Regularity of a free boundary problem
Define \(v\) to be a Newtonian potential: \[ v(x)=\int \limits_\Omega E_n(x-y) f(y)dy, \] \noindent where \[ E_n=\begin{cases} \frac 1 {2 \pi} log |x |\quad \quad \quad n=2, \\ \frac 1 {(2-n)\omega _n}|x |^{2-n} \quad n \geq 3, \end{cases} \] \(\omega _n\) is the area of the unit sphere in \(\mathbb{R}^n\) and \(\Omega\) is a bounded open set in \(\mathbb{R}^n\). The authors assume that \(f\) is a real analytic function in some neighbourhood of \(x_0 \in \partial \Omega\) and \(v\) is analytically continuable from the complement of \(\Omega\) in some neighbourhood of \(x_0\). The main problem is to study under which conditions one can conclude that \(\partial \omega \cap B_\rho (x_0) (0<\rho \leq r)\) is a real analytic surface.
scientific article; zbMATH DE number 1542107
scientific article