Domain dependence of eigenvalues of elliptic type operators (original) (raw)
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We prove estimates for the variation of the eigenvalues of uniformly elliptic operators with homogeneous Dirichlet or Neumann boundary conditions upon variation of the open set on which an operator is defined. We consider operators of arbitrary even order and open sets admitting arbitrary strong degeneration. The main estimate is expressed via a natural and easily computable distance between open sets with continuous boundaries. Another estimate is obtained via the lower Hausdorff-Pompeiu deviation of the boundaries, which in general may be much smaller than the usual Hausdorff-Pompeiu distance. Finally, in the case of diffeomorphic open sets we obtain an estimate even without the assumption of continuity of the boundaries.
We present a general spectral stability theorem for nonnegative selfadjoint operators with compact resolvents, which is based on the notion of a transition operator, and some appli-cations to the study of the dependence of the eigenvalues of uniformly elliptic operators upon domain perturbation. Copyright © 2006 V. I. Burenkov and P. D. Lamberti. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-erly cited.
Domain perturbations and estimates for the solutions of second order elliptic equations
Journal de Mathématiques Pures et Appliquées, 2002
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.