We study an eigenvalue problem for the Laplace operator on a planar region with a growing crack. We impose the Neumann boundary condition on the crack and the Dirichlet boundary condition elsewhere. One tip of the crack is fixed at the boundary. We obtain the full asymptotic expansions of the first two eigenvalues of that operator as the other tip of the crack reaches the boundary.
"Eigenvalue Problems on Domains with Cracks I." Tokyo J. Math. 28 (2) 341 - 380, December 2005. https://doi.org/10.3836/tjm/1244208195