On differentiability of eigenvalues of second order elliptic operators on non-smooth domains

The regularity of eigenvalues of elliptic operators upon deformations of a given bounded domain is a classical problem in elliptic PDEs which has been focused by many authors. We establish a theorem on CrCr dependence of algebraically simple eigenvalues and eigenfunctions with respect to perturbations of C1C1 class of non-smooth domains and of CrCr class of coefficients of elliptic operators. Moreover, we also compute the first variation of these eigenvalues in relation to both parameters for non-smooth domains. As a byproduct, we extend Hadamard's formula to second order elliptic operators for domains of C2C2 class and other non-smooth ones.