Title: Critical Perturbations and Solvability for Elliptic Equations

Abstract: In this talk we will present recent results concerning solvability of certain Boundary Value Problems associated to a general linear second order elliptic equation, under the assumption that the equation is close, in some critical Lebesgue spaces for the coefficients, to an equation for which solvability is already known.

Our main contribution is to be able to treat lower order terms in the equation in critical spaces, which prevent us from obtaining any regularity of solutions to the homogeneous equation as in the celebrated De Giorgi-Nash-Moser result for purely second order equations. We develop instead an abstract approach in order to define the relevant solution operators (layer potentials), and build the perturbation theory based only on the ellipticity condition. We are optimistic that this approach can be used to treat more general operators, such as systems.

This is joint work Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi.