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Nodal Set Openings on Perturbed Rectangular Domains
Thomas Beck, Marichi Gupta, Jeremy L. Marzuola
We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio , we identif…
Two-phase free boundary problems in convex domains
Thomas Beck, David Jerison, Sarah Raynor
We study the regularity of minimizers of a two-phase free boundary problem. For a class of n-dimensional convex domains, we establish the Lipschitz continuity of the minimizer up t…
Localization of the first eigenfunction of a convex domain
Thomas Beck
We study the first Dirichlet eigenfunction of the Laplacian in a -dimensional convex domain. For domains of a fixed inner radius, estimates of Chiti imply that the ratio of the…
Nodal line estimates for the second Dirichlet eigenfunction
Thomas Beck, Yaiza Canzani, Jeremy L. Marzuola
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by…
The torsion function of convex domains of high eccentricity
Thomas Beck
The torsion function of a convex planar domain has convex level sets, but explicit formulae are known only for rectangles and ellipses. Here we study the torsion function on convex…
Uniform level set estimates for ground state eigenfunctions
Thomas Beck
We study the behaviour of the first eigenfunction of the Dirichlet Laplacian on a planar convex domain near its maximum. We show that the eccentricity and orientation of the superl…