Approximation of Dirichlet Eigenvalues on Domains with Small Holes

Approximation formulas for the eigenvalues of the Laplacian with Dirichlet boundary conditions on domains with small holes are derived and discussed. The corresponding results for the free and the supported vibrating plate follow. A conceptually simple proof is given based on Courant′s min-max principle. The author′s approximation for the capacity of small balls leads to a highly accurate eigenvalue approximation formula for domains with spherical holes. General holes are treated by means of a harmonic correction method, an isoperimetric inequality relating capacity to volume, and a Poincaré inequality for capacity potentials. In addition we provide L∞-bounds for the eigenfunctions.