Limit behaviour of a singular perturbation problem for the Biharmonic operator

Serena Dipierro, Aram L. Karakhanyan, Enrico Valdinoci

Research output: Contribution to journalArticle

Abstract

We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. (arXiv:1808.07696, 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.

Original languageEnglish
Pages (from-to)1-35
JournalApplied Mathematics and Optimization
DOIs
Publication statusE-pub ahead of print - 25 Jul 2019

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Biharmonic Operator
Limit Behavior
Singular Perturbation Problems
Monotonicity Formula
Biharmonic
Forcing Term
Zero set
Parameter Perturbation
Free Boundary Problem
Behavior of Solutions
Level Set
Combustion
Counterexample
Regularity
Model

Cite this

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abstract = "We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. (arXiv:1808.07696, 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.",
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Limit behaviour of a singular perturbation problem for the Biharmonic operator. / Dipierro, Serena; Karakhanyan, Aram L.; Valdinoci, Enrico.

In: Applied Mathematics and Optimization, 25.07.2019, p. 1-35.

Research output: Contribution to journalArticle

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AU - Karakhanyan, Aram L.

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