Mathematical Sciences: Singular Integral Operators, Decompostions of Function Spaces Related to Wavelets, and Boundary Elements in Nonsmooth Domains

9303363 Torres在这个项目上所做的工作将发展小波型分解，用于研究算子的谱和连续性质。具体地说，将考虑带有粗糙符号的伪微分算子。运算符的分析将通过构建其光滑度与运算符的符号相匹配的分子来进行。我们将构造抛物空间的类似分解，并用于研究其核具有各向同性奇点的算子。以前关于离散空间和乘子算子的工作将得到进一步发展。此外，还将研究非光滑区域中的边界元方法。将采用奇异积分和层势技术。要考虑的问题包括弹性静力学中的牵引力边值问题和不稳定凝固中的移动边界问题的Galerkin程序的构造。为了帮助这项研究的应用，将努力将函数空间限制在允许有效的数值实现的那些函数空间。为此，Holder空间的层势方法和齐型空间的分解技术的最新发展将被纳入到该项目中。小波分析在奇异积分算子中的应用使人们对这些变换的连续性有了新的认识。此外，小波分解的非常离散的性质允许在其数值逼近中有意想不到的精度。边界元方法研究的第二部分继续努力了解在没有标准光滑性假设的情况下可以在多大程度上进行微分方程式的数学处理。这些假设往往不能反映描述物理世界现象的问题中存在的条件。***

9303363 Torres Work done on this project will develop wavelet type decompositions for the study of spectral and continuity properties of operators. In particular, pseudodifferential operators with rough symbols will be considered. The analysis of the operators will be carried out through the construction of molecules whose smoothness matches that of the symbols of the operators. Similar decompositions for parabolic spaces will be constructed and used to study operators whose kernels have non- isotropic singularities. Previous work on discrete spaces and multiplier operators will be developed further. Work will also be done on boundary element methods in nonsmooth domains. Singular integral and layer potential techniques will be employed. The problems to be considered include the construction of Galerkin procedures for the traction boundary value problem from elastostatics and for moving boundary problems in unstable solidification. To aid in applications of this research, efforts will be made to confine the function spaces to those that permit efficient numerical implementation. For this purpose, the latest developments on the method of layer potentials in Holder spaces and decomposition techniques for spaces of homogeneous type will be incorporated into the project. The application of wavelet analysis to singular integral operators has led to new insights into the continuity properties of these transformations. In addition, the very discrete nature of the wavelet decompositions allows for unexpected accuracy in their numerical approximation. The second part of the research on boundary element methods continues efforts to understand the degree by which mathematical treatment of differential equations may be carried out in the absence of standard smoothness assumptions. These assumptions often do not reflect conditions present in problems describing phenomena taken from the physical world. ***