On an eigenvalue for the Laplace operator in a disk with Dirichlet boundary condition on a small part of the boundary in a critical case
Rustem Rashitovich Gadyl'shin, S. V. Rep’evskii, E. A. Shishkina · Proceedings of the Steklov Institute of Mathematics · 2016
A boundary-value problem of finding eigenvalues is considered for the negative Laplace operator in a disk with Neumann boundary condition on almost all the circle except for a small arc of vanishing length, where the Dirichlet boundary condition is imposed. A complete asymptotic expansion with respect to a parameter (the length of the small arc) is constructed for an eigenvalue of this problem that converges to a double eigenvalue of the Neumann problem.