On Riesz minimal energy problems

In R n , n ⩾ 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 ∞ -manifolds Γ ℓ , ℓ ∈ L , each Γ ℓ being charged with Borel measures with the sign α ℓ ≔ ± 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 Γ ≔ ⋃ ℓ ∈ L Γ ℓ . 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.