# The PDEs and Applications seminar: The minimal regularity Dirichlet problem for degenerate elliptic PDEs beyond symmetric coefficients

- Date: –12:00
- Location: Ångströmlaboratoriet, Lägerhyddsvägen 1 64119
- Lecturer: Andrew Morris, University of Birmingham
- Organiser: Matematiska institutionen
- Contact person: Kaj Nyström
- Seminarium

Abstract: We prove that the Dirichlet problem for degenerate elliptic

equations with nonsymmetric coefficients on Lipschitz domains is

solvable when the boundary data is in weighted $L^p$ for some

$p<\infty$. The result is achieved without requiring any structure on

the coefficient matrix, thus allowing for nonsymmetric coefficients, in

which case $p>2$ becomes necessary. We build on the groundbreaking

result obtained by Hofmann, Kenig, Mayboroda and Pipher for uniformly

elliptic equations, by allowing for the bound and ellipticity on the

coefficients to degenerate under the control of a Muckenhoupt weight. We

also adopt an alternative strategy, which is outlined in their work,

although the crucial technical estimate is not at all an obvious

extension of the uniformly elliptic theory. In this approach, a Carleson

measure estimate for bounded solutions is established directly. This

allows us to avoid good-$\lambda$ inequalities entirely, and thus apply

a Dhalberg--Kenig--Stein pull-back based on an $L^2$-Hodge decomposition

instead of an $L^{2+\epsilon}$-version. The result is then combined with

an oscillation estimate for solutions, which allows us to avoid the

method of $\epsilon$-approximability, to deduce that the degenerate

harmonic measure is in the $A_\infty$-class with respect to weighted

Lebesgue measure on the domain boundary. This is joint work with Steve

Hofmann and Phi Le.