Least supersolution approach to regularizing free boundary problems (Q1000559)

In the interesting paper under review, the author studies a free boundary problem obtained as a limit for \(\varepsilon \to 0\) of the regularizing family of semilinear elliptic equations \(\Delta u = \beta_\varepsilon(u) F(\nabla u)\), where \(\beta _\varepsilon\) approximates the Dirac delta function at the origin and \(F\) is a Lipschitz continuous function bounded away from \(0\) and infinity. The least supersolution approach is used to construct solutions satisfying geometric properties of the level surfaces uniformly in \(\varepsilon.\) This allows to prove that the free boundary of a limit has the ``right'' (in the measure theoretical sense) weak geometry. A classification of the global profiles of the blow-up analysis is carried out by means of suitable barriers with curvature, and the limit functions are proven to be viscosity and pointwise solution (\({\mathcal H}^{n-1}\) almost everywhere) to the free boundary problem. Finally, it is proved that the free boundary is a \(C^{1,\alpha}\)-surface around \({\mathcal H}^{n-1}\)-almost everywhere point.