We will review work with Tatiana Toro yielding a characterization of those domains for which the harmonic measure has a density whose logarithm has vanishing mean oscillation.2000 Mathematics Subject Classification: 31B25, 35R35, 42B35, 51M25.

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Résumé. On généralise la démonstration du théorème du disque topologique de Reifenberg pour inclure le cas d'ensembles ayant des trous, et on donne des conditions suffisantes sur l'ensemble E pour l'existence de paramétrage de E par un plan affine ou une variété de dimension d. L'une de ces conditions porte sur la sommabilité des carrés des nombres de P. Jones β 1 (x, r), et s'applique en particulier aux ensembles localement Ahlfors-réguliers età l'existence de très grand morceaux d'images bi-Lipschitziennes de R d .Abstract. We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set E for the existence of a bi-Lipschitz parameterization of E by a d-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers β 1 (x, r).In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of R d .

In the late 1950s and early 1960s, the work of De Giorgi [DeGi] and Nash [N], and then Moser [Mo], initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients. Weak solutions in a domain 0, a priori only in a Sobolev space W All the results mentioned above were carried out for elliptic operators L=div A{ where the matrix A=(a ij ) has bounded measurable coefficients and is symmetric. However, it turns out that the symmetry of the matrix is not needed to get these results: Morrey [Mor] first observed this in connection with the De Giorgi Nash Moser theory; for the results in [CFMS], this fact has not been formally observed until now. With appropriate reformulation in terms of adjoint operators, and adjoint Green's functions, the results of [CFMS] are valid without the symmetry assumption (see Section 1).

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.  2003 Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ.-Dans cet article, on démontre la conjecture proposée par les auteurs dans Free boundary regularity for harmonic measures and Poisson kernels (Ann. of Math. 150 (1999) 369-454) concernant la régularité de la frontière libre pour le noyau de Poisson au-dessous du seuil de continuité. On prouve que si Ω est un domaine corde-arc Reifenberg plat tel que le logarithme du noyau de Poisson appartienne à VMO, alors le vecteur unitaire normal à la frontière appartient aussi à VMO.