Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

Bonder, Julián Fernández; Groisman, Pablo; Rossi, Julio D.

doi:10.1007/s10231-006-0009-y

Cited by 20 publications

(22 citation statements)

References 21 publications

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“…The author also studies the problem of maximizing the first five Wentzell eigenvalues subject to a volume constraint, for which (3) is a special case. Shape optimization problems for Steklov eigenvalues with mixed boundary conditions have also been studied [BGR07].…”

Section: Related Workmentioning

confidence: 99%

Computational Methods for Extremal Steklov Problems

Akhmetgaliyev¹,

Kao²,

Osting³

2017

SIAM J. Control Optim.

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Abstract. We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the p-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet-and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the p-th Steklov eigenvalue is unique (up to dilations and rigid transformations), has p-fold symmetry, and an axis of symmetry. The p-th Steklov eigenvalue has multiplicity 2 if p is even and multiplicity 3 if p ≥ 3 is odd.

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Section: Related Workmentioning

confidence: 99%

Computational Methods for Extremal Steklov Problems

Akhmetgaliyev¹,

Kao²,

Osting³

2017

SIAM J. Control Optim.

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“…Both the theories of boundary trace in Sobolev spaces and of Hardy inequalities have a large number of applications, especially to boundary value problems for partial differential equations and non linear analysis. They have been developed, via different methods and in different settings, by various authors ( see, for example, [4], [24], [25], [28] or the references on this topic in the monographs [3], [34], [36], [42]). In particular, the lack of extremals in (3) and in (6) has inspired many mathematicians to consider possible extra terms on the left hand side.…”

Section: Introductionmentioning

confidence: 99%

Sharp Hardy inequalities in the half space with trace remainder term

Alvino¹,

Volpicelli²,

Ferone³

2012

Nonlinear Analysis: Theory, Methods & Applications

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Abstract. In this paper we deal with a class of inequalities which interpolate the Kato's inequality and the Hardy's inequality in the half space. Starting from the classical Hardy's inequality in the half spacewe show that, if we replace the optimal constantwith a smaller one, 2 ≤ β < n, then we can add an extra trace-term equals to that one that appears in the Kato's inequality. The constant in the trace remainder term is optimal and it tends to zero when β goes to n, while it is equal to the optimal constant in the Kato's inequality when β = 2.

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“…We will also need the following lemma, that has been proven in [10]. We only make a sketch of the proof for the reader's convenience.…”

Section: Dimension Reduction Proof Of Theorem 11mentioning

confidence: 99%

An Optimization Problem Related to the Best Sobolev Trace Constant in Thin Domains

Bonder¹,

Rossi

Schönlieb

2008

Commun. Contemp. Math.

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Let Ω ⊂ R N be a bounded, smooth domain. We deal with the best constant of the Sobolev trace embedding W 1,p (Ω) → L q (∂Ω) for functions that vanish in a subset A ⊂ Ω, which we call the hole, i.e. we deal with the minimization problem S A = inf u p W 1,p (Ω) / u p L q (∂Ω) for functions that verify u| A = 0. It is known that there exists an optimal hole that minimizes the best constant S A among subsets of Ω of the prescribed volume.In this paper, we look for optimal holes and extremals in thin domains. We find a limit problem (when the thickness of the domain goes to zero), that is a standard Neumann eigenvalue problem with weights and prove that when the domain is contracted to a segment, it is better to concentrate the hole on one side of the domain.

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Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

Cited by 20 publications

References 21 publications

Computational Methods for Extremal Steklov Problems

Computational Methods for Extremal Steklov Problems

Sharp Hardy inequalities in the half space with trace remainder term

An Optimization Problem Related to the Best Sobolev Trace Constant in Thin Domains

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