Akman, Murat and Banerjee, Agnid and Vega Garcia, Mariana Smit (2019) On a Bernoullitype overdetermined free boundary problem. Working Paper. arXiv. (Unpublished)

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Abstract
In this article we study a Bernoullitype free boundary problem and generalize a work of Henrot and Shahgholian to $\mathcal{A}$harmonic PDEs. These are quasilinear elliptic PDEs whose structure is modeled on the $p$Laplace equation for a fixed $1<p<\infty$. In particular, we show that if $K$ is a bounded convex set satisfying the interior ball condition and $c>0$ is a given constant, then there exists a unique convex domain $\Omega$ with $K\subset \Omega$ and a function $u$ which is $\mathcal{A}$harmonic in $\Omega\setminus K$, has continuous boundary values $1$ on $\partial K$ and $0$ on $\partial\Omega$, such that $\nabla u=c$ on $\partial \Omega$. Moreover, $\partial\Omega$ is $C^{1,\gamma}$ for some $\gamma>0$, and it is smooth provided $\mathcal{A}$ is smooth in $\mathbb{R}^n \setminus \{0\}$. We also show that the super level sets $\{u>t\}$ are convex for $t\in (0,1)$.
Item Type:  Monograph (Working Paper) 

Uncontrolled Keywords:  Quasilinear elliptic equations, pLaplacian, Degenerate elliptic equations, Free boundary problems, Bernoullitype free boundary problems 
Divisions:  Faculty of Science and Health > Mathematical Sciences, Department of 
Depositing User:  Elements 
Date Deposited:  23 Jul 2021 07:09 
Last Modified:  23 Jul 2021 08:15 
URI:  http://repository.essex.ac.uk/id/eprint/25832 
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