Open PDE & Analysis Seminar

Prof Guy David (Université Paris-Saclay (Orsay))

An elliptic operator on the complement of a Cantor set $K$ whose elliptic measure is proportional to the Hausdorff measure on $K$.

Friday 19th June 2020, 3pm (Paris time)

Abstract : We consider the one-dimensional Cantor set $K$ in the plane also known as the one-quarter Cantor set, and define an elliptic operator on $R^2 \setminus K$ for which the elliptic measure, with pole at infinity, is exactly proportional to the standard invariant measure $\mu$ on $K$. The operator has the nice form $L = div a \nabla$, but with an appropriate scalar function $a$. It was known that for the Laplacian, the harmonic measure and $\mu$ are mutually singular, and the case of $L$ could have been hard to compute. But in fact, we can simply construct the Green function so that $L$ has the desired properties.