Mathematical Sciences: Partial Differential Equations Under Minimal Smoothness Conditions

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This award supports mathematical research in the area of partial differential equations. The work concerns boundary value problems in domains with relatively coarse boundaries, that is boundaries on which discontinuities in the derivative are allowed such as those with corners. Both elliptic and parabolic equations will be treated. Although considerable work has been done in the area in the past few years, little attention has been given to mixed boundary value problems. In particular attention will be given to the question of regularity: if the boundary components are known to be integrable of a certain order then one would like to establish the best power integral of the gradient. A second line of work focuses on spectral properties of elliptic operators. Here the problem is that of locating spectral resonances in domains which consists of a cavity joined to an unbounded exterior domain by a thin tube. The point of this type of investigation is to quantify the extent to which the operator may be modelled by separate elliptic operators on the individual parts. Finally, work will continue on questions involving inverse boundary value problems. In its more practical setting, one wants to know the extent to which quantities such as charge on the surface of a body uniquely determines current across the surface. The goal of this work is to look at cases where the underlying differential operator may have nonsmooth coefficients. Partial differential equations form a basis for mathematical 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.