Mathematical Sciences: Harmonic Analysis and Partial Differential Equations

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This project continues mathematical research leading to the development of various aspects of harmonic analysis with a view toward applications to problems in linear and nonlinear partial differntial equations. The main emphasis will be on the study of elliptic boundary value problems on domains with rough boundaries. More precisely, the boundaries are not required to have a continuously turning tangent. They may have corners but not cusps. These are effectively the minimal conditions one might impose in physical representations where imprecise measurements always lead to a loss of smoothness. Results on homogeneous equations in any dimension are considered well developed. Not so are the inhomogeneous problems. In fact, in two dimensions there is a Dirichlet problem for which the fundamental estimate comparing the gradient with the boundary values does not hold. Work will now be done on the corresponding inhomogeneous Neumann problem to see if the estimates can be obtained for functions in appropriate Sobolev spaces. A second line of research, more geometric, concerns the question of estimating a Riemannian metric in terms of the spectrum of it s associated Laplacian. In particular, work will be done in estimating the metric of a three-manifold in terms of its spectral data with no restriction on the manifold's conformal class. This will be done by constructing a function whose quadratic norm dominates the Riemannian measure. The construction requires the solution of a specific differential inequality on the manifold. Partial differential equations form the backbone of mathematical modeling in the physical sciences. Phenomena which involve continuous change such as that seen in motion, materials and energy are known to obey certain general laws which are expressible in terms of the interactions and relationships between partial derivatives. The key role of mathematics is not to state the relationships, but rather, to extract qualitative and quantitative meaning from them.