On The Asymptotic Behaviour of Some Problems of the Calculus of Variations

Chipot, Michel

doi:10.1007/bf03377383

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Recall that u ℓ is a minimizer on Ω ℓ , whereas the following estimate is on Ω ℓ0 . See [5] for a very similar argument. Lemma 3.6 There exist constants l and c 4 , independent of ℓ, such that for all l ≤ ℓ 0 ≤ ℓ,…”

Section: Lemma 33 Let H(t) a Nonnegative Bounded Function Defined On ...mentioning

confidence: 83%

“…The majority of works on the ℓ → +∞ problem makes use of the boundary value problem itself, i.e., the PDE plus boundary condition. One exception to this rule are the recent papers [5,7], in which the authors consider instead a sequence of problems in the calculus of variations posed on elongated domains, see also [6]. This is the approach we adopt here as well.…”

Section: Introductionmentioning

confidence: 99%

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On Problems in the Calculus of Variations in Increasingly Elongated Domains

Dret

Mokrane

2018

Chin. Ann. Math. Ser. B

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Laboratoire d'équations aux dérivées partielles non linéaires et histoire des mathématiques, ENS, B.P. 92, Vieux Kouba, 16050 Alger, Algérie and USTHB, Faculté des mathématiques, Département d'analyse, Laboratoire d'analyse mathématique et numérique des équations aux dérivées partielles, Bab Ezzouar, Alger, Algérie. AbstractWe consider minimization problems in the calculus of variations set in a sequence of domains the size of which tends to infinity in certain directions and such that the data only depend on the coordinates in the directions that remain constant. We study the asymptotic behavior of minimizers in various situations and show that they converge in an appropriate sense toward minimizers of a related energy functional in the constant directions. MSC 2010: 35J25, 35J35, 35J62, 35J92, 49J45, 74K99. Keywords: Calculus of variations, domains becoming unbounded, asymptotic behavior, exponential rate of convergence.

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Section: Lemma 33 Let H(t) a Nonnegative Bounded Function Defined On ...mentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

On Problems in the Calculus of Variations in Increasingly Elongated Domains

Dret

Mokrane

2018

Chin. Ann. Math. Ser. B

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“…The question of analyzing the behavior of minimizers of various variational problems passing from truncated to infinite cylindrical domains, in terms of minimizers for their cross sections has been treated in in [7,6,5,3,4,8]. In the current context we take strong advantage of the Euler equation to establish comparisons.…”

Section: Introductionmentioning

confidence: 99%

On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

Chipot

Dávila

Pino

2016

J. Fixed Point Theory Appl.

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The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen-Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.

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On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded

Esposito

Roy

2021

ASY

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We analyze the asymptotic behavior of the eigenvalues of nonlinear elliptic problems under Dirichlet boundary conditions and mixed (Dirichlet, Neumann) boundary conditions on domains becoming unbounded. We make intensive use of Picone identity to overcome nonlinearity complications. Altogether the use of Picone identity makes the proof easier with respect to the known proof in the linear case. Surprisingly the asymptotic behavior under mixed boundary conditions critically differs from the case of pure Dirichlet boundary conditions for some class of problems.

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On The Asymptotic Behaviour of Some Problems of the Calculus of Variations

Cited by 4 publications

References 8 publications

On Problems in the Calculus of Variations in Increasingly Elongated Domains

On Problems in the Calculus of Variations in Increasingly Elongated Domains

On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded

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