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SUMMARY:Exterior Steklov problems: numerical tools and spectral properties
  - Denis Grebenkov (École Polytechnique)
DTSTART:20260416T143000Z
DTEND:20260416T153000Z
UID:TALK244042@talks.cam.ac.uk
DESCRIPTION:In this talk\, we present an overview of recent progress in nu
 merical methods&nbsp\;for studying spectral properties of generalized Stek
 lov problems associated&nbsp\;with the Laplace and Helmholtz equations in 
 the exterior of a compact set&nbsp\;with Lipschitz boundary [1].\nThe firs
 t part is devoted to the Steklov problem in the exterior of an infinitely&
 nbsp\;thin compact set embedded in three-dimensional space [2]. Such probl
 ems arise\,&nbsp\;in particular\, in sloshing phenomena in hydrodynamics a
 nd in the small-target&nbsp\;asymptotic analysis of various partial differ
 ential equations with mixed Robin\,&nbsp\;Neumann\, and Dirichlet boundary
  conditions [3]. We discuss the geometric structure&nbsp\;of low-energy ei
 genfunctions\, as well as two-sided bounds on the principal&nbsp\;eigenval
 ue.\nIn the second part\, we consider a generalized Steklov problem for th
 e Helmholtz&nbsp\;equation with real\, purely imaginary\, and complex para
 meters. Its reformulation&nbsp\;in terms of boundary layer potentials enab
 les efficient numerical computation&nbsp\;for a wide class of exterior pla
 nar domains with piecewise smooth boundaries.&nbsp\;The resulting numerica
 l evidence leads to several conjectures relevant to spectral&nbsp\;geometr
 y\, including corrections to Weyl's law and the asymptotic behavior of&nbs
 p\;eigenfunctions [4].\nReferences:\n[1] L. Bundrock\, A. Girouard\, D. S.
  Grebenkov\, M. Levitin\, and I. Polterovich\,&nbsp\;The exterior Steklov 
 problem for Euclidean domains (submitted\; preprint 2511.09490v2).\n[2] D.
  S. Grebenkov and R. Maurette\, Reactive capacitance of&nbsp\;flat patches
  of arbitrary shape\, Phys. Rev. E 113\, 034112 (2026).\n[3] D. S. Grebenk
 ov and M. J. Ward\, The asymptotic analysis of&nbsp\;some PDE and Steklov 
 eigenvalue problems with partially reactive&nbsp\;patches in 3-D (submitte
 d\; preprint 2509.17394v1).\n[4] K. A. Patil\, N. Nigam\, and D. S. Greben
 kov\, Generalized exterior&nbsp\;Steklov-Helmholtz eigenvalue problems in 
 the plane (in preparation).
LOCATION:Seminar Room 1\, Newton Institute
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