Mathematical Sciences: Symmetry for PDE, Quantitative Properties of Solutions of PDE, and Unique Continuation

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Three projects will be the focus of mathematical work done on problems arising in the theory of nonlinear partial differential equations. The first is concerned with questions related to symmetry in overdetermined boundary value problems. The symmetry occurs in solutions of certain equations in which the existence of positive solutions implies that the domain is a ball and the solution is radially symmetric. Work will be done examining the degree to which the positivity assumption may be dropped while one can still infer symmetry of the domain. Related to this investigation are questions concerning averages of functions over a fixed set as the set is subject to rigid motions through space. If, on assuming that the averages are zero, the function must be zero, one says that the Pompeiu property holds. The problem of deciding the validity of the property is equivalent to showing the existence of solutions of the eigenvalue problem for the Laplacian. Work will be done in looking for geometric properties of sets which complement this analytic result. The second project concerns questions from potential theory in which knowledge of quantitative properties of solutions of the relevant operator are sought. One particular issue is the problem of giving geometric conditions on the boundary of a domain which characterize the regular points for the heat operator. Related work will consider conditions on the boundary from which one may measure the extent of nontangential limits of solutions. In the third project, work will concentrate on a new approach to uniqueness properties of elliptic and non-elliptic operators that is not based on the classical Carleman method. Recent studies have concentrated on operators containing unbounded lower order terms. The object is to determine when solutions of the homogeneous equation which equal zero on an open set must equal zero everywhere. A 1939 result of Carleman has influenced all subsequent results in this area. New discoveries using a blend of geometric and variational ideas will be employed to extend the present theory to cover larger classes of operators.