Mathematical Sciences: Bergman Space Inequalities

小行星9401498 该奖项支持专注于积分不等式发展的数学研究，这些不等式提供了关于椭圆型偏微分方程解的内部光滑性如何由其边界函数的大小控制的精确信息。 具体而言，该项目涉及的努力，以找到必要和充分的条件对权重，使不等式之间的衍生物的解决方案偏微分方程和他们的边界值举行。 也就是说，这些功能的各种权力之间的积分。 这些不等式通常被称为双范数不等式，由于几个原因而有用。 一个是它们给出了近似解如何收敛到正确解的信息。 在这项工作中，基础域将有相对课程的边界，迫使域的几何形状在分析中发挥重要作用。 这些不等式的扩展被称为双调和算子也是预期的。 积分不等式传统上在微分方程的研究中起着中心作用。 它们给出了关于解和导数的平均值的关键信息，通常提供没有具体形式（除了数值估计）的存在结果。 ***

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. ***