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

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.