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Elliptic Partial Differential Equations 

WS 2023/24    09.10.2023 - 02.02.2024

Lectures

Lecture notes

Exercises

Deadline for submission of homeworks is always Friday. Joint submission of homework by groups of students is not allowed.
At least 50% of points for homework is required for admission to the exam. 
This percentage is calculated as A/M, where A is the number the earned points and M is the maximum possible number of points. 
The exercises marked with *  are taken into account in A, but not in M.

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

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. 
Weak Dirichlet problem for operators with lower order terms (uniqueness and existence). 
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.
C¹ boundary. 
Solvability of the classical Dirichlet problem.

5.* Harnack inequality for equations in divergence form 
Derivation of the Harnack inequality (Theorem of Moser) 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

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