This 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.
