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arXiv · 2206.10331

A free boundary problem for the p-Laplacian with nonlinear boundary conditions

Paolo Acampora·Emanuele Cristoforoni
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Abstract

We study a nonlinear generalization of a free boundary problem that arises in the context of thermal insulation. We consider two open sets $\Omega\subseteq A$, and we search for an optimal $A$ in order to minimize a non-linear energy functional, whose minimizers $u$ satisfy the following conditions: $\Delta_p u=0$ inside $A\setminus\Omega$, $u=1$ in $\Omega$, and a nonlinear Robin-like boundary $(p,q)$-condition on the free boundary $\partial A$. We study the variational formulation of the problem in SBV, and we prove that, under suitable conditions on the exponents $p$ and $q$, a minimizer exists and its jump set satisfies uniform density estimates.

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Paolo Acampora, Emanuele Cristoforoni. 2022-06-21. A free boundary problem for the p-Laplacian with nonlinear boundary conditions. https://doi.org/10.1007/s10231-023-01350-x

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