Qualitative behavior of solutions of elliptic free boundary problems (Q1204817)

The purpose of this important paper is to study a question concerning the general free boundary problem for the following situation: Find a curve \(\Gamma\) and a function \(u\in C^ 2(\Omega)\cap C^ 1(\Omega\cup\Gamma)\cap C^ 0(\overline\Omega)\) such that \[ Qu=0\quad\text{ in } \quad\Omega,\qquad u=1\quad\text{ on } \quad\Gamma^* \] and, for a fixed \(\lambda>0\) \[ u=0,\qquad |\nabla u|=\lambda\quad \text{ on } \quad\Gamma, \] where \(Q\) is the second- order elliptic operator given by \[ Qu\equiv au_{xx}+2bu_{xy}+cu_{yy}\quad \text{ in } \quad \Omega, \] \(a,b,c\) depend on \(x,y,u_ x\), and \(u_ y\) and \(ac-b^ 2>0\), \(\Omega=\Omega(\Gamma^*,\Gamma)\) is the region between \(\Gamma^*\) and \(\Gamma\), \(\Gamma^*\) is a given Jordan curve in \(\mathbb{R}^ 2\) and \(\Gamma\) is a Jordan curve in \(\mathbb{R}^ 2\) which surrounds \(\Gamma^*\). The qualitative behavior of the free boundary is compared with that of the fixed boundary. Main results: Using curves of constant gradient direction \((\Gamma\) and \(u\) constitute a solution of this free boundary problem), the geometry of the free boundary \(\Gamma\) is compared with the geometry of the fixed boundary \(\Gamma^*\). In particular, \(\Gamma\) is shown to have a simpler geometry than \(\Gamma^*\) does.