Nonlinear Neumann Boundary Conditions for Quasilinear Degenerate Elliptic Equations and Applications

We prove comparison results between viscosity sub- and supersolutions of degenerate elliptic and parabolic equations associated to, possibly nonlinear, Neumann boundary conditions. These results are obtained under more general assumptions on the equation (in particular the dependence in the gradient of the solution) and they allow applications to quasilinear, possibly singular, elliptic or parabolic equations. One of the main applications is the extension of the so-called level set approach for equations set in bounded domains with nonlinear Neumann boundary conditions. In such a framework, the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary.