Boundary Value Problems in Unbounded Semmes-Kenig-Toro Domains

Tuesday, September 10, 2019 - 2:00pm
Math Sci 111
José María Martell
(ICMAT CSIC-UAM-UC3M-UCM, Spain) Miller Scholar

The method of layer potentials is a powerful tool that allows one to solve boundary value problems for the Laplacian (and for more general elliptic systems) by reducing the original problem to showing that a certain singular integral operator is invertible. S. Hofmann, M. Mitrea, and M. Taylor considered the case of bounded Semmes-Kenig-Toro domains with an outer unit normal which is close to the space of vanishing mean oscillation functions. This in turn allowed them to use the Fredholm theory to obtain the desired invertibility. 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 Lamé system of ellipticity. 
Joint work with J.J. Marín, D. Mitrea, I. Mitrea, and M. Mitrea.   

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