Mathematical Sciences: Harmonic Analysis and Elliptic PDE

9401081 Pipher该奖项支持对调和分析和椭圆型偏微分方程式三个领域中出现的问题进行数学研究。第一类是关于非光滑区域上的高阶椭圆算子。尽管关于定义在非光滑区域上的方程的研究已经做了大量的工作，但对于确定边界数据和获得唯一解的最优空间的理解还有待进一步的研究。第二组问题是二阶椭圆理论，部分是由与加藤的平方根问题有关的例子引起的。对于某些复椭圆型方程的正则性问题，其根本目的是解决正则性问题，这一问题目前只对一维方程得到了完全解决。总体而言，如果没有严格的对称性假设，我们所知的并不多。第三个项目涉及多参数伸缩族下的不变算子。这些运营商可以被视为产品型运营商。为了得到简单可积函数上算子的精确估计和界，人们需要一个不同的观点。虽然关于有限伸缩族的研究已经完成了一些工作，但是关于理解这类算子在一般k参数伸缩族上的有界性的问题。偏微分方程式是建立物理世界数学模型的基础。数学分析的作用与其说是创建方程，不如说是提供有关解的定性和定量信息。这可能包括回答有关唯一性、平稳性和成长性的问题。此外，分析经常开发出近似解的方法和对这些近似的精度的估计。***

9401081 Pipher This award supports mathematical research on problems arising in three areas of harmonic analysis and elliptic partial differential equations. The first concerns higher order elliptic operators on non-smooth domains. Although considerable work has already been done on equations defined on non-smooth domains, it remains to understand the optimal spaces for prescribing boundary data and obtaining unique solutions. The second set of questions lies in second order elliptic theory, motivated partially by examples connected with the square root problem of Kato. The question of solving the regularity problem, which is the essential goal, for certain complex elliptic equations has only been completely solved for the one-dimensional equation. In general, without heavy symmetry assumptions, not much is know. The third project concerns operators invariant under a multiparameter family of dilations. These operators may be viewed as product-type operators. To obtain sharp estimates and bounds for the operators on functions which are simply integrable, one needs a different point of view. While some work has been accomplished on restricted families of dilations, the issue of understanding the boundedness of such operators with general k-parameter dilations. Partial differential equations form a basis formathematical 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. ***