Mathematical Sciences: Partial Differential Equations in Nonsmooth Domains

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The mathematical work supported by this grant lies in the area of systems of partial differential equations. More specifically, boundary value problems in domains with nonsmooth boundaries for elliptic and parabolic systems and higher order equations will be investigated. In addition work on homogenization will be carried out. One of the particular problems of interest involves elliptic systems of elastostatics and parabolic systems with boundary data in the Lebesgue spaces. It is not known, in dimension four and greater, what range of power norms of boundary data will yield solutions. 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.