Nodal sets of Laplacian eigenfunctions with an eigenvalue of multiplicity 2

We study the effects of a domain deformation to the nodal set of Laplacian eigenfunctions when the eigenvalue is degenerate. In particular, we study deformations of a rectangle that perturb one side and how they change the nodal sets 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 large class of boundary deformations and study how these properties change along each eigenvalue branch for small perturbations. We show that internal crossings of the nodal set break under generic deformations and obtain estimates on the location and regularity of the nodal sets on the perturbed rectangle.