Mathematical Sciences: Subelliptic Partial Differential Equations and Harmonic Analysis

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Work on this project will address several questions concerning subelliptic partial differential equations, especially aspects of analytic hypoellipticity. Related work on the representations of nilpotent Lie groups and harmonic analysis will also be carried out. A differential operator is said to be hypoelliptic if the smoothness of a solution of the corresponding inhomogeneous equation is related to that of the inhomgeneous term (often referred to simply as the right-hand-side of the equation). By analytic, one means that analytic solutions are guaranteed when the inhomogeneous term is analytic. A few cases where analytic hypoellipticity for elliptic operators occurs are known and a few where it fails are also known. The vast middle ground is untouched and represents the main theme of this research. Additionally, in those cases where hypoellipticity fails, it remains to determine exactly what order of smoothness one can expect to possess and to relate it to underlying geometric invariants of the domain on which the operator is defined. The work begins in three-dimensional Euclidean space, but has natural extensions to similar question on nilpotent groups like the Heisenberg group, as well as to compact manifolds with boundary. This research has many deep connections with the foundations of partial differential equations. On the surface it appears highly abstract and distant from the physical world. In practice, the fundamental results derive from the study of specific examples which occur in various applications. To give, a priori, information about the solution of a complex partial differential equation can be of immense value to any user of mathematics.