Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator
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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.
KeywordsBiharmonic operator Singular perturbation problems Monotonicity formula
Mathematics Subject Classification31A30 31B30 35R35
- 8.Dipierro, S., Karakhanyan, A., Valdinoci, E.: A free boundary problem driven by the biharmonic operator. arXiv:1808.07696 (2018)