Yuwei Hu, Jun Zheng, Porosity of the free boundary in a minimum problem

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DOI: 10.23952/jnva.8.2024.1.06

Volume 8, Issue 1, 1 February 2024, Pages 109-124

Abstract. Given a bounded domain $\Omega \subset \mathbb{R} ^N(N\geq 2)$ , a positive constant $\lambda$ , and functions $q$ , $h\in L^\infty (\Omega )$ , we study geometric properties of non-negative minimizers of the minimum problem
$\mathcal{J} (u)=\int_{\Omega } \left(A\left(| \nabla u|\right)+qF\left(u^+\right)+hu+\lambda \chi _{\left\{u>0\right\}}\right) \text{d}x \to \min$
over certain class $\mathcal{K}$ in the framework of Orlicz-Sobolev spaces, where $u^+$ denotes the positive part of $u$ , $\chi_{\{\cdot\}}$ is the standard characteristic function, and the functions $A$ and $F$ satisfy the structural conditions of Lieberman-Tolksdorf’s type. In particular, $F$ is allowed to grow with a subcritical exponent. By using the technique of blow-up and the Harnack’s inequality, we firstly prove the non-degeneracy of non-negative minimizers near the free boundary $\Gamma ^+:=\partial\{u>0\}\cap \Omega$ , and then we show that the free boundary $\Gamma ^+$ is locally porous. Furthermore, we also prove that $\{u>0\}$ has a uniformly positive density.

How to Cite this Article:
Y. Hu, J. Zheng, Porosity of the free boundary in a minimum problem, J. Nonlinear Var. Anal. 8 (2024), 109-124.