Mathematical Sciences: Harmonic Analysis and Elliptic PDE

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9401081 Pipher This award supports mathematical research on problems arising in three areas of harmonic analysis and elliptic partial differential equations. The first concerns higher order elliptic operators on non-smooth domains. Although considerable work has already been done on equations defined on non-smooth domains, it remains to understand the optimal spaces for prescribing boundary data and obtaining unique solutions. The second set of questions lies in second order elliptic theory, motivated partially by examples connected with the square root problem of Kato. The question of solving the regularity problem, which is the essential goal, for certain complex elliptic equations has only been completely solved for the one-dimensional equation. In general, without heavy symmetry assumptions, not much is know. The third project concerns operators invariant under a multiparameter family of dilations. These operators may be viewed as product-type operators. To obtain sharp estimates and bounds for the operators on functions which are simply integrable, one needs a different point of view. While some work has been accomplished on restricted families of dilations, the issue of understanding the boundedness of such operators with general k-parameter dilations. Partial differential equations form a basis formathematical modeling of the physical world. The role of mathematical analysis is not so much to create the equations as it is to provide qualitative and quantitative information about the solutions. This may include answers to questions about uniqueness, smoothness and growth. In addition, analysis often develops methods for approximation of solutions and estimates on the accuracy of these approximations. ***