The free boundary for a semilinear non-homogeneous Bernoulli problem

Abstract

In the classical homogeneous one-phase Bernoulli-type problem, the free boundary consists of a “regular” part and a “singular” part, as Alt and Caffarelli have shown in their pioneer work (Alt and Caffarelli, 1981 [1]) that regular points are C1,γ in two-dimensions. Later, Weiss (1999) [34] first realized that in higher dimensions a critical dimension d⁎ exists so that the singularities of the free boundary can only occur when d⩾d⁎.In this paper, we consider a non-homogeneous semilinear one-phase Bernoulli-type problem, and we show that the free boundary is a disjoint union of a regular and a singular set. Moreover, the regular set is locally the graph of a C1,γ function for some γ∈(0,1). In addition, there exists a critical dimension d⁎ so that the singular set is empty if d<d⁎, discrete if d=d⁎ and of locally finite Hd−d⁎ Hausdorff measure if d>d⁎. As a byproduct, we relate the existence of viscosity solutions of a non-homogeneous problem to the Weiss-boundary adjusted energy, which provides an alternative proof to existence of viscosity solutions for non-homogeneous problems.