On The Attainable Eigenvalues of the Laplace Operator

Bucur, Dorin; Buttazzo, Giuseppe; Figueiredo, Isabel N.

doi:10.1137/s0036141097329135

Cited by 37 publications

(51 citation statements)

References 11 publications

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“…For other recent results on minimization problems for functions of eigenvalues, we refer e.g. to [5], [32] and the review papers [2], [4], [17]. Finally, we point out that the results in section 2 are mostly valid in any dimension while the results of the section 3 are more specifically two-dimensional.…”

Section: Introductionmentioning

confidence: 85%

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Henrot

Oudet

2003

Archive for Rational Mechanics and Analysis

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Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditionsAntoine HENROT, Edouard OUDET AbstractIn this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the "stadium", convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties of the minimizer (regularity, geometric properties).

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Section: Introductionmentioning

confidence: 85%

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Henrot

Oudet

2003

Archive for Rational Mechanics and Analysis

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“…iii) The problem is of a very special type, involving only the first two eigenvalues of the Laplace operator, where neither geometrical constraints nor monotonicity of the cost are required (see [38]). …”

Section: Optimal Dirichlet Regionsmentioning

confidence: 99%

“…In the scalar case (with Ω convex and Σ = ∅, otherwise the same modifications seen above apply, see [23] for a general discussion) the interpretation of (38) in terms of transportation problem is related to the distance defined by…”

Section: Relationships Between Optimal Mass and Optimal Transportationmentioning

confidence: 99%

Optimal Shapes and Masses, and Optimal Transportation Problems

Buttazzo

Pascale

2003

Optimal Transportation and Applications

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Summary. We present here a survey on some shape optimization problems that received particular attention in the last years. In particular, we discuss a class of problems, that we call mass optimization problems, where one wants to find the distribution of a given amount of mass which optimizes a given cost functional. The relation with mass transportation problems will be discussed, and several open problems will be presented.

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“…The study of shape optimization problems for the eigenvalues of an elliptic operator is a fascinating field that has strong relations with several applications as for instance the stability of vibrating bodies, the propagation of waves in composite media, the thermic insulation of conductors. Investigation of such problems is important also for study of the qualitative properties of the eigenvalues [2]. In the work we consider some inverse problems relatively domain.…”

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confidence: 99%

“…It is known [3] that in the considered case the eigenfunctions u j , j = 1, 2, ... of the problem (1), (2) belong to the class For the sake of simplicity we denote by u the first normalized eigenfunction of the problem (1), (2) corresponding to the first eigenvalue λ 1 .…”

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confidence: 99%

Some shape optimization problems for eigenvalues

Gasimov¹

2008

J. Phys. A: Math. Theor.

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Abstract. In the work we consider some inverse problems relatively domain for the Laplace operator. The aim of the work is to find a domain under the condition that, the function, expressed by spectral data of the problem is given in the boundary of this domain. Introduction. Solution of a wide class of practical problems is reduced to the minimization of the functionals related with eigenvalues [1]. The study of shape optimization problems for the eigenvalues of an elliptic operator is a fascinating field that has strong relations with several applications as for instance the stability of vibrating bodies, the propagation of waves in composite media, the thermic insulation of conductors. Investigation of such problems is important also for study of the qualitative properties of the eigenvalues [2]. In the work we consider some inverse problems relatively domain. In its mathematical formulation the problem consists in taking an elliptic operator and considering its eigenvalues λ k as a functionals of the domain D where the problem is solved. In the work these problems are reduced to the minimization of the functionals including spectral data of the corresponding operators. Usually in such problems obtained optimality conditions for domain involve eigenfunctions that makes difficult their use in the determination of the domain. Here we reduce the considering shape optimization problems to variational formulation, get some optimality conditions and a formula for the eigenvalue corresponding to the optimal domain. Note that the obtained formulas don't include an eigenfunction. It makes them interesting both from practical and theoretical points of view. For the sake of simplicity we consider only Laplace operator, but the results may be extended for other elliptic operators.

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On The Attainable Eigenvalues of the Laplace Operator

Cited by 37 publications

References 11 publications

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Minimizing the Second Eigenvalue of the Laplace Operator with Dirichlet Boundary Conditions

Optimal Shapes and Masses, and Optimal Transportation Problems

Some shape optimization problems for eigenvalues

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