Contents/conteúdo

Mathematics Department Técnico Técnico

Colloquium  RSS

05/07/2001, 17:00 — 18:00 — Amphitheatre Pa1, Mathematics Building
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.

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The Mathematics Colloquium is a series of monthly talks organized by the Department of Mathematics of IST, aiming to be a forum for the presentation of mathematical ideas or ideas about Mathematics. The Colloquium welcomes the participation of faculty, researchers and undergraduate or graduate students, of IST or other institutions, and is seen as an opportunity of bringing together and fostering the building up of ideas in an informal atmosphere.

Organizers: Conceição Amado, Lina Oliveira e Maria João Borges.