Joint BCAM-UPV/EHU Analysis and PDE seminar: Layer potentials and Boundary Value Problems in unbounded domains

Abstract

S. Hofmann, M. Mitrea, and M. Taylor considered boundary value problems in bounded Semmes-Kenig-Toro using the method of layer potentials. This method allows one to solve boundary value problems for the Laplacian and other elliptic operators once it is shown that a certain singular integral operator is invertible. The previous authors established the desired invertibility using the Fredholm theory and exploiting the compactness of the boundary-to-boundary double layer, fact that follows from the extra cancellation of its kernel based on the good oscillation properties of the outer unit normal. In this talk we will study the case of unbounded domains, where the Fredholm theory is not expected to work. We assume that the outer unit normal has sufficiently small oscillation and we establish the desired invertibility by using a Neumann series. Our theory works for the Laplacian and, more generally, for other elliptic systems with constant complex coefficients such as the complex version of the Lam system of ellipticity.

Joint work with J.J. Marín, D. Mitrea, I. Mitrea, and M. Mitrea.



Link to the session: 
https://us06web.zoom.us/j/99649860282?pwd=SE0vemtYMFlwbFBNTXQyOTBONG0vZz09

More info at https://sites.google.com/view/apdebilbao/home