### 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 language | English |
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Pages (from-to) | 1-35 |

Journal | Applied Mathematics and Optimization |

DOIs | |

Publication status | E-pub ahead of print - 25 Jul 2019 |

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**Limit behaviour of a singular perturbation problem for the Biharmonic operator.** / Dipierro, Serena; Karakhanyan, Aram L.; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Limit behaviour of a singular perturbation problem for the Biharmonic operator

AU - Dipierro, Serena

AU - Karakhanyan, Aram L.

AU - Valdinoci, Enrico

PY - 2019/7/25

Y1 - 2019/7/25

N2 - 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.

AB - 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.

KW - Biharmonic operator

KW - Monotonicity formula

KW - Singular perturbation problems

UR - http://www.scopus.com/inward/record.url?scp=85069692222&partnerID=8YFLogxK

U2 - 10.1007/s00245-019-09598-7

DO - 10.1007/s00245-019-09598-7

M3 - Article

SP - 1

EP - 35

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

SN - 0095-4616

ER -