Nonlinear oblique derivative problems for singular degenerate parabolic equations on a general domain (Q1881343)

The authors establish comparison and existence results for viscosity solutions of the problem \[ u_t+F(t,x,u,Du,D^2u)=0 \quad \text{in }(0,T)\times\Omega,\qquad B(x,Du)=0 \quad \text{on }(0,T)\times\partial\Omega \] with a suitable initial condition, where \(\Omega\subset {\mathbb R}^n\) is a bounded and \(C^1\)-smooth domain. The nonlinearity \(F(t,x,u,Du,D^2u)\) covers a wide class of singular degenerate parabolic equations which includes, for instance, the mean curvature flow equation. Regarding the fully nonlinear oblique derivative boundary condition, a typical example is one of the form \[ {{\partial u}\over{\partial\nu}}=a(x)| Du| ,\qquad a\in C^\infty(\overline\Omega;(0,1)) \] which appears as the interpretation of the standard capillary condition in the level set approach to motion of hypersurfaces with velocity depending on the normal direction and the curvature.