A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions

¹ Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido 060-0810, Japan.
² Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan.

^* Corresponding author: qingliu@fukuoka-u.ac.jp

Received: 18 March 2019
Accepted: 27 November 2019

Abstract

This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.