Back to the home page
of A.Grigor'yan

# Analysis of Elliptic Differential Equations

SS 2016 11.04.2016 -
22.07.2016

## Klausur

Do
21.07.2016 12:00-14:00 T2-208

## Lectures

Di 16-18 C01-142 Do 12-14 T2-208
Lecture notes

## Contents of the course

**0. Introduction**

Elliptic equations of 2nd order in divergence and non-divergence
forms.

The divergence theorem. Physical origin of the equations in divergence form.

Generators of diffusion processes. Origin of the equations in non-divergence
form.

**1. Weak Dirichlet problem for divergence form**

Distributions and Sobolev spaces.

Weak Dirichlet problem for operators
with measurable coefficients.

Solvability of the weak Dirichlet problem.

Equation with lower order terms. Sobolev inequality.

Theorem of Lax-Milgram.

The Fredholm alternative.

Estimates of *L*^{µ}-norm
of solutions.

**2. Higher order derivatives of weak solutions**

Existence of 2nd order weak derivatives.

Existence of higher order weak derivatives.

Equations with lower order terms.

Sobolev embedding theorem and existence of classical derivatives.

Strong solutions for equations in non-divergence form.

**
3. Hölder continuity for equations in divergence form
**

Faber-Krahn inequality.

Mean value
inequality for subsolutions.

Poincare inequality.

Weak Harnack inequality for positive supersolutions.

Oscillation inequality.

Hölder continuity of weak solutions (Theorem of de Giorgi).

Weak solutions of inhomogeneous equations.

Application to quasi-linear equations. A fixed point theorem of Leray-Schauder.

**4.
Boundary behavior**

Hölder continuity up to a flat boundary.

Boundary as a graph.

*C*^{1}
boundary.

Solvability of the classical Dirichlet problem.

**5. Harnack inequality**

Derivation of the Harnack inequality from the weak Harnack inequality.

**6*.
****
Hölder estimates for equations in non-divergence form**
**
**

Strong and classical solutions.

Maximum principle of Alexandrov-Pucci.

Weak Harnack inequality.

Oscillation lemma and Hölder continuity (Theorem of Krylov-Safonov).

## Literature

- Evans L.C., Partial differential equations, AMS, 1997, 2008, 2010
- Gilbarg D., Trudinger N., Elliptic partial differential equations of
second order, Springer, 1983
- Grigor'yan A., Heat kernel and Analysis on manifolds, AMS, 2009.
- Jost J., Partial differential equations, Graduate Texts in Mathematics 214,
Springer, 2013.
- Landis E.M., The second order equations of elliptic and parabolic type,
AMS, 1998.