School on Elliptic PDE

Oct 15 2012 - 11:30
Oct 25 2012 - 13:30
Venue: 

Institute of Mathematical Sciences (ICMAT), Campus de Cantoblanco, Madrid (Spain)

Short description of the event: 

The School has two parts:
1. “A brief introduction to the method of boundary layer potentials”
Marius Mitrea, University of Missouri-Columbia, USA. October 15-18, 2012
2. “Topics in Geometric Analysis and Applications to PDEs”.
Dorina Mitrea, University of Missouri-Columbia, USA. October 22-25, 2012

The School has two parts:

1. “A brief introduction to the method of boundary layer potentials”
Marius Mitrea, University of Missouri-Columbia, USA. October 15-18, 2012

One of the most effective strategies for solving boundary value problems for large classes of partial differential equations is the method of boundary layer potentials. Its essence consists of reducing the entire original problem in a given domain D to solving an integral equation formulated entirely on the boundary of D. This course is designed as a rapid introduction to this type of technology, with special emphasis on the role played by the tools of modern Harmonic Analysis, and in particular the theory of Singular Integral Operators of Calderón-Zygmund type, to this subject.
Topics:

How to implement the method of boundary layer potentials for the most basic PDE of mathematical physics (Laplacian, Lame, Stokes, general systems).
A quick survey of Calderón-Zygmund theory (with applications to jump relations and non-tangential maximal function estimates).
Tools for solving integral equations involving singular integral operators and applications to well-posedness (a little bit of Fredholm theory and geometric measure theory).

2. “Topics in Geometric Analysis and Applications to PDEs”.
Dorina Mitrea, University of Missouri-Columbia, USA. October 22-25, 2012

Variation of area formulas for hypersurfaces in a Riemannian manifold are basic results in Riemannian geometry with fundamental implications in the theory of minimal surfaces. The goal of this course is to develop self-contained proofs of these results, corresponding to hypersurfaces in ℝn, which have the attractive feature that they completely avoid the heavy differential geometrical jargon which typically accompanies much of the work on this topic. This requires that we pedantically develop a number of alternative tools

Topics:

Hypersurfaces of class Ck in ℝn (parametrizations, oriented surfaces, descriptions of unit normals in terms of parametrizations, integration, surface to surface change of variables).
Domains of class Ck in ℝn.
Domains satisfying a uniform exterior/interior ball condition; the nearest point to the boundary function.
A distinguished extension of the outward unit normal to a domain of class C2.
Mean curvature and Gauss curvature.
Minimal surfaces, the first and second variation of area formulas.
Applications to PDEs.