Asymptotic analysis and numerical computation of the Laplacian eigenvalues using the conformal mapping

We consider the Dirichlet and Neumann eigenvalues of the Laplacian for a simply connected, planar domain. The eigenvalues are the characteristic values of layer potential operators involving the Hankel function. First, we obtain an asymptotic formula for the Laplace eigenvalues with respect to the perturbation of the domain. The results are based on the Gohberg--Sigal theory for operator valued functions. Second, we propose a numerical computation scheme to compute the Laplacian eigenvalues based on the matrix formulation of the layer potential operators with a geometric basis associated with the exterior conformal mapping of the domain. This provides a way to compute Laplace eigenvalues, $\lambda$, by finding roots of polynomials in $\lambda$ and $\log\lambda$. We also derive a fully computable a priori error estimate with no assumption on the domain's convexity. This shows the relation between the domain's regularity and the convergence rate of the proposed method to compute Laplace eigenvalues.