Study of viscosity solutions for partial differential equations with subdifferential

本研究的目的是将粘性解的概念推广到具有次微分的多值非线性偏微分方程解的概念，并研究解的存在唯一性等性质。粘性解理论已被广泛地应用于一类带褶皱的非线性偏微分方程组，其中障碍问题是粘性解的一类重要应用。重点研究了以下三点：1.利用次微分方法建立了各种偏微分方程组，并应用粘性解理论对其进行了求解。将粘性溶液配方与已知配方进行了比较，并研究了其优越性。证明了各种非线性偏微分方程解的可解性，并用粘性解讨论了各种方法之间的关系，推广了以障碍问题为典型例子的二阶非线性次微分偏微分方程粘性解的概念，证明了粘性解的唯一性、存在性和稳定性。此外，我们还研究了次微分项的Yoshida逼近的收敛问题。

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.