Applications to the optimal control and differential game via the viscosity solution theory

(1)得到了具有退化系数的Hamilton-Jacobi （HJ）方程下半连续（lsc）粘度解的唯一性和表示公式。这是Ishii和Ramaswamy结果的“lsc”版本。(2)证明了状态约束（简称SC）问题的追逃对策黏性解的唯一性和值函数的时间离散逼近的收敛性。这些结果推广了Alziary的结果。(3)证明了最小时间问题lsc黏度解的唯一性。为了证明Dirichlet型边界条件与SC型边界条件的等价性，我们一般证明了一个函数是相关PDE的lsc黏度解，当且仅当它满足动态规划原理。这扩展了p - l的相应结果。狮子用于连续粘度溶液。(4)构造了SC问题的ε-最优反馈控制。为了构造它们，我们使用来自相关HJ方程的信息。此外，我们需要中卷积近似、超微分的单调公式和小域SC问题的值函数的收敛性。(5)我们得到了包含梯度超线性增长项的全非线性二阶均匀椭圆偏微分方程粘度解的比较原理和(b)内部Holder连续性。(a)推广了Caffarelli、Crandall、Kocan和Swiech的结果，(b)推广了Caffarelli的结果。为了证明(b)，我们需要一个广义的Alexandroff-Bakelman-Pucci极大原理，一个相关极值偏微分方程经典超解的精确构造，以及一个修正的立方分解引理。

(1) We obtain the uniqueness and representation formula for lower semicontinuous (lsc for short) viscosity solutions of Hamilton-Jacobi (HJ for short) equations with degenerate coefficients.This is a "lsc" version of a result of Ishii and Ramaswamy.(2) We show the uniqueness of viscosity solutions of pursuit-evasion games of the state constraint (SC for short) problem and the convergence of time-discrete approximations to the value function. These results generalize those of Alziary.(3) We show the uniqueness of lsc viscosity solutions for the minimum time problem. To show the equivalence between the Dirichlet type boundary condition and the SC type one, we prove in general that a function is a lsc viscosity solution of the associated PDE if and only if it satisfies the Dynamic Programming Principle. This extends the corresponding result by P.-L. Lions for continuous viscosity solutions.(4) We construct ε-optimal feedback controls for the SC problem. To construct them, we use the information from the associated HJ equation. Furthermore, we need the inf-convolution approximations, the monotone formula for super-differentials and the convergence of value functions for SC problems of smaller domains.(5) We obtain (a) the comparison principle and (b) interior Holder continuity of viscosity solutions of fully nonlinear, second-order, uniformly elliptic PDEs which involve superlinear growth terms for the gradient. The result (a) extends that of Caffarelli, Crandall, Kocan and Swiech while the result (b) generalizes that of Caffarelli.To show (b), we need a generalized Alexandroff-Bakelman-Pucci maximum principle, a precise construction of classical supersolutions of the associated extremal PDEs, and a modified cube decomposition lemma.