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

TL;DR

This paper investigates comparison principles for fully nonlinear parabolic equations with Ventcell boundary conditions, establishing conditions under which viscosity solutions are well-defined and unique in various domains.

Contribution

It extends the theory of viscosity solutions to include Ventcell boundary conditions for both first- and second-order equations, with new comparison results under natural assumptions.

Findings

Comparison results for second-order equations with strict ellipticity near the boundary.

Comparison results for first-order equations with quasiconvexity and coercivity.

Adaptation of the twin blow-up method to Ventcell boundary conditions.

Abstract

In 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.