NONLINEAR OBLIQUE BOUNDARY-VALUE-PROBLEMS FOR HESSIAN EQUATIONS IN 2 DIMENSIONS

We study nonlinear oblique boundary value problems for nonuniformly el liptic Hessian equations in two dimensions, These are equations whose principal part is given by a suitable symmetric function of the eigenv alues of the Hessian matrix D(2)u Of the solution u, An interesting fe ature of our second derivative estimates is the need for certain stron g structural hypotheses on the boundary condition, which are not neede d in the uniformly elliptic case, Restrictions of this type are natura l in our context; we present examples showing that second derivative b ounds may fail if we do not assume such conditions.