Description

In this project we consider perturbation and asymptotic problems for elliptic differential equations. We consider several different types of perturbation: domain (regular and singular perturbation for electromagnetic, degenerate, Steklov, nonlinear and higher order problems, corner singularities, etc.), mass and geometry (eigenvalue bounds and optimization), coefficients (regularity and stability, constant/nonconstant cases). The main aim of the project is to exploit the interplay between potential theory and calculus of variations and, on a higher scale, to involve more prominently geometric ideas in unprecedented ways: we will not only study perturbation and asymptotic problems in Riemannian settings, but also apply geometric techniques for the study of problems in Euclidean spaces. Apart from actual perturbation problems, we also consider more abstract, foundational questions that are necessary to improve the understanding of the geometrical and functional structure, such as: the role of the mass from a geometric point of view; domain perturbation in a general Riemannian setting; reducible operators for solving general BVPs; numerical computation of potentials; regularity properties of layer potentials; etc.