Viscosity Solutions of Nonlinear Partial Differential Equations (15 credits)

Teacher:

Start of the course:

First week in October 1, 2012 15:15-17:00, in room 3733 of the Department of Mathematics.

Date and Times: will be announced at first meeting.

Goal

To learn the basics of partial differential equations, both linear and nonlinear theory. The course will treat PDE of elliptic type, and to some extent applications, starting from the linear case and ending up with fully nonlinear equations.

Prerequisites

General knowledge in analysis. Some elementary course in PDE.

As a general background in PDE (e.g. to put the contents of the course in a broader context) we recommend

J. Rauch: Partial Differential Equations, Springer 1991.

However no previous knowledge of PDE will be assumed, only basic analysis (including some point set topology and some integration theory and functional analysis).

Examination:

Part I: Homeworks

Part II: Homeworks, and a presentation/oral exam.

Part I: Linear equations

This part of the course will treat the classical theory of linear elliptic PDE with variable and in general nonsmooth coefficients.

Literature:

D. Gilbarg and N. Trudinger: Elliptic Partial Differential equations of Second Order, 2nd ed., Springer 1983. Chapter 1-7.

Topics:

Laplace's equation (Ch. 2 in GT).

Classical maximum principle, gradient estimates, Harnack inequality (Ch. 3).

Poisson's equation with H\"older estimates (Ch.4).

Topics in functional analysis and Sobolev spaces (Ch. 5 and 7).

Schauder estimates and classical solutions (Ch. 6).

Part II: Nonlinear equations

This part treats modern theory of PDE, with geometric methods.

Literature:

L.A. Caffarelli and X. Cabre': Fully Nonlinear Elliptic Equations.

Topics:

Tangent paraboloids and second order differentiability, Viscosity solutions of elliptic equations; Examples,

Alexandroff estimate and maximum principle, Harnack Inequality, Uniqueness of solutions, Concave equations, Dirichlet problem.