RESEARCH ON THE THEORY OF VISCOSITY SOLUTIONS OF DIFFERENTIAL EQUATIONS AND ITS APPLICATIONS

以研究微分方程黏性解的理论及其应用为主题，研究了边值问题的黏性解、弱KAM理论、黏性解的正则性、优化问题、微分方程的几种渐近问题、相边界的曲率流动和运动、质量输运问题、工程和经济问题。基于之前所做的调查，我们已经成功地获得了许多新的观察，上面列出的每一个主题。我们对弱KAM理论中的Aubry集及其在渐近问题中的应用的研究是有意义的。

On the theme of researching the theory of viscosity solutions of differential equations and its applications, we investigated viscosity solutions of boundary value problems, weak KAM theory, regularity of viscosity solutions, optimizations problems, several kinds of asymptotic problems in differential equations, curvature flows and motions of phase boundaries, mass transportation problems, problems in engineering and economics. Based on the investigations done before, we have succeeded to obtain many, new observations on each of subjects listed above. Our contributions to research on Aubry sets in weak KAM theory and its application to asymptotic problems are significant.