Joint Study on Viscosity Solutions and Their Applications

研究了一阶和二阶偏微分方程粘性解的基本性质及其应用。特别地，我们在曲面演化的水平集方法和微分方程确定性最优控制的基本理论和应用方面得到了几个结果。分析了计算机芯片制造中腐蚀的一些数学模型，表明水平集方法为研究这些模型提供了正确的途径，从而证明了以前的数值计算是正确的。这项研究是由H.Ishii和L.C.Evans共同完成的。对于周期函数为正弦函数的半线性抛物型偏微分方程解，在适当的标度下，证明了在适当的尺度下，这些方程的解覆盖到每个水平集通过平均曲率运动演化的函数。这一结果是由H.Ishii.3获得的。我们统一了本斯、梅里曼、奥舍算法和门限增长动态…Griffeath等人对门限动力学的更多模拟，并将这种门限动力学应用于各向异性平均曲率运动和曲率无关运动的近似格式。这项研究是由H.Ishii、G.E.Pir**和P.E.Souganidis.共同完成的。我们证明了有限维空间中Hamilton-Jacobi方程的Ergoc问题的通常形式不足以处理无限维空间中的遍历问题。这项研究是由M.Arisawa、H.Ishii和P.L.Lions共同完成的。我们研究了界面能不光滑时曲线的表面能驱动运动，并将粘性解理论推广到包括晶态能在内的非光滑能上。这是由Y.Giga和M.-H.Giga.7完成的。我们研究了状态约束的微分对策，证明了当边界条件被适当解释时，值函数是Bellman方程的唯一连续粘性解。这是由M.Bardi、S.Koike和P.Soravia共同完成的。较少

We studied basic properties and their applications of viscosity solutions of first and second order partial differential equations. In particular, we obtained several results in the fundamental theory and applications of level set approach to evolutions of surfaces and deterministic optimal control of differential equations.1. We analyzed some of mathematical models of etching in manufacturing of computer chips, showed that the level set approach gives the right way to study these models, and thus justified the former numerical computations. This research was done jointly by H.Ishii and L.C.Evans.2. We showed for semilinear parabolic partial differential equations with periodic functions like the sine function as nonlinear term that under appropriate scalings solutions of these equations coverge to those functions of which every level sets evolve by mean curvature motion. This result was obtained by H.Ishii.3. We unified the Bence, Merriman, Osher algorithm and the threshold growth dyn … More amics by Griffeath and others to the threshold dynamics and applied this threshold dynamics to yield approximation schemes for anisotropic mean curvature motions and curvature-independent motions. This research was done jointly by H.Ishii, G.E.Pir** and P.E.Souganidis.4. We showed that the usual formulation of ergodc problems for Hamilton-Jacobiequations in finite-dimensional spaces is not sufficient to treat the ergodic problems in infinite-dimensional spaces. This research was done jointly by M.Arisawa, H.Ishii, and P.L.Lions.5. We studied surface energy-driven motion of curves when the interface energy is not smooth and extended the theory of viscosity solutions to cover the motion by the nonsmooth energy including crystalline energy. This was done by Y.Giga and M.-H.Giga.7. We studied differential games with state constraints and showed that the value function is a unique continuous viscosity solution of the Bellman equation when the boundary condition is appropriately interpreted. This was done jointly by M.Bardi, S.Koike, and P.Soravia. Less