Tatiana Toro

Harmonic measure on locally flat domains

We will review work with Tatiana Toro yielding a characterizati on of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation.

Poisson kernel characterization of Reifenberg flat chord arc domains

Abstract In this paper we prove the conjecture stated by the authors in Free boundary regularity for harmonic measures and Poisson kernels (Ann. of Math. 150 (1999) 369–454) concerning the free boundary regularity problem for the Poisson kernel below the continuous threshold. We show that if Ω is a Reifenberg flat chord arc domain, and the logarithm of the Poisson kernel has vanishing mean oscillation then the unit normal vector to the boundary also has vanishing mean oscillation.

Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions

In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonic measure

Free boundary regularity below the continuous threshold: 2-phase problems

\omega^+

A∞ implies NTA for a class of variable coefficient elliptic operators

of a domain

The Cauchy problem for Schrödinger flows into Kähler manifolds

\Omega=\Omega^+\subset\mathbb{R}^n

Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps

and the harmonic measure

Potential Analysis Meets Geometric Measure Theory

\omega^-

Stability of Lewis and Vogel’s result

of

Free boundary regularity for harmonic measures and Poisson kernels

\Omega^-