Alec Payne

Generic boundary behavior of Dirichlet eigenfunctions

We extend Uhlenbeck's classical results on the generic properties of Laplacian eigenfunctions. In particular, we find a residual set of $C^k$ domains in $\mathbb{R}^n$, $k>n$, such that the nodal set of any Dirichlet eigenfunction intersects the boundary orthogonally along a $C^{k-1}$ embedded submanifold, whenever the...