Mathematical Sciences: Bergman Space Inequalities

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9401498 Wilson This award supports mathematical research focusing on the development of integral inequalities which give precise information on how the interior smoothness of solutions of elliptic partial differential equations is controlled by the size of their boundary functions. Specifically, the project concerns efforts to find necessary and sufficient conditions on pairs of weights so that inequalities between derivatives of solutions of partial differential equations and their boundary values hold. That is, between the integral of various powers of these functions. These are usually referred to as two-norm inequalities and are useful for several reasons. One is that they give information on how approximate solutions can be expected to converge to the correct one. In this work, the underlying domains will have relatively course boundaries, forcing the geometry of the domain to play a fundamental role in the analysis. Expansion of these inequalities to what are called biharmonic operators is also expected. Integral inequalities have traditionally played a central role in the study of differential equations. They give criticalinformation about average values of solutions and derivatives, often providing existence results where no concrete form (except for numerical estimates) is possible. ***