November 9 Lecture. See the remarks on sub-supersolutions. (
, pdf file
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