Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions

Abstract

In this article, we are interested in viscosity solutions for second-order fully nonlinear parabolic equations having a $L^1$ dependence in time and associated with nonlinear Neumann boundary conditions. The main contributions of our study are, not only to treat the case of nonlinear Neumann boundary conditions, but also to revisit the theory of viscosity solutions for such equations and to extend it in order to take in account singular geometrical equations.In particular, we provide comparison results, both for the cases of standard and geometrical equations, which extend the known results for Neumann boundary conditions even in the framework of continuous equations.