Date and Time
-
Location
Math Sci 110
Organizers
Speaker
Simon Bortz (University of Alabama)

It has been known for quite some time that various conditions on the oscillation of the matrix $A$ in elliptic/parabolic operators of the form 

 $L = -div A \nabla$ or $L = \partial_t - \div A \nabla$ 

are sufficient to guarantee either the $L^p$ solvability of the Dirichlet problem or the logarithm of the density of the elliptic/parabolic measure, $k$,  is in BMO or (locally) Hölder continuous. Conditions such as the global Hölder continuity of $A$ are classical, whereas other conditions quantifying the oscillation of $A$ in terms of Carleson measures are more recent. Only very recently in joint works with Toro and Zhao, and later with Egert and Saari, was it shown that the BMO norm of $\log k$ could be controlled by these Carleson conditions. These new results are elliptic and heavily relied on work of David, Li and Mayboroda.

 

In forthcoming work with Egert and Saari, we adapt these new results (B., Toro, Zhao and B. Egert, Saari) to the parabolic setting. Therefore we needed to also adapt the work of David, Li and Mayboroda to the parabolic setting and, in doing so, sharpened their results. It seems very likely that these sharper estimates will allow one to treat the classical results (Hölder continuous coefficients) and the modern results (Carleson conditions) with a unified method. 

 

I will delve into the ideas underpinning these results.