PDE seminar

Summer 2008

Instructor: Dr. Dung Le

COURSE: Regularity theory for nonlinear PDEs

PREREQUISITES: Basic knowledge on Functional Analysis and Sobolev spaces.

TIME & PLACE: 3 weeks in July 2008-Department of Mathematics. Monday through Friday from 9-10am and 10-11:30am of informal discussion. Exact time and location will be announced later.

TEXTS:

Linear and Quasilinear equations of Parabolic type by O.A. Ladyzhenskaya, V.A. Solinikov and N.N. Ural'ceva

Elliptic Partial Differential Equations of Second Order by D. Gilbarg and N.S. Trudinger

Direct Methods in the Calculus of Variations by E. Giusti

Journal articles (will be provided)

Contact email: dle@sphere.math.utsa.edu

PURPOSE: Nonlinear PDEs occur in many real life applications and their analysis requires ad-hoc methods. However, before one can carry out a complete study of these PDEs, one has to face up to the fundamental regularity problem. Assuming a solution already exists, how good is it? What are possible a-priori estimates? Answers to these questions may lead to the existence of solutions and our understanding of their behaviors (multi-bumps, collapsing, blow up, quenching, dynamics...). Up to date, the methods and ideas coined by Moser and DiGiorgi prove to be the corner stones. The seminar attempts to "softly" guide its audience through the technicalities of the above texts describing the classical methods and their generalizations applying to scalar equations. Open problems and new techniques for systems of equations will be presented. In particular, a brief description of open problems proposed by the instructor in his NSF supported proposals will also be discussed.

REQUIREMENTS: Lectures will be conducted in English. Everyone is expected to come to class well prepared (by reading the material ahead of time and mastering the material that has been covered previously). Students are more than welcome to ask questions and discuss problems. Don?t be shy!