Regularity estimates for some free boundary problems of obstacle-type

We study regularity estimates for solutions to implicit constraint obstacle problems and penalized boundary obstacle problems. We first prove regularity estimates for the solution and the free boundary in the classical stochastic impulse control problem. We show that the free boundary partial {u < Mu}, where Mu is the implicit constraint obstacle, can be decomposed into a union of regular points, singular points, and degenerate points with corresponding regularity and measure theoretic estimates. We then turn to generalizing our analysis to the fully nonlinear problem with obstacles admitting a general modulus of semi-convexity omega(r). We prove that solutions to the fully nonlinear stochastic impulse control problem are C^{omega(r)} up to C^{1,1}. Finally we turn our attention to study both nonuniform and uniform estimates for the penalized boundary obstacle problem, Delta^{1/2}u^{epsilon} =beta_{epsilon} (u^{epsilon}). We obtain sharp estimates for the solution in both the nonuniform and uniform theory.