Study of viscosity solutions for partial differential equations with subdifferential

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The aim of present research is to extend the notion of viscosity solutions to multi-valued nonlinear partial differential equations (PDE's) with subdifferential and to investigate the existence, uniqueness and other properties of solutions. The theory of viscosity solutions has been applied to wide class of frilly nonlinear PDE's. Among them, the obstacle problem is an important class of applications of viscosity solutions. We have put emphasis of the research on following three points:1. We formulate various PDE's by subdifferential and apply the theory of viscosity solutions for it.2. We compare the formulation by viscosity solution with known ones and study an advantage for it.3. We prove solvability of various nonlinear PDE's and discuss the relation between the methods by viscosity solution.It is shown that the notion of viscosity solutions are extended for nonlinear second order PDE's with subdifferential whose typical example is obstacle problem, and unique, existence theorem and stability of the solution are proved. Moreover, we investigate convergence of Yoshida approximation for subdiffrential term.