
Minimax Theory and its Applications 05 (2020), No. 2, 181198 Copyright Heldermann Verlag 2020 Hopf Formulas for Nonlinear Obstacle Problems Emmanuel N. Barron Dept. of Mathematics and Statistics, Loyola University, Chicago, IL 60660, U.S.A. ebarron@luc.edu Robert Jensen Dept. of Mathematics and Statistics, Loyola University, Chicago, IL 60660, U.S.A. rjensen@luc.edu [Abstractpdf] A Hopf formula is derived for\\[1mm] \centerline{$\max\{u_t+H(Du),\,h(t,x)u\}=0,\quad u(T,x)=g(x) \geq h(t,x),$}\\[1mm] where $g$ is assumed convex and $x \mapsto h(t,x)$ is also convex. This generalizes a formula without time dependent obstacle due to Subbotin. A Hopf formula for a concave obstacle is also derived. In addition, the Hopf formula for the obstacle problem with quasiconvex $g$ is established. Next we consider the double obstacle problem. Assume the two obstacles $g_1(x) \leq g_2(x)$ are given functions, both convex or both concave. The nonlinear double obstacle variational inequality $\max\{\min\{u_t+H(Du),g_2u\},g_1u\}=0$ on $(\infty,T)\times \mathbb{R}^n$, with terminal data either $g_2$ in the convex case and $g_1$ in the concave case has a viscosity solution given by a Hopf type formula. These formulas are derived by using differential games with stopping times. Keywords: Differential games, stopping times, Hopf formula, double obstacle. MSC: 49K35, 49K45, 49L25, 49L20, 90C47. [ Fulltextpdf (140 KB)] for subscribers only. 