Mathematical Sciences: Partial Differential Equations, GroupRepresentations and Number Theory

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Ehrenpreis 8805909 This project will encompass three areas of mathematical analysis, each of which represents a continuation of earlier research. The first concerns the extensions of solutions of partial differential equations from domains which have the shape of convex wedges to surrounding domains when the wedges share common boundary elements. In its general form the goal is to determine the relations among the Cauchy data on the boundary which imly the extendability of solutions. Solutions have been obtained for constant coefficient equations in low dimentions. Work will also be done on problems of scattering theory. A recent concept of nonabelian mechanics in which the poisson bracket of space variables makes the space and momentum into seperate Lie algebras (rather than vanishing), when suitably paired. This approach leads to a new notion of completely integrable systems which can be applied to scattering problems on nonabelian groups. Work planned seeks to develop this research in higher dimensions. Efforts will also continue on the question of determining when a left-invariant operater on a Lie group can be surjective on the infinitely differentiable functions. This work derives from earlier research on the so-called Lewy operater - a partial differential equation with no solutions. Initial results using the Husenberg group suggest that a general framework for deciding on the surjectivity of differential operators and hence the conditions under which one can expect solutions of the corresponding inhomogeneous equations.