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Analysis of Elliptic Differential Equations 

SS 2016    11.04.2016 - 22.07.2016


Do  21.07.2016   12:00-14:00   T2-208


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 Lµ-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
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).


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  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, 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.