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

小行星9303363托雷斯 在这个项目上所做的工作将开发小波型分解的研究频谱和连续性的运营商。 特别地，将考虑具有粗糙符号的伪微分算子。 算子的分析将通过构建其光滑度与算子符号相匹配的分子来进行。 抛物空间的类似分解将被构造并用于研究其核具有非各向同性奇点的算子。 以前的工作离散空间和乘子算子将进一步发展。 工作也将在非光滑区域的边界元方法。 将采用奇异积分和层势技术。 要考虑的问题包括建设的Galerkin程序的牵引边值问题，从弹性静力学和不稳定凝固的移动边界问题。 为了帮助这项研究的应用，将努力限制的功能空间，允许有效的数值实现。 为此目的，在保持器空间中的层势方法和齐次型空间的分解技术的最新发展将被纳入该项目。 小波分析的应用奇异积分算子导致了新的见解，这些变换的连续性。 此外，小波分解的非常离散的性质允许在其数值近似意想不到的精度。 边界元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微分方程的数学处理的程度。 这些假设通常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. ***