Asymptotics of Robin eigenvalues on sharp infinite cones
Konstantin Pankrashkin
Carl von Ossietzky Universität Oldenburg, GermanyMarco Vogel
Carl von Ossietzky Universität Oldenburg, Germany
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Abstract
Let be a bounded domain with Lipschitz boundary. For and , consider the infinite cone
and the operator acting as the Laplacian on with the Robin boundary condition at , where is the outward normal derivative and . We look at the dependence of the eigenvalues of on the parameter : this problem was previously addressed for only (in that case, the only admissible are finite intervals). In the present work, we consider arbitrary dimensions and arbitrarily shaped “cross-sections” and look at the spectral asymptotics as becomes small, i.e., as the cone becomes “sharp” and collapses to a half-line. It turns out that the main term of the asymptotics of individual eigenvalues is determined by the single geometric quantity
More precisely, for any fixed and , the -th eigenvalue of exists for all sufficiently small and satisfies
The paper also covers some aspects of Sobolev spaces on infinite cones, which can be of independent interest.
Cite this article
Konstantin Pankrashkin, Marco Vogel, Asymptotics of Robin eigenvalues on sharp infinite cones. J. Spectr. Theory 13 (2023), no. 1, pp. 201–241
DOI 10.4171/JST/452