Boundary Deformations of Elliptic Eigenvalue Problems

This thesis explores spectral properties of elliptic eigenvalue problems on domains subject to boundarydeformations and is dedicated to two main problems that provide deeper insight into the sensitive relationshipbetween domain geometry and eigenfunction behavior. In Chapter 2, we study the effects of a domain deformation to the nodal set of Laplacian eigenfunctionsdefined on a rectangle and corresponding to an eigenvalue of multiplicity 2. We establish geometric properties,such as number of nodal domains, presence of crossings, and boundary intersections, of nodal sets for a largeclass of boundary deformations and analyze how these properties change for small perturbations. We showthat internal crossings of the nodal set break under generic deformations and obtain estimates on the locationand regularity of the nodal sets on the perturbed rectangle. The contents of Chapter 2 have been published inAnnales Mathematiques du Québec. In Chapter 3, we study the eigenvalues and eigenfunctions of a differential operator, Dε , that governsthe asymptotic behavior of an unsupervised learning algorithm known as Locally Linear Embedding. Whendefined over a compact Riemannian manifold with boundary, the differential operator is elliptic over a movingsubmanifold of codimension 0. We show that a natural regularity condition on the eigenfunctions imposes aconsistent Neumann-type boundary condition on the elliptic region and use the Frobenius method to estimatepointwise behavior. We then determine the limiting sequence of eigenvalues analytically and compare themto numerical predictions. The contents of Chapter 3 are available at https://arxiv.org/abs/2501.09572. Finally, in Chapter 4, we explore the spectral behavior of a differential operator that, up to a local signchange, resembles Dε on the interval. The sign change is applied to terms that are localized in a boundarylayer, and we seek to understand how it affects the possible eigenfunctions. Contrasting Chapter 3, we showthat eigenfunctions of this operator solve a singular Sturm-Liouville equation and share a Dirichlet boundarycondition on the elliptic region. Using asymptotics of hypergeometric functions, we then determine thelimiting sequence of eigenvalues and compare to that of Dε .