6 papers · 1 filter
The Dirichlet-to-Neumann map, the boundary Laplacian, and Hörmander's rediscovered manuscript
Alexandre Girouard, Mikhail Karpukhin, Michael Levitin +1
How close is the Dirichlet-to-Neumann (DtN) map to the square root of the corresponding boundary Laplacian? This question has been actively investigated in recent years. Somewhat s…
Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems
Alexandre Girouard, Mikhail Karpukhin, Jean Lagacé
We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and ot…
Robin spectrum: two disks maximize the third eigenvalue
Alexandre Girouard, Richard S. Laugesen
The third eigenvalue of the Robin Laplacian on a simply-connected planar domain of given area is bounded above by the third eigenvalue of a disjoint union of two disks, provided th…
Hypersurfaces of Euclidean space with prescribed boundary and small Steklov eigenvalues
Bruno Colbois, Alexandre Girouard, Antoine Métras
Given a smooth compact hypersurface with boundary , we prove the existence of a sequence of hypersurfaces with the same boundary as , such that each Stek…
The Steklov and Laplacian spectra of Riemannian manifolds with boundary
Bruno Colbois, Alexandre Girouard, Asma Hassannezhad
Given two compact Riemannian manifolds with boundary and such that their respective boundaries and admit neighborhoods and which are isometric,…
Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space
Bruno Colbois, Alexandre Girouard, Katie Gittins
We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on th…