Regularity of solutions of obstacle problems for elliptic equations with oblique boundary conditions

Much has been written about various obstacle problems in the context of variational inequalities. In particular, if the obstacle is smooth enough and if the coefficients of associated elliptic operator satisfy appropriate conditions, then the solution of the obstacle problem has continuous rst derivatives. For a general class of obstacle problems, we show here that this regularity is attained under minimal smoothness hypotheses on the data and with a one-sided analog of the usual modulus of continuity assumption for the gradient of the obstacle. Our results apply to linear elliptic operators with Holder continuous coefficients and, more generally, to a large class of fully nonlinear operators and boundary conditions.