Back to the home page of A.Grigor'yan
SS 2016 11.04.2016 - 22.07.2016
Do 21.07.2016 12:00-14:00 T2-208
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
Mean value inequality for subsolutions.
Weak Harnack inequality for positive supersolutions.
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.
Hölder continuity up to a flat boundary.
Boundary as a graph.
Solvability of the classical Dirichlet problem.
5. Harnack inequality
Derivation of the Harnack inequality from the weak Harnack inequality.
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).