Mathematical Sciences: Harmonic Analysis and Elliptic PDE

9401081 piphher该奖项支持在谐波分析和椭圆偏微分方程三个领域出现的问题的数学研究。第一类是关于非光滑域上的高阶椭圆算子。尽管在非光滑域上定义的方程已经做了大量的工作，但仍然需要了解规定边界数据和获得唯一解的最优空间。第二组问题是二阶椭圆理论，部分原因是与加藤平方根问题有关的例子。求解某些复杂椭圆方程的正则性问题是其根本目标，而求解一维方程的正则性问题目前还没有完全解决。一般来说，如果没有严格的对称假设，我们知道的就不多。第三个项目涉及多参数扩张族下的算子不变量。这些操作符可以被视为产品类型操作符。为了得到简可积函数上算子的锐利估计和界，我们需要一个不同的观点。虽然一些工作已经完成了对限制族的扩张，但理解具有一般k参数扩张的算子的有界性问题。偏微分方程是物理世界数学建模的基础。数学分析的作用与其说是创建方程，不如说是提供关于解的定性和定量信息。这可能包括回答关于独特性、平滑性和增长性的问题。此外，分析常常发展出解的近似方法和对这些近似精度的估计。＊＊＊

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. ***