The Saint-Venant Inequality for the Laplace Operator with Robin Boundary Conditions

Bucur, Dorin; Giacomini, Alessandro

doi:10.1007/s00032-015-0243-0

Cited by 15 publications

(16 citation statements)

References 20 publications

(24 reference statements)

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“…Assume that f ∈ L ∞ (D), f ≥ 0, Ω is Lipschitz and u ≥ 0 is a minimizer in (2.2). Following the ideas developed in [4,5], we observe that the function u extended by zero on D \ Ω belongs to SBV (D). Consequently, one can relax the shape optimization problem, by replacing the couple (Ω, u) with a new unknown v ∈ SBV (D, R + ), the set Ω being identified with {v > 0}.…”

Section: Further Remarksmentioning

confidence: 95%

Symmetry breaking for a problem in optimal insulation

Bucur

Buttazzo

Nitsch

2017

Journal de Mathématiques Pures et Appliquées

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We consider the problem of optimally insulating a given domain Ω of R d ; this amounts to solve a nonlinear variational problem, where the optimal thickness of the insulator is obtained as the boundary trace of the solution. We deal with two different criteria of optimization: the first one consists in the minimization of the total energy of the system, while the second one involves the first eigenvalue of the related differential operator. Surprisingly, the second optimization problem presents a symmetry breaking in the sense that for a ball the optimal thickness is nonsymmetric when the total amount of insulator is small enough. In the last section we discuss the shape optimization problem which is obtained letting Ω to vary too.

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Section: Further Remarksmentioning

confidence: 95%

Symmetry breaking for a problem in optimal insulation

Bucur

Buttazzo

Nitsch

2017

Journal de Mathématiques Pures et Appliquées

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“…A quantity of interest is generalized torsional rigidity, denoted by S β in these papers, and with N = 2 by Q steady in [22]. The generalization of the St Venant Inequality for the case β > 0 is proved in [7]. When N = 2 and β > 0 there is good numerical evidence, reported in [22], suggesting that a result corresponding to our Theorem 6 should be available for S β = Q steady .…”

Section: The Generalized Torsion Problemmentioning

confidence: 88%

Inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional rectangular parallelepipeds

Keady¹,

Wiwatanapataphee²

2018

Mathematical Inequalities &Amp; Applications

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This document consists of two papers, both submitted, and supplementary material. The submitted papers are here given as Parts I and II. Their abstracts are given at pages 2 and 37 respectively. The supplementary material is given in appendices.The supplementary material includes material directed at answering questions stated in, or implicit in, the submitted papers. The maple code referenced in the supplements is available via https://sites.google.com/site/keadyperthunis/home/papers One of the questions is as follows. Let φ 1 (z) = arctan(1/z)/z. (φ 1 is, itself, completely monotone.) Is the inverse of φ 1 , denoted µ in the following, also completely monotone? Various approaches have been considered. Direct calculation of the higher derivatives of µ -with maple to handle the lengthy expressions -and an inductive argument is considered in work presented at page 25. Some complex variable approaches are instigated at page 33.Abstract Any function f from (0, ∞) onto (0, ∞) which is decreasing and convex has an inverse g which is positive and decreasing -and convex. When f has some form of generalized convexity we determine additional convexity properties inherited by g. When f is positive, decreasing and (p, q)-convex, its inverse g is (q, p)-convex. Related properties which pertain when f is a Stieltjes function are developed. The results are illustrated with the Stieltjes function f (x) = arctan(1/ √ x)/ √ x, a function which arises, via a transcendental equation, in an application presented in Part II.

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“…The inequality (1.15) is also called Faber-Krahn inequality for Robin Laplacian. When (M, g) is Euclidean space, the result is due to Bossel in [7] for n = 2 and Daners in [17] for all dimensions (See also [16,9,10]). For bounded domains in Riemannian manifolds with Ric(g) ≥ n −1, the inequality (1.15) is due to Chen-Cheng-Li in [12].…”

Section: Andmentioning

confidence: 99%

“…In 2015, by using free discontinuity techniques, Bucur et al in [10] proved the Saint-Venant inequality for Robin Laplacian in Euclidean spaces. In 2019, Alvino et al in [2] obtained the same Saint-Venant inequality for Robin Laplacian via a Talenti type comparison result in Euclidean spaces.…”

Section: Introductionmentioning

confidence: 99%

Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

Chen,

Li,

Wei

2021

Preprint

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The Saint-Venant Inequality for the Laplace Operator with Robin Boundary Conditions

Cited by 15 publications

References 20 publications

Symmetry breaking for a problem in optimal insulation

Symmetry breaking for a problem in optimal insulation

Inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional rectangular parallelepipeds

Comparison results for solutions of Poisson equations with Robin boundary on complete Riemannian manifolds

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