Flexibility of Steklov eigenvalues via boundary homogenisation

Abstract

Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use a framework of variational eigenvalues and provide a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains. This is the case even though any actual maximisers (except for simply connected surfaces) are always far from being planar themselves. In particular, it yields sharp upper bound for the first Steklov eigenvalue of doubly connected planar domains.

Applications of possibly hidden symmetry to Steklov and mixed Steklov problems on surfaces

“…Using recent results of Dorin Bucur and Mickaël Nahon [6], we show in Subsection 2.3 that any Steklov eigenvalue bound that is valid for surfaces with smooth boundary is also valid for surfaces with Lipschitz boundary and also for weighted Steklov eigenvalue problems on surfaces. Mikhail Karpukhin and Jean Lagacé [26] independently obtained the same result for manifolds of all dimensions. Thus for the simplicity of the presentation, we state our results as well as previous results (including those that originally assumed stronger regularity on ∂Ω) for unweighted Steklov problems on surfaces with Lipschitz boundary, keeping in mind that the statements still hold if one considers the weighted Steklov problem with a non-negative, nontrivial L ∞ weight.…”