arXiv subjects

Lukas Bundrock

Publications and source records attributed to Lukas Bundrock.

3 recordsLinked to original sources

arXiv2026

The exterior Steklov problem for Euclidean domains

We investigate the Steklov eigenvalue problem in an exterior Euclidean domain. First, we present several formulations of this problem and establish the equivalences between them. Next, we examine various properties of the exterior Steklov eigenvalues and eigenfunctions. One of our main findings is an Escobar-type lower bound for the first exterior Steklov eigenvalue on convex domains in dimensions three and higher. This bound is expressed in terms of the principal curvatures of the boundary and is sharp, with equality attained for a ball. Moreover, it implies the existence of a sequence of convex domains with fixed volume and the first exterior Steklov eigenvalues tending to infinity. This contrasts with the interior case, as well as with the two-dimensional exterior case, for which we show that an analogue of the Weinstock isoperimetric inequality holds.

math.SP↗

arXiv2025

Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains

We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $α$ is positive, and the superconducting generation regime ($α<0$), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all $p$ and all small real $α$, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as $α\to 0$ for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions.

math.AP↗

arXiv2024

Geometric Optimization of the First Robin Eigenvalue in Exterior Domains

This paper addresses the geometric optimization problem of the first Robin eigenvalue in exterior domains, specifically the lowest point of the spectrum of the Laplace operator under Robin boundary conditions in the complement of a bounded domain. In contrast to the Laplace operator on bounded domains, the spectrum of this operator is not purely discrete. The discrete nature of the first eigenvalue depends on the parameter of the Robin boundary condition. In two dimensions, D. Krejcirik and V. Lotoreichik show that the ball maximizes the first Robin eigenvalue among all smooth, bounded, simply connected sets with given perimeter or given area, provided the eigenvalue is discrete. We extend these findings to higher dimensions. The discrete spectrum of the Laplace operator under Robin boundary conditions can be characterized through the Steklov eigenvalue problem in exterior domains, a topic studied by G. Auchmuty and Q. Han. Assuming that the lowest point of the spectrum is a discrete eigenvalue, we show that the ball is a local maximizer among nearly spherical domains with prescribed measure. However, in general, the ball does not emerge as the global maximizer for the first Robin eigenvalue under either prescribed measure or prescribed perimeter.

math.AP↗