Abstract

In this talk, we present an overview of recent progress in numerical methods for studying spectral properties of generalized Steklov problems associated with the Laplace and Helmholtz equations in the exterior of a compact set with Lipschitz boundary [1]. The first part is devoted to the Steklov problem in the exterior of an infinitely thin compact set embedded in three-dimensional space [2]. Such problems arise, in particular, in sloshing phenomena in hydrodynamics and in the small-target asymptotic analysis of various partial differential equations with mixed Robin, Neumann, and Dirichlet boundary conditions [3]. We discuss the geometric structure of low-energy eigenfunctions, as well as two-sided bounds on the principal eigenvalue. In the second part, we consider a generalized Steklov problem for the Helmholtz equation with real, purely imaginary, and complex parameters. Its reformulation in terms of boundary layer potentials enables efficient numerical computation for a wide class of exterior planar domains with piecewise smooth boundaries. The resulting numerical evidence leads to several conjectures relevant to spectral geometry, including corrections to Weyl’s law and the asymptotic behavior of eigenfunctions [4]. References: [1] L. Bundrock, A. Girouard, D. S. Grebenkov, M. Levitin, and I. Polterovich, The exterior Steklov problem for Euclidean domains (submitted; preprint 2511.09490v2). [2] D. S. Grebenkov and R. Maurette, Reactive capacitance of flat patches of arbitrary shape, Phys. Rev. E 113 , 034112 (2026). [3] D. S. Grebenkov and M. J. Ward, The asymptotic analysis of some PDE and Steklov eigenvalue problems with partially reactive patches in 3-D (submitted; preprint 2509.17394v1). [4] K. A. Patil, N. Nigam, and D. S. Grebenkov, Generalized exterior Steklov-Helmholtz eigenvalue problems in the plane (in preparation).