## Regularity properties of a free boundary near contact points with the fixed boundary.(English)Zbl 1050.35157

This paper concerns the regularity properties of solutions of a free boundary problem. Let $$\Omega$$ be a domain in the half space $$\mathbb{R}^n_+= \{x\in\mathbb{R}^n: x_1> 0\}$$, $$n\geq 2$$, and suppose that a function $$u$$ satisfies $\Delta u=\chi_\Omega\quad\text{in}\quad B^+,\quad u=|\nabla u|=0\quad\text{in}\quad B^+\setminus\Omega,\quad\text{and}\quad u= 0\quad\text{on}\quad \Pi\cap B,$ where $$B$$ is the unit ball with center at the origin, $$B^+= \{x\in B: x_1> 0\}$$, $$\Pi= \{x_1= 0\}$$, and $$\chi_\Omega$$ denotes the characteristic function of $$\Omega$$. The free boundary $$\Gamma(u)$$ is defined by $\Gamma(u)= \{x: u(x)=|\nabla u(x)|= 0\}\cap \partial\Omega.$ The main purpose of this paper is to study the behavior of the free boundary near the contact points $$\Gamma_0(u)= \Gamma(u)\cap\Pi$$ with neither $$u\geq 0$$ nor Lipschitz regularity assumption on $$\partial\Omega$$. The authors show, among other things, that the free boundary is the graph of a $$C^1$$ function near the contact points. The interior points of the free boundary have been well studied by L. A Caffarelli, L. Karp and H. Shahgholian [Ann. Math. (2) 151, 269–292 (2000; Zbl 0960.35112)].

### MSC:

 35R35 Free boundary problems for PDEs 35J60 Nonlinear elliptic equations 35B65 Smoothness and regularity of solutions to PDEs

Zbl 0960.35112
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### References:

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