On Riesz minimal energy problems

In Rn, n⩾2, we study the constructive and numerical solution of minimizing the energy relative to the Riesz kernel , where 1>α>n, for the Gauss variational problem, considered for finitely many compact, mutually disjoint, boundaryless (n−1)-dimensional C∞-manifolds Γℓ, ℓ∈L, each Γℓ being charged with Borel measures with the sign αℓ≔±1 prescribed. We show that the Gauss variational problem over an affine cone of Borel measures can alternatively be formulated as a minimum problem over an affine cone of surface distributions belonging to the Sobolev–Slobodetski space H−ε/2(Γ), where ε≔α−1 and . This allows the application of simple layer boundary integral operators on Γ and, hence, a penalty approximation. 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. To the discretized problem, a gradient-projection method is applied. Numerical results are presented to illustrate the approach.