  We consider general second order uniformly elliptic operators subject to
homogeneous boundary conditions on open sets $\phi (\Omega)$ parametrized by
Lipschitz homeomorphisms $\phi $ defined on a fixed reference domain $\Omega$.
Given two open sets $\phi (\Omega)$, $\tilde \phi (\Omega)$ we estimate the
variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm
$\|\tilde \phi -\phi \|_{W^{1,p}(\Omega)}$ for finite values of $p$, under
natural summability conditions on eigenfunctions and their gradients. We prove
that such conditions are satisfied for a wide class of operators and open sets,
including open sets with Lipschitz continuous boundaries. We apply these
estimates to control the variation of the eigenvalues and eigenfunctions via
the measure of the symmetric difference of the open sets. We also discuss an
application to the stability of solutions to the Poisson problem.
