Joint Study on Viscosity Solutions and Their Applications

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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