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Hadamard type asymptotics for eigenvalues of the Neumann problem for elliptic operators
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
2016 (English)In: Journal of Spectral Theory, ISSN 1664-039X, E-ISSN 1664-0403, Vol. 6, no 1, 99-135 p.Article in journal (Refereed) Published
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Abstract [en]

This paper considers how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain. The proximity of two domains is measured in terms of the norm of the difference between the two resolvents corresponding to the reference domain and the perturbed domain, and the size of eigenfunctions outside the intersection of the two domains. This construction enables the possibility of comparing both nonsmooth domains and domains with different topology. An abstract framework is presented, where the main result is an asymptotic formula where the remainder is expressed in terms of the proximity quantity described above when this is relatively small. As an application, we develop a theory for the Laplacian in Lipschitz domains. In particular, if the domains are assumed to be C-1,C-alpha regular, an asymptotic result for the eigenvalues is given together with estimates for the remainder, and we also provide an example which demonstrates the sharpness of our obtained result.

Place, publisher, year, edition, pages
EUROPEAN MATHEMATICAL SOC , 2016. Vol. 6, no 1, 99-135 p.
Keyword [en]
Hadamard formula; domain variation; asymptotics of eigenvalues; Neumann problem
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-129508DOI: 10.4171/JST/120ISI: 000376418300005OAI: oai:DiVA.org:liu-129508DiVA: diva2:940087
Available from: 2016-06-20 Created: 2016-06-20 Last updated: 2016-06-20

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Kozlov, VladimirThim, Johan
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