Regularity properties of a free boundary near contact points with the fixed boundary.
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 \textit{L. A Caffarelli}, \textit{L. Karp} and \textit{H. Shahgholian} [Ann. Math. (2) 151, 269--292 (2000; Zbl 0960.35112)].
scientific article; zbMATH DE number 1911421
scientific article