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.