Some Comparison Results for First-Order Hamilton-Jacobi Equations and Second-Order Fully Nonlinear Parabolic Equations with Ventcell Boundary Conditions

Abstract

International audienceIn this article, we consider fully nonlinear, possibly degenerate, parabolic equations associated with Ventcell boundary conditions in bounded or unbounded, smooth domains. We first analyze the exact form of such boundary conditions in general domains in order that the notion of viscosity solutions makes sense. Then we prove general comparison results, both for first- and second-order equations, under rather natural assumptions on the nonlinearities: $(i)$ in the second-order case, the only restrictive assumption is that the equation has to be strictly elliptic in the normal direction, in a neighborhood of the boundary; $(ii)$ in the first-order one, quasiconvexity assumptions have to be imposed both on the equation and the boundary condition, the equation being coercive in the normal direction. Our method is inspired by the ``twin blow-up method'' of Forcadel-Imbert-Monneau, that we adapt to a scaling consistent with the Ventcell boundary condition