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 first 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 Hölder
continuous coefficients and, more generally, to a large class of fully nonlinear
operators and boundary conditions.
