Problems in Local and Fractional Elliptic Partial Differential Equations Involving Surfaces

The design of surfaces and shapes satisfying certain mechanical and geometrical properties is a problem of major importance in data science, industry, engineering and computer science. Computational-based algorithms to construct the so-called surfaces of minimum mean curvature variation have been playing a major role for the past 40 years in problems such as data interpolation, reconstruction of corrupted images, or the design of ship hulls and car surfaces. This project will provide the necessary mathematical foundation and tools for new, more efficient computational and artificial intelligence algorithms. Scientific models of climatology, plasma physics, population dynamics, spread of diseases, viscoelasticity and social media dynamics are described by mathematical equations involving long-range interactions or action at a distance. In these problems, the main feature is nonlocality: what happens far away has an instantaneous influence in the current state of the system. Moreover, in many cases, the nonlocal interactions have a distinct boundary, separation surface or interface (for example, the temperature over the surface of the ocean interacts with the temperature on the surface of the land, and both regions are separated through the beach, which is the common interface). This research will analyze the influence of these boundaries and interfaces in several nonlocal models, providing a systematic approach for imposing boundary conditions and precise analytical results describing the solutions. The outcomes will be crucial for computational implementation and reliable artificial intelligence models of climatology, population growth, and spread of diseases. The project includes opportunities for graduate student training, collaboration with researchers from other US institutions and publication of results in scientific journals. The project will establish (1) the analytical theory of surfaces of minimum mean curvature variation, (2) the creation of a robust, exportable method to impose nonhomogeneous boundary conditions for nonlocal equations in bounded domains, and (3) the analysis of regularity estimates for fixed interface local and nonlocal transmission problems emerging from the theory of fibered materials and long-range interaction models. The main questions that will be answered are well-posedness of boundary value problems and regularity properties of solutions. The methodology combines tools from the theory of partial differential equations, the calculus of variations, geometric analysis and functional analysis. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.