Singularities in Boundary Value Problems with Applications to Mathematical Physics

边值问题中的奇异性及其在数学物理上的应用/ / / / / / / / / /一是非光滑和奇摄动域数学物理边值问题的谱和渐近分析及其边界约简（如相应的边界积分方程）。特别地，我们将导出线性和非线性边值问题在最小系数假设下的解的显式渐近表示。第二个主题是描述和分析具有奇异性的边值问题的多层数值方法。基于求解边值问题和相关积分方程的一种新的近似方法，导出了一类新的具有奇异性的积分算子的非解析型公式。弹性、电磁场理论、声学、气动和流体力学等许多问题的解都具有由非光滑系数和边界、快速振荡和数据不连续引起的奇异性。本研究项目的目标是获得描述数学物理中某些问题解的重要性质的结果，特别是它们的奇点。在这个项目中发展的技术涉及到泛函分析、偏微分方程和数值分析中的问题，并将有助于这些问题的解决。许多要调查的主题应该是学生感兴趣的，并且它的目的是让一些学生参与这个项目。

Singularities in Boundary Value Problems with Applications to Mathematical PhysicsVladimir Maz'yaOhio StateUniversity AbstractThis project involves research in two areas of the analysis of partial differential equations. One is the spectral and asymptotic analyses of boundary value problems of mathematical physics in non-smooth and singularly perturbed domains as well as their boundary reductions (e.g. to the corresponding boundary integral equations). In particular, explicit asymptotic representations will be derived for solutions of linear and nonlinear boundary value problems under minimal assumptions on the coefficients.A second topic is the description and analysis of multilevel numerical methods for boundary value problems with singularities. New classes of non-analytic cubature formulas for integral operators with singularities will be derived based on a new approximation method for solving boundary value problems and associated integral equations.Many problems of elasticity, electromagnetic field theory, acoustics, aero- and hydrodynamics possess solutions with singularities caused by non-smooth coefficients and boundaries, fast oscillations and discontinuities of the data. The goal of this research project is to obtain results that describe the significant properties of the solutions of some problems in mathematical physics - especially their singularities. The techniques to be developed in this project involve issues in, and will contribute to, functional analysis, partial differential equations and numerical analysis. Many of the topics to be investigated should be of interest to students and it is intended to involve a number of students in the project.