Joint Study on Viscosity Solutions and Their Applications
Project/Area Number  07044094 
Research Category 
GrantinAid for international Scientific Research

Allocation Type  Singleyear Grants 
Section  Joint Research 
Research Institution  Chuo University 
Principal Investigator 
ISHII Hitoshi Chuo University, Faculty of Science and Engineering Professor, 理工学部, 教授 (70102887)

CoInvestigator(Kenkyūbuntansha) 
LIONS P.L. Universite de Paris, Dauphine, CEREMADE, CEREMADE, 教授
SONER H.M. Carnegie Mellon University, Department of Mathematics, 数学科, 教授
SOUGANIDIS P.E. University of Wisconsin, Madison, Department of Mathematics, 数学科, 教授
GRANDALL M.G. University of California, Santa Barbara, Department of Mathematics, 数学科, 教授
EVANS L.C. University of California, Berkeley, Department of Mathematics, 数学科, 教授
小池 茂昭 埼玉大学, 理学部, 助教授 (90205295)
儀我 美一 北海道大学, 理学部, 教授 (70144110)
GIGA Y. Hokkaido University, Faculty of Science
KOIKE S. Saitama University, Faculty of Science

Project Period (FY) 
1995

Project Status 
Completed(Fiscal Year 1995)

Budget Amount *help 
¥2,100,000 (Direct Cost : ¥2,100,000)
Fiscal Year 1995 : ¥2,100,000 (Direct Cost : ¥2,100,000)

Keywords  Viscosity Solutions / Evolution of surfaces / Singular Perturbations / Optimal Control / ReactionDiffusion systems / HamiltonJacobi Equations / Degenerate Elliptic Equations / Degenerate Parabolic Equations 
Research Abstract 
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 curvatureindependent 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 HamiltonJacobiequations in finitedimensional spaces is not sufficient to treat the ergodic problems in infinitedimensional spaces. This research was done jointly by M.Arisawa, H.Ishii, and P.L.Lions. 5. We studied surface energydriven 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

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