# Riesz minimal energy problems on Ck−1,1-manifolds

Harbrecht, Helmut and Wendland, Wolfgang L. and Zorii, Natalia. (2014) Riesz minimal energy problems on Ck−1,1-manifolds. Mathematische Nachrichten, Vol. 287, H. 1. pp. 48-69.

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Official URL: http://edoc.unibas.ch/dok/A6212117

In $mathbb R^n$, $ngeqslant 2$, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel $|{bf x}-{bf y}|^{alpha-n}$, where $1>alpha(alpha-1)/2$, each $Gamma_ell$ being charged with Borel measures with the sign $alpha_ell:=pm1$ 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 $H^{-{varepsilon}/{2}}(Gamma)$, where $varepsilon:=alpha-1$ and $Gamma:=bigcup_{ellin L}Gamma_ell$. 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 sparsify the system matrix. Numerical results are presented to illustrate the approach.