Local porosity of the free boundary in a minimum problem

Abstract

<abstract><p>Given certain set $ \mathcal{K} $ and functions $ q $ and $ h $, we study geometric properties of the set $ \partial\{x\in\Omega:u(x) > 0\} $ for non-negative minimizers of the functional $ \mathcal{J} (u) = \int_{\Omega }^{} \, \left(\frac{1}{p}| \nabla u| ^p+q(u^+)^\gamma +hu\right)\text{d}x $ over $ \mathcal{K} $, where $ {\Omega \subset} \mathbb{R} ^n(n\geq 2) $ is an open bounded domain, $ p\in(1, +\infty) $ and $ \gamma \in (0, 1] $ are constants, $ u^+ $ is the positive part of $ u $ and $ \partial\{x\in\Omega :u(x) > 0\} $ is the so-called free boundary. Such a minimum problem arises in physics and chemistry for $ \gamma = 1 $ and $ \gamma \in(0, 1) $, respectively. Using the comparison principle of $ p $-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.</p></abstract>

Cite this paper

@article{hu2023local,
  title = {{Local porosity of the free boundary in a minimum problem}},
  author = {Hu, Yuwei and Zheng, Jun},
  journal = {Electronic Research Archive},
  volume = {31},
  number = {9},
  pages = {5457--5465},
  year = {2023},
  doi = {10.3934/era.2023277}
}

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