10. Čechovská přednáška: prof. Dr.-Ing. Wolfgang L. Wendland, Dr. h. c. (Universität Stuttgart) - "On Riesz minimal energy problems on C^(k−1,1)-manifolds"
Jde o desátou prednášku konanou v rámci cyklu reprezentačních přednášek organizovaných na počest prof. Eduarda Čecha, jednoho z nejvýznamnějších českých matematiků novodobé historie a zakladatele Matematického ústavu AV ČR.
Abstract
The classical Gauss problem of minimizing the potential energy of a given conductor in Rn, n >= 2, is one of the basic problems modeling equilibrium states in electrostatics and more general physical phenomena. Here we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel |x − y|^(alpha−n), where 1 (alpha − 1)⁄2, each Gamma_i charged with Borel measures with the sign alpha_i := ±1 prescribed.We show that the Gauss variational problem over a convex set of Borel measures can alternatively be formulated as a minimum problem over the corresponding set of surface distributions belonging to the Sobolev-Slobodetski space H^(−epsilon⁄2)(Gamma), where epsilon := alpha − 1 and Gamma is the union of {Gamma_l, l in L}. An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Gamma. A corresponding numerical method is based on the Galerkin-Bubnov discretization with piecewise constant boundary elements. Wavelet matrix compression is applied to sparcify the system matrix. Numerical results are presented to illustrate the approach. Finally we present a concept how to extend the Riesz energy problem to the strongly singular case −1