9400912 Nirenberg This award supports mathematical research on problems arising in the field of nonlinear partial differential equations. They include second order elliptic equations in general domains in Euclidean n-dimensional space as well as on manifolds. Regularity conditions will not be assumed and the differential operators are not necessarily self-adjoint. Other work will focus on free boundary problems, in particular singular limits for nonlinear problems and Liouville-type theorems for application to nonlinear boundary value problems. A second line of investigation involves statistical mechanics for wave equations. Specifically, statistical/mechanical probabilistic methods will be employed to study classical and non-classical nonlinear waves. A new (integrable) shallow water equation which has interesting connections with geodesic flow on creased manifolds of otherwise constant curvature. Partial differential equations form a basis for mathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***
