Uniform convergence for elliptic problems on varying domains

Abstract. The paper deals with the genericity of domain-dependent spectral properties of the Laplacian-Dirichlet operator. In particular we prove that, generically, the squares of the eigenfunctions form a free family. We also show that the spectrum is generically non-resonant. The results are obtained by applying global perturbations of the domains and exploiting analytic perturbation properties. The work is motivated by two applications: an existence result for the problem of maximizing the rate of exponential decay of a damped membrane and an approximate controllability result for the bilinear Schrödinger equation. Mathematics Subject Classification.

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“…In the sequel of the paper, we make use several times of the following result, whose proof can be found in [3], Theorem 7.3.…”

Section: Notations and Abstract Genericity Resultsmentioning

confidence: 99%

“…We claim that each Λ k (t) converges, as t → +∞, to an eigenvalue of the Laplacian-Dirichlet operator onΩ. Indeed, the compact convergence of Ω t toΩ implies the strong convergence of the corresponding resolvent operators (see, for instance, [3]). This, in turns, guarantees that |λ …”

Section: Proposition 22 Let M > 2 and ωmentioning

confidence: 96%

The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

Privat

Sigalotti

2009

ESAIM: COCV

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“…We show that the Leray-Schauder degree is persistent under domain perturbation. The proof is similar to the argument for periodic solutions of parabolic equations in [10,Theorem 7.1] or the argument for L ∞ solutions of elliptic equations in [2,Theorem 8.2]. However, we also consider perturbation of the nonlinear terms.…”

Section: Domain Perturbation For Bounded Solutions Of Semilinear Equamentioning

confidence: 81%

“…In Particular, if deg(I − Q, U, 0) = 0, then (5.1) has a C 0 (R, L 2 ( )) solution in U (see [21,Theorem 4.3.2]). We follow [2] to state a sequence of results below. Q, B(u, ε), 0).…”

Section: Domain Perturbation For Bounded Solutions Of Semilinear Equamentioning

confidence: 99%

Persistence of bounded solutions of parabolic equations under domain perturbation

Ngiamsunthorn

2011

J. Evol. Equ.

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The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L 2 and the L p -settings. For the L p -theory, we also prove the Hölder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation. (2000): Primary 35B20; Secondary 35K10 Mathematics Subject Classification

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“…Let f ∈ L ∞ (R N ). Then there exists a unique u ∈ H 1 0,loc ( ) ∩ L ∞ (R N ) (see [7]) solution of the Poisson equation…”

Section: Introductionmentioning

confidence: 98%

Regularity in Capacity and the Dirichlet Laplacian

Biegert

Warma

2006

Potential Anal

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Given an open set in R N , we prove that every function u in H 1 0 ( ) ∩ C( ) is zero everywhere on the boundary ∂ if and only if is regular in capacity. If in addition is bounded, then it is regular in capacity if and only if the mappingdenotes the Perron solution of the Dirichlet problem. Let R be the set of all open subsets of R N which are regular in capacity. Then one can define metrics d l and d g on R only involving the resolvent of the Dirichlet Laplacian. Convergence in those metrics will be defined to be the local/global uniform convergence of the resolvent of the Dirichlet Laplacian applied to the constant function 1. We prove that the spaces (R, d g ) and (R, d l ) are complete and contain the set W of all open sets which are regular in the sense of Wiener (or Dirichlet regular) as a closed subset. Mathematics Subject Classifications (2000) 35J67 · 31C15.

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Uniform convergence for elliptic problems on varying domains

Cited by 26 publications

References 28 publications

The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

The squares of the Laplacian-Dirichlet eigenfunctions are generically linearly independent

Persistence of bounded solutions of parabolic equations under domain perturbation

Regularity in Capacity and the Dirichlet Laplacian

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