Aspects of Subelliptic Partial Differential Equations

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Abstract Christ Research will be conducted on a variety of questions concerning subelliptic partial differential equations and related topics. Among these are (i) Local and global analytic hypoellipticity of sums of squares of vector fields satisfying the bracket condition, and in the d-bar Neumann problem. (ii) Global regularity in the d-bar Neumann problem and for the Bergman projection in the infinitely differentiable (but not finite type) category. (iii) Characterization of optimal exponents of Gevrey regularity for sums of squares of vector fields. (iv) Local solvability of partial differential operators with double characteristics and smooth coefficients. All these topics are related to certain eigenvalue problems that arise from limits and/or reductions of partial differential operators; key issues are whether eigenvalues exist, and the extent to which the limiting problems control the original operators. The emphasis will be on low-dimensional problems, and on those connected to complex analysis in several variables and to harmonic analysis on Lie groups. Since Isaac Newton's insight that the fundamental laws of nature should relate the rates at which observed quantities change, rather than describe those quantities directly, most basic physical laws have been formulated as partial differential equations. To understand such equations and their solutions has thus become one of the fundamental goals of mathematical research. Partial differential equations are rarely solvable by explicit formulas, except in very symmetric situations. On the other hand, in physical applications it is often useful to understand certain qualitative features of solutions: whether any solution exists (which in practice may mean that some situation is or is not physically viable), and what kinds of singularities, if any, solutions can exhibit (shock waves, vortices, and wave fronts are typical physical manifestations of mathematical singularities). These issues can often be understand in situations where formulas for the solutions cannot be constructed. However, partial differential operators vary greatly in character, and each type must be investigated on its own terms.