Spectral stability of higher order uniformly elliptic operators

Details Spectral stability of higher order uniformly elliptic operators Spectral stability of higher order uniformly elliptic operators

2012-04-12

arxiv.org/abs/1204.2848v1

Sobolev spaces in mathematics. II, Int. Math. Ser. (N. Y.), 9, p. 69-102, Springer, New York, 2009

Mathematics

Spectral Theory

Scientific paper

10.1007/978-0-387-85650-6_5

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.

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