On properties of operators of classical potential theory. (Russian) Zbl 0699.31010

Dyakin, V. V.; Raevskij, V. Ya.

Mat. Zametki 45, No. 2, 138-140 (1989).

The authors consider the operator in \(L_ 2(S)\) \[ (B_{\mu})(x)=\int_{S}\mu (y)(\partial /\partial n_ x)(1/| x- y|)dS_ y,\quad x\in S, \] which is the direct value on S of the normal derivative of the single layer potential, the adjoint one \(B^*\) representing the direct value on S of the double layer potential. Here S is a Lyapunov surface confining one or more finite simply connected domains. A completely continuous operator U in complex Hilbert space is quasi-Hermitian (symmetrizable) by definition if there exists a positive selfadjoint operator T such that \(TU=U^*T\); hence U is a selfadjoint operator with respect to the scalar product (T.,.). The main result is quasi-Hermiticity of B, where T may be choosen in the form \[ (T\sigma)(x)=\int_{S}(\sigma (y)/| x-y|)dS_ y,\quad x\in S, \] which is the value on S of the single layer potential. The main consequences of this property are quasi-Hermiticity of \(B^*\), completeness in \(L_ 2(S)\) of eigenfunctions of B and \(B^*\), orthogonality of B’s eigenfunctions with respect to the scalar product (T.,.), and the fact that the eigensubspace of \(B^*\) is an image of B’s eigensubspace under the map T.
In addition, validity of these properties is proved for the case of multiply-connected domains.

Reviewer: V.Ya.Raevskij

MSC:

31B15	Potentials and capacities, extremal length and related notions in higher dimensions
47A70	(Generalized) eigenfunction expansions of linear operators; rigged Hilbert spaces
47B15	Hermitian and normal operators (spectral measures, functional calculus, etc.)

Keywords:

single layer potential; double layer potential