Techniques in Linear and Nonlinear Partial Differential Equations

The sliding method has been introduced, and the method of moving planes has been greatly extended and improved. These methods have been applied to a number of problems for fully nonlinear second order elliptic equations 1 to treat symmetry and monotonicity of solutions in domains - even with nonsmooth boundaries 2 prove existence, uniqueness, monotonicity of traveling waves in a cylinder, for problems arising in combustion theory and in biology, including extensions to higher dimension of work of Kolmogoroff, Petrovsky, Piskounoff. Separate work studied flame propagation as a limit problem leading to a free boundary problem. Regularity of the free boundary has been studied by methods which should prove useful in other problems. Work on statistical mechanics, on deriving hydrodynamic scaling limit equation of Ginzburg-Landau. A rigorous proof was given of the Euler equations for conservation laws from Hamiltonian systems. The question of existence of breathers, existence of time-periodic solutions of hyperbolic equations on the line, was studied. Nonexistence of certain solutions depending analytically on a parameter was shown. Formal solutions and various estimates were established. Variational methods have been used for nonlinear equations, heat flow methods have been used in the study of harmonic maps.