Domain perturbations, capacity and shift of eigenvalues
Journées équations aux dérivées partielles (1999), article no. 8, 10 p.

After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator H. If H is subjected to a domain perturbation the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in Capacity in abstract Hilbert spaces and applications to higher order differential operators (Comm. P. D. E., 24:759–775, 1999) and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an L 2 -space. Moreover, a similar capacitary upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.

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André Noll. Domain perturbations, capacity and shift of eigenvalues. Journées équations aux dérivées partielles (1999), article  no. 8, 10 p. https://proceedings.centre-mersenne.org/item/JEDP_1999____A8_0/

[Agm65] S. Agmon. Elliptic Boundary Value Problems. Van Nostrand Mathematical Studies, 1965. | MR | Zbl

[AM95] W. Arendt and S. Monniaux. Domain perturbation for the first eigenvalue of the Dirichlet Schrödinger operator. Oper. Theory Adv. Appl., 78 : 9-19, 1995. | MR | Zbl

[CF78] I. Chavel and E.A. Feldman. Spectra of domains in compact manifolds. J. Funct. Anal., 30 : 198-222, 1978. | MR | Zbl

[CF88] I. Chavel and E.A. Feldman. Spectra of manifolds less a small domain. Duke Math. J., 56 : 399-414, 1988. | MR | Zbl

[Cou95] G. Courtois. Spectrum of manifolds with holes. J. Funct. Anal., 134 : 194-221, 1995. | MR | Zbl

[Dava] E.B. Davies. Lp spectral theory of higher order elliptic differential operators. preprint 1996.

[Davb] E.B. Davies. Uniformly elliptic operators with measurable coefficients. preprint 1997.

[DMN97] M. Demuth, I. Mcgillivray, and A. Noll. Capacity and spectral theory. In M. Demuth, E. Schrohe, B.-W. Schulze, and J. Sjöstrand, editors, Spectral Theory, Microlocal Analysis, Singular Manifolds, volume 14 of Advances in Partial Differential Equations, pages 12-77. Akademie Verlag, Berlin, 1997. | MR | Zbl

[FOT94] M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet Forms and Symmetric Markov Processes, volume 19 of Studies in Mathematics. Walter de Gruyter Co, Berlin, 1994. | MR | Zbl

[GZ94] F. Gesztesy and Z. Zhao. Domain perturbations, Brownian motion and ground states of Dirichlet Schrödinger operators. Math. Z., 215 : 143-150, 1994. | MR | Zbl

[Hör83a] L. Hörmander. The Analysis of Linear Partial Differential Operators I. Grundlehren der mathematischen Wissenschaften. Springer Verlag, Berlin-New York, 1983. | Zbl

[Hör83b] L. Hörmander. The Analysis of Linear Partial Differential Operators II. Grundlehren der mathematischen Wissenschaften. Springer Verlag, Berlin-New York, 1983. | Zbl

[Joh49] F. John. On linear partial differential equations with analytic coefficients. Commun. Pure Appl. Math., 2 : 209-253, 1949. | MR | Zbl

[Maz85] V.G. Maz'Ja. Sobolev Spaces. Springer Series in Soviet Mathematics. Springer Verlag Berlin, New York, 1985. | MR

[McG96] I. Mcgillivray. Capacitary estimates for Dirichlet eigenvalues. J. Funct. Anal., 139 : 244-259, 1996. | MR | Zbl

[MR84] P.J. Mckenna and M. Rao. Lower bounds for the first eigenvalue of the Laplacian with Dirichlet boundary conditions and a theorem of Hayman. Appl. Anal., 18 : 55-66, 1984. | MR | Zbl

[Nol97] A. Noll. A generalization of Dynkin's formula and capacitary estimates for semibounded operators. In M. Demuth and B.-W. Schulze, editors, Differential Equations, Asymptotic Analysis and Mathematical Physics, volume 100 of Mathematical Research, pages 252-259. Akademie Verlag, Berlin, 1997. | MR | Zbl

[Nol98] A. Noll. Domain perturbations, shift of eigenvalues and capacity. Technical report, TU Clausthal, 1998. Submitted to Comm. P. D. E.

[Nol99] A. Noll. Capacity in abstract Hilbert spaces and applications to higher order differential operators. Comm. P. D. E., 24 : 759-775, 1999. | MR | Zbl

[Oza81] S. Ozawa. Singular variation of domains and eigenvalues of the Laplacian. Duke Math. J., 48(4) : 767-778, 1981. | MR | Zbl

[Oza82] S. Ozawa. The first eigenvalue of the Laplacian on two-dimensional manifolds. Tohoku Math. J., 34 : 7-14, 1982. | MR | Zbl

[Oza83] S. Ozawa. Electrostatic capacity and eigenvalues of the Laplacian. J. Fac. Sci. Univ. Tokyo, 30 : 53-62, 1983. | MR | Zbl

[Rau75] J. Rauch. The mathematical theory of crushed ice. In Partial Differential Equations and Related Topics, volume 446 of Lect. Notes in Math., pages 370-379. Springer, Berlin, 1975. | MR | Zbl

[RT75a] J. Rauch and M. Taylor. Electrostatic screening. J. Math. Phys., 16 : 284-288, 1975. | MR

[RT75b] J. Rauch and M. Taylor. Potential and scattering theory on wildly perturbed domains. J. Funct. Anal., 18 : 27-59, 1975. | MR | Zbl

[Szn98] A.-S. Sznitman. Brownian motion, obstacles and random media. Springer Monographs in Mathematics. Berlin. Springer-Verlag, 1998. | MR | Zbl

[Tay76] M. Taylor. Scattering length and perturbations of -Δ by positive potentials. J. Math. Anal. Appl., 53:291-312, 1976. | MR | Zbl

[Tay79] M. Taylor. Estimate on the fundamental frequency of a drum. Duke Math. J., 46:447-453, 1979. | MR | Zbl