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An optimization problem with volume constrain for a degenerate quasilinear operator

Julian Fernandez Bonder·Sandra Martinez·Noemi Wolanski
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Abstract

We consider the optimization problem of minimizing $\int_Ω|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial\{u>0\}\cap Ω$, is smooth.

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Julian Fernandez Bonder, Sandra Martinez, Noemi Wolanski. 2006-02-17. An optimization problem with volume constrain for a degenerate quasilinear operator. https://arxiv.org/abs/math/0602389

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