An optimization problem with volume constrain for a degenerate quasilinear operator

Details An optimization problem with volume constrain for a degenerate quasilinear operator An optimization problem with volume constrain for a degenerate quasilinear operator

2006-02-17

arxiv.org/abs/math/0602389v1

Mathematics

Analysis of PDEs

Scientific paper

We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$ 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, $\partial\{u>0\}\cap \Omega$, is smooth.

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