CAREER: Analysis of Partial Differential Equations in non-smooth media

The present project is focused on the impact of irregular geometry and internal medium inhomogeneity on the behavior of solutions to fundamental boundary value problems. Despite over two centuries of intensive research and a massive body of results, today partial differential equations on domains are thoroughly understood only when both the coefficients and the domain exhibit smoothness. However, the dominance of irregularities in real life and numerous problems in modern mathematics demand for an extension of the classical 'smooth' theory. Three major directions for this extension pursued by the PI are: (a) the study of boundary problems with rough data on non-smooth domains; (b) the second-order elliptic operators with non-smooth coefficients; (c) elliptic PDEs of higher order. The scope of the proposed work incorporates a full range of sharp well-posedness and regularity results, asymptotic formulas, introduction and analysis of new measures, capacities and new function spaces, numerical experiment, localization phenomena.

Given the ubiquitousness of irregularities in nature and in man-made mechanisms, their understanding is essential to science, engineering, industry, and to society in general. Indeed, most classical models only provide a nice, smooth approximation of the objects and processes. However, perfectly uniform smooth systems do not exist in nature, and every real object inadvertently possesses singularities (a sharp edge of the boundary, an abrupt change of the medium, a defect of the construction). They often have a decisive effect on the occurring phenomena, but cannot be captured in a smooth set-up (e.g., the fractal structure of lungs). The present project combines the mathematical pursuit and a rigorous educational program towards the general goal of advancing analysis of partial differential equation in non-smooth media and preparing a diverse array of mathematicians working in related areas. The latter objective will, in particular, be achieved through a novel Workshop for Women in Analysis and PDE.