Nonlinear oblique boundary value problems for Hessian equations in two dimensions

Abstract

We study nonlinear oblique boundary value problems for nonuniformly elliptic Hessian equations in two dimensions. These are equations whose principal part is given by a suitable symmetric function of the eigenvalues of the Hessian matrix D2u of the solution u. An interesting feature of our second derivative estimates is the need for certain strong structural hypotheses on the boundary condition, which are not needed in the uniformly elliptic case. Restrictions of this type are natural in our context; we present examples showing that second derivative bounds may fail if we do not assume such conditions.