Istituto di Matematica Applicata e Tecnologie Informatiche     
Savare' G., Schimperna G. Domain perturbations and estimates for the solutions of second order elliptic equations. Preprint ercim.cnr.ian//2001-1230, 2001.
We study the dependence of the variational solution of the inhomogeneous Dirichlet problem for a second order elliptic equation with respect to perturbations of the domain. We prove optimal $L^2$ and energy estimates for the difference of two solutions in two open sets in terms of the ``distance'' between them and suitable geometrical parameters which are related to the regularity of their boundaries. We derive such estimates when at least one of the involved sets is uniformly Lipschitz: due to the connection of this problem with the regularity properties of the solutions in the $L^2$ family of Sobolev-Besov spaces, the Lipschitz class is the reasonably weakest one compatible with the optimal estimates.
Subject Dirichlet problem, uniformly Lipschitz domain, Hausdorff distance, domain perturbation
35J20, 35B30

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