Minimax Theory and its Applications 05 (2020), No. 2, 181--198
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
[Abstract-pdf]
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 $\max\{\min\{u_t+H(Du),g_2-u\},g_1-u\}=0$ 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.
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