# MATH 521: Topics in Real Analysis, Fall 2002

### Description:

The general 2nd order linear elliptic partial differential equation (on a bounded domain in  Rn) has the form

Sij aij(x) D x i D xj u  +  S k bk(x)Dx k u  +  c(x)u   =   f(x) ,

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

S ij Dxi(aij(x) Dx j u)  +  Sk bk(x)Dx 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 Dxi( 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
Prerequisites for the course include some knowledge of basic analysis as in Math 425
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.

### Meets:  (CHANGE!)

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 1992.
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 ( ps file , pdf file )
Holder continuity via De Giorgi: Sec. 4.3 of Han-Lin
Nonlinear Equations:  Sec. 4.5 of Han-Lin..

### Corrections:

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

November 9 Lecture. See the remarks on sub-supersolutions. ( ps file , pdf file )

### Instructor:

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

### Homepage:

http://math.rice.edu/~hardt/521F02/