The Cr dependence problem of eigenvalues of the Laplace operator on domains in the plane

The Cr dependence problem of multiple Dirichlet eigenvalues on domains is discussed for elliptic operators by regarding Cr+1-smooth one-parameter families of C1 perturbations of domains in Rn. As applications of our main theorem (Theorem 1), we provide a fairly complete description for all eigenvalues of the Laplace operator on disks and squares in R2 and also for its second eigenvalue on balls in Rn for any n≥3. The central tool used in our proof is a degenerate implicit function theorem on Banach spaces (Theorem 2) of independent interest.