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Hopf Formulas for Nonlinear Obstacle Problems

机译：霍普夫公式非线性障碍问题

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摘要

A Hopf formula is derived for max{u_t+H(Du), h(t,x)-u｝=0, u(T,x)=g(x)≥h(t, x), where g is assumed convex and x → 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) ≤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 （-∞,T）×?~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.
机译：霍普夫公式推导为马克斯{u_t + H (Du),t h (x) - u} = 0, u (t, x) = g h (x)≥(t, x), g在哪里假定凸和x→h (t, x)也凸。这个概括一个没有时间相关的公式由于Subbotin障碍。凹障碍也是派生。霍普夫公式的障碍问题quasiconvex g。双障碍问题。障碍g_1里面(x)≤g_2 (x)给出函数,两个凸或凹。障碍变分不等式终端数据要么g_2凸的情况和g_1里面凹有粘性解决方案由一种霍普夫公式。利用微分游戏公式推导停车时间。

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