Hopf Formulas for Nonlinear Obstacle Problems
Keywords:
Differential games, stopping times, Hopf formula, double obstacle.Abstract
A Hopf formula is derived for max{ut+H(Du), h(t, x)−u} = 0, u(T, x) = g(x) ≥ h(t, x), where g is assumed convex and x 7→ 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 g1(x) ≤ g2(x) are given functions, both convex or both concave. The nonlinear double obstacle variational inequality max{min{ut +H(Du), g2 −u}, g1 −u} = 0 on (−∞, T)×R n, with terminal data either g2 in the convex case and g1 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.
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