Seminar 19.10.2023: Martin Dindos

[quittner]

Seminár z kvalitatívnej teórie diferenciálnych rovníc
Seminar on Qualitative Theory of Differential Equations

Thursday 19.10.2023 at 14:00 Lecture room M-223

Martin Dindoš (University of Edinburgh):
New progress on solvability of Regularity problem for elliptic operators with coefficients satisfying large Carleson condition

Abstract:
Recall that when $Lu=div(A\nabla u)$ and coefficients $A$ are elliptic and satisfy Carleson condition
(that is $|\nabla A|^2\delta$ is a Carleson measure) then the corresponding elliptic measure is $A_\infty$
and hence the $L^p$ Dirichlet problem is solvable for some large $p<\infty$.
There is also an analogous smallness result, namely that on $C^1$ domains $L^p$ Dirichlet problem
is solvable for all $1<p<\infty$, provided $|\nabla A|^2\delta$ is a vanishing Carleson measure.

In the case of Regularity problem, the analogous smallness result was established in Dindos-Pipher-Rule.
We have recently (jointly with S. Hofmann and J. Pipher) established an n-dimensional reduction
that relates solvability of Regularity problem to the solvability of Regularity problem for a block-form operator.

This reduction has allowed us to fully resolve the question of solvability of Regularity problem
with coefficients satisfying large Carleson condition in all dimensions and also the Neumann problem
in dimension 2 in the interval for $1<p<1+\epsilon$.