2026/07/19 by Antonio Farah
#math.AP
We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form wΔw = h(∇ w). We assume the existence of a star-shaped domain D such that h < 0 in D, h = 0 on ∂ D, and h > 0 in Dc. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation Δu = f(u), and f behaves like -γu-(γ+ 1) for some γ∈ (0,2). In this case, known as the Alt-Phillips equation for negative power potentials, h(ρ) = c(|ρ|2 - 1). We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as γ converges to 2, the free boundary converges to a minimal surface.