Colóquio 

05/07/2001, 17:00 — 18:00 — Anfiteatro Pa1, Pavilhão de Matemática
Roland Duduchava, Academy of Sciences, Tbilisi

Boundary Value Problems and the Green Formula

The Green formula plays an outstanding role in boundary value problems (BVPs) for linear partial differential equations (PDEs) in n-dimensional domains. With their help it is possible to establish existence and uniqueness of a solution, obtain a representation of this solution via layer potentials and to derive an equivalent boundary integral equation (BIE) on the boundary of the domain.

The simplest versions of the Green formula are readily obtained by “partial integration” (the Gauss formula). We describe how to get all possible versions of the Green formula by pure algebraic operations for BVPs when the basic equation in the domain is an arbitrary PDE with \(N \times N\) matrix coefficients and the boundary conditions are prescribed by quasi-normal partial differential operators with vector \(1 \times N\) coefficients.

If the basic system possesses a fundamental solution, a representation formula for the solution is derived. Exact boundedness properties of the relevant layer potentials, mapping function spaces on the boundary (Bessel potential, Besov, Zygmund spaces) into appropriate weighted function spaces in the domain, are established.

Some related topics, such as the Green formula for a BVP on a smooth open surface in the \( n\)-space, Calderon projections, Plemelji formulae etc. will be discussed as well.