Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

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Abstract Chanillo 9623079 The project consists of ten problems, all proposed in the field of Partial Differential Equations(PDE). Eight of the proposed problems pertain to questions arising in non-linear PDE, and two problems are proposed to study over-determined systems of linear partial differential equations. The non-linear problems all arise from various situations that arise in Physics and Geometry. There is a stronger focus in the proposal on systems of PDE than on single equations. The problems are selected in part because solutions to them and understanding them require some understanding of classical Harmonic analysis. Very often the behaviour of a physical system is described by partial differential equations. Thus it is important to know how to solve these equations. Very often there are many, many solution families and thus one may really not know all the possible solutions. However if by some technique we can cut down the list of solutions and show that only solutions of a certain category are the the only possibility then the task of determining all solutions become easier. For example if an equation exhibits symmetry one could try to show that all solutions of the equation will have this symmetry manifest. This will automatically rule out large classes of objects and focus our study on a small class which will then tell us everything about the equation. Recently there has been a great surge of interest in solutions of a type of equation called the Ginzburg-Landau equation (GL equations), named after the Russian physicists, Ginzburg and Landau who first used them in theories of superconductivity in the 1950's. The GL equations also appear in string theories. A major part of our project is devoted to classiying all the solutions of the Ginzburg-Landau equation.