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arXiv · 0706.4446

An optimization problem with volume constrain in Orlicz spaces

Sandra Martinez
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Abstract

We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla u|) dx$ in the class of functions $W^{1,G}(\Omega)$, with a constrain on the volume of $\{u>0\}$. The conditions on the function $G$ allow for a different behavior at 0 and at $\infty$. 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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Sandra Martinez. 2007-06-29. An optimization problem with volume constrain in Orlicz spaces. https://doi.org/10.1016/j.jmaa.2007.09.061

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