Isoperimetric inequalities for the logarithmic potential operator

Abstract

In this paper we prove that the disc is a maximiser of the Schatten $p$-norm of the logarithmic potential operator among all domains of a given measure in ${\mathbb{R}}^2$, for all even integers $2\le p<\infty$. We also show that the equilateral triangle has the largest Schatten $p$-norm among all triangles of a given area. For the logarithmic potential operator on bounded open or triangular domains, we also obtain analogies of the Rayleigh-Faber-Krahn or P{ó}lya inequalities, respectively. The logarithmic potential operator can be related to a nonlocal boundary value problem for the Laplacian, so we obtain isoperimetric inequalities for its eigenvalues as well.

revised version with corrected formulations and arguments; to replace the previous version

References in corpus (2)

• Schatten classes on compact manifolds: Kernel conditions
• Eigenvalues and Eigenfunctions of Double Layer Potentials

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