Newtonian potential theory for unbounded sources and applications to free boundary problems (Q1357381)

The authors study comprehensively the most important inverse problems in the Newtonian potential theory in \(\mathbb{R}^n\). They formulate this problem in the following terms: Let \(D\) be an open set in \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), \(n\geq 2\), and let \(\sigma\) be a positive, bounded function in \(\mathbb{R}^n\). Given a harmonic function \(v\) in \(D\), find a function \(f\) and an open set \(\Omega\) satisfying the system: \[ \Delta f(x)=-\sigma(x) \chi_\Omega(x),\quad x\in\mathbb{R}^n,\quad f(x)= v(x),\quad x\in D\setminus\Omega; \] \(\Delta\) is the Laplace operator in the distributional sense and \(\chi_\Omega\) is the characteristic function of \(\Omega\). The main object of the paper is the regularity of the free boundary \(\Gamma=\partial\Omega\). The authors present results of other authors, their own ones from the last 15 years, compare and comment them and show their interconnections. The paper contains five sections, where the authors present their most important results.