A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator

Lamberti, Pier Domenico; Cristoforis, Massimo Lanza de

doi:10.1007/s00009-007-0128-8

Cited by 34 publications

(90 citation statements)

References 9 publications

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“…In the first one the dependence of λ n [φ(Ω)] on a given diffeomorphism φ is investigated. This has been done by many authors: see, e.g., Hale [11], Henry [12], Lamberti and Lanza de Cristoforis [16], Sokolowsky and Zolésio [23] and the references therein. In particular differentiability and analyticity results with respect to φ and the corresponding formulas for derivatives and asymptotic expansions have been obtained.…”

Section: Introductionmentioning

confidence: 97%

Spectral stability of Dirichlet second order uniformly elliptic operators

Burenkov

Lamberti

2008

Journal of Differential Equations

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We prove sharp stability results for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Dirichlet boundary conditions upon domain perturbation. The main results concern estimates for the variation of the eigenvalues via the Hausdorff distance between the domains or the Lebesgue measure of their symmetric difference. Our analysis includes domains with Lipschitz boundaries as well as domains with boundary degenerations of power type

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Section: Introductionmentioning

confidence: 97%

Spectral stability of Dirichlet second order uniformly elliptic operators

Burenkov

Lamberti

2008

Journal of Differential Equations

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“…There is a vast literature concerning this problem in the 20th century: Hadamard [22] in 1908, Courant and Hilbert [13] in the German edition of 1937, Polya and Szëgo [43] in 1951, Garabedian and Schiffer [19,20] in 1952-1953, Polya and Schiffer [42] in 1953, Schiffer [46] in 1954 and thereafter [2,18,3,27,39,[36][37][38]40,47,16,44,14,17,21]. For more recent advances, we cite the works [29,11,15,35,6,26,25,30,31,4,8,9,33,10,34,32]. We also mention three interesting works on generic properties of eigenvalues and eigenfunctions due to Uhlenbeck [48,49] and Pereira [41] which are closely related to the issue of this paper.…”

Section: Introductionmentioning

confidence: 99%

On differentiability of eigenvalues of second order elliptic operators on non-smooth domains

Haddad

Montenegro

2015

Journal of Differential Equations

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The regularity of eigenvalues of elliptic operators upon deformations of a given bounded domain is a classical problem in elliptic PDEs which has been focused by many authors. We establish a theorem on C r dependence of algebraically simple eigenvalues and eigenfunctions with respect to perturbations of C 1 class of non-smooth domains and of C r class of coefficients of elliptic operators. Moreover, we also compute the first variation of these eigenvalues in relation to both parameters for non-smooth domains. As a byproduct, we extend Hadamard's formula to second order elliptic operators for domains of C 2 class and other non-smooth ones.

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“…Proof Let Δ φ 2 , J φ be the pull‐backs to Ω of the operators

Δ_{φ (Ω)}^{2}

, J φ (Ω) , that is, the operators defined by the pairings

Δ_{φ}^{2} [u] [η] MathClass-rel= Δ_{φ (Ω)}^{2} [u MathClass-bin\circ φ^{(MathClass-bin- 1)}] [η MathClass-bin\circ φ^{(MathClass-bin- 1)}]

, J φ [ u ][ η ] = J φ (Ω) [ u ∘ φ ( − 1) ][ η ∘ φ ( − 1) ] for all u , η ∈ V (Ω). The proof of the analyticity of Λ F , s follows by the abstract results in and applied to the operator

{((), Δ_{φ}^{2})}^{(MathClass-bin- 1)} MathClass-bin\circ {scriptJ}_{φ}

. See also .…”

Section: An Analyticity Resultsmentioning

confidence: 99%

“…Let

u_{l} MathClass-rel= v_{l} MathClass-bin\circ \overset{φ}{true}

for all l ∈ F . By proceeding as in , we have that

normald {MathClass-rel|}_{φ MathClass-rel= \overset{φ}{true}} Λ_{F MathClass-punc, s} [ψ] MathClass-rel= MathClass-bin- λ_{F}^{s MathClass-bin+ 1} [\overset{φ}{true}] ((), \binom{| F | - 1}{s - 1}) {MathClass-op\sum}_{l MathClass-rel\in F} Δ_{\overset{φ}{true}}^{2} ([], normald {MathClass-rel|}_{φ MathClass-rel= \overset{φ}{true}} ((), {((), Δ_{φ}^{2})}^{(MathClass-bin- 1)} MathClass-bin\circ {scriptJ}_{φ}) [ψ] (u_{l})) [u_{l}] MathClass-punc.

The proof of (2.8) will follow by combining (2.9) with the following formula

\begin{matrix} aligned \\ right & left Δ_{\overset{MathClass-op̃}{φ}}^{2} [d |_{φ = \overset{MathClass-op̃}{φ}} ({(Δ_{φ}^{2})}^{MathClass-open(- 1 MathClass-close)} \circ J_{φ})] \end{matrix}

…”

Section: An Analyticity Resultsmentioning

confidence: 99%

Shape deformation for vibrating hinged plates

Buoso

Lamberti

2013

Math Methods in App Sciences

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We consider the biharmonic operator subject to homogeneous intermediate boundary conditions of Steklov-type. We prove an analyticity result for the dependence of the eigenvalues upon domain perturbation and compute the appropriate Hadamard-type formulas for the shape derivatives. Finally, we prove that balls are critical domains for the symmetric functions of multiple eigenvalues subject to volume constraint

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A Real Analyticity Result for Symmetric Functions of the Eigenvalues of a Domain-Dependent Neumann Problem for the Laplace Operator

Cited by 34 publications

References 9 publications

Spectral stability of Dirichlet second order uniformly elliptic operators

Spectral stability of Dirichlet second order uniformly elliptic operators

On differentiability of eigenvalues of second order elliptic operators on non-smooth domains

Shape deformation for vibrating hinged plates

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