Perlen-Kolloquium: Wolfgang L. Wendland (Univerität Stuttgart)

On the Gauss problem with Riesz potential

This is a lecture on joint work with H. Harbrecht (U. Basel, Switzerland), G. Of (TU. Graz, Austria) and N. Zorii (Nat. Academy Sci. Kiev, Ukraine).

In Rⁿ , 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, which is considered for finitely many compact, mutually disjoint, boundaryless (n−1)–dimensional Lipschitz manifolds Γ<sub>l, l ∈ L, each Γ_l being charged with Borel measures with the sign α_l = ±1 prescribed.

For Newton potentials, i.e. the special case α = 2, this problem goes back to C. F. Gauss who used it as the model for electrostatic fields. Nowadays it is of interest for the storage of charges as produced by solar electricity modules. The more general Riesz kernels have also applications for finding numerical integration formulae on manifolds.

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. A corresponding numerical method is based on the Galerkin–Bubnov discretization with piecewise constant boundary elements. For n = 3 and α = 2, multipole approximation and in the case 1 < α < 3 = n wavelet matrix compression is applied to sparsify the system matrix. For a subclass of these problems, a dual formulation results in linear boundary integral equations although the original problem is nonlinear. Numerical results are presented to illustrate the approach. 

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