9225000 Serrin Work supported by this award will be done in three principal directions: the study of radial solutions of nonlinear elliptic equations; the asymptotic behavior and stability of nonlinear holonomic oscillators and the relation of Newtonian mechanics to invariance principles. In the first direction, emphasis will be placed on establishing existence or nonexistence of ground states, especially when the governing nonlinearities have algebraic growth both in the solution variable and its gradient. A ground state is a non-negative solution (not trivial) which tends to zero off compact sets. Work on holonomic oscillators, which arise particularly in Lagrangian mechanics, concerns the stabilizing effect of various damping modes and nonautonomous restoring forces. Particular attention will be paid to the effect of intermittently applying damping mechanisms, specifically, to determine necessary and sufficient conditions under which such damping causes transient effects to decay. Finally, the derivation of equations of motion from invariance principles continues earlier work of the same nature. It is expected that extensions to chemical mixtures and micropolar materials will be carried out. These are cases not well understood. 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. ***
