On a Free Boundary Problem

This paper considers a discontinuous semilinear elliptic problem: \ -\Delta u=g(u)H(u-\mu )\quad \text{in }\Omega,\qquad u=h\quad \text{on }% \partial \Omega, \] −Δu=g(u)H(u−μ) in Ω, u=h on ∂Ω, where H is the Heaviside function, μ a real parameter and Ω the unit ball in ℝ². We deal with the existence of solutions under suitable conditions on g, h, and μ. It is shown that the free boundary, i.e. the set where u=μ, is sufficiently smooth.