Abstract:

The Perron method and Wiener's criterion characterise solvability for classical Dirichlet problems. We will see how these tools provide for the construction of elliptic measures. And how the theory of Muckenhoupt weights is relevant in modern approaches to minimal regularity Dirichlet problems. As an application, we will consider recent joint work, with Steve Hofmann and Phi Le, concerning degenerate elliptic equations with nonsymmetric coefficients. Our approach is based on proving a Carleson measure estimate for bounded solutions, relying on weighted Kato-type square function estimates. An oscillation estimate for solutions then allows us to prove that the associated elliptic measure is absolutely continuous, in a quantitative scale-invariant sense, with respect to the surface measure on the domain boundary.

Bio:

Andrew is a harmonic analyst who specialises in developing functional calculus methods to investigate solutions to partial differential equations in rough geometric contexts. His current research is partially supported by the Royal Society International Exchange grant Eliminating symmetry and permitting singularity in periodic homogenization.

Zoom Link:

https://anu.zoom.us/j/84591706600?pwd=ZTIzMGM1L1RnNlo1b2RVa0pwekFBUT09
Meeting ID: 845 9170 6600
Password: 890679