Riesz minimal energy problems on $C^{k−1,1}$-manifolds

In $\mathbb{R}^n, n\ge 2$, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel $|x − y|^{α−n}$, where 1 < α < n, for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless (n−1)-dimensional $C^{k−1,1}$-manifolds $Γ_l$, l ∈ L, where k > (α−1)/2, each $Γ_l$ being charged with Borel measures with the sign $α_l := ±1$ prescribed. We show that the Gauss variational problem over a cone of Borel measures can alternatively be formulated as a minimum problem over the corresponding cone of surface distributions belonging to the Sobolev–Slobodetski space $H^{−ε/2}(Γ)$, where ε := α−1 and $Γ := \bigcup_{l∈L} Γ_l$. An equivalent formulation leads in the case of two manifolds to a nonlinear system of boundary integral equations involving simple layer potential operators on Γ. 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.