Skip to content
DraconiticResearch
Search Sources API
201,683 studies31 sourcesUpdated 22 min ago
Peer-reviewedCC BYJournal of Spectral Theory

The exterior Steklov problem for Euclidean domains

  • Lukas Bundrock1
  • Alexandre Girouard2
  • Denis S. Grebenkov3
  • Michael Levitin4
  • Иосиф Викторович Полтерович5
5 affiliations
  1. 1The University of Alabama, Tuscaloosa, USA
  2. 2Université Laval, Québec, Canada
  3. 3CNRS – École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France
  4. 4University of Reading, UK
  5. 5Université de Montréal, Canada
Open at sourceDOICite

Abstract

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.

Related

  1. Discreteness of the Steklov Spectrum for Exterior Non-Compact Free-Boundary Minimal Surfaces with Regular Ends of Finite Total Curvature
    Igor Kozyrev
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  2. A sharp curvature lower bound for the first exterior p-harmonic Steklov eigenvalue
    Yating Niu, Tao Wang
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. On the spectrum of the biharmonic Steklov operator with Neumann boundary conditions acting on forms of the unit Euclidean ball
    Rodolphe Abou Assali
    HALPreprintCC BY 4.0Full textAnalysis & PDEsGeometry & topologyHALarXiv
  4. Sharp bounds for higher mixed Steklov-Robin eigenvalues on domains with holes
    Sagar Basak
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  5. Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains
    Hermann Douanla, Hubert Nnang, Joel Fotso Tachago +1
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@article{bundrock2026exterior,
  title = {{The exterior Steklov problem for Euclidean domains}},
  author = {Lukas Bundrock and Alexandre Girouard and Denis S. Grebenkov and Michael Levitin and Иосиф Викторович Полтерович},
  journal = {Journal of Spectral Theory},
  year = {2026},
  month = oct,
  doi = {10.4171/jst/632},
  url = {https://doi.org/10.4171/jst/632},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The exterior Steklov problem for Euclidean domains
AU  - Lukas Bundrock
AU  - Alexandre Girouard
AU  - Denis S. Grebenkov
AU  - Michael Levitin
AU  - Иосиф Викторович Полтерович
PY  - 2026
DA  - 2026/10/01
JO  - Journal of Spectral Theory
DO  - 10.4171/jst/632
UR  - https://doi.org/10.4171/jst/632
AB  - 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.
SN  - 1664-039X
SN  - 1664-0403
ER  -