International Congress of Mathematicians

Harmonie Aleasure and "Locally Flat" Domains 705[21] is that Dahlberg's theorem extends to this case, i.e., that doj = kda withlogfc £ BAIO, and, in fact, OJ £ A 00 (da).We say that O C R" +1 is a "o"-chord-arc domain" if 0 if oJ-Reifenberg flat, 0 isof locally finite perimeter, the boundary of 0 is Ahlfors regular and the BAIO normof the unit normal n is bounded by Ö. We say that 0 is "vanishing chord-arc" if, inaddition, it is Reifenberg vanishing and n £ VMO(da).Remark. S. Semmes [22] has proved that, if 0 is a ^-chord-arc domain (underthe definition used above), then(1 + ^ r W " < a(B(r, Q) n 90) < (1 + y/S)a n r n ,where a n is the volume of the unit ball in R" and Ö < 8 n . Moreover, a combinationof the results in [22] and [16] shows that, if 0 is a oJ-Reifenberg flat domain whichis of locally finite perimeter, and for which a(B(r,Q) n 90) < (1 + 8)a n r n , thenthe BAIO norm of n is bounded by CVô for Ô < b~ n . Thus, the two notions introducedof "vanishing chord-arc" domains in the plane are the same, and a domainis vanishing chord-arc exactly when it is of locally finite perimeter, has an Ahlforsregular boundary, it is Reifenberg vanishing and satisfies n £ VAIO.isOur potential-theoretic result, which extends the work of Jerison-Kenig [10],Theorem 3. ([15]) If ii is a vanishing chord-arc domain, then OJ (OJ°°) has theproperty that doj = kda (doj°° = h da) with logfc £ VAIO (log h £ VAIOJ.This was proved by a combination of real variable arguments, potential-theoreticarguments, and the estimates in [10].In order to understand possible converses of this, extending the work of Pommerenketo higher dimensions, we will recall precisely the Alt-Caffarelli [1] resultwhich we alluded to earlier. In the language that we have introduced, their localregularity theorem can be stated as follows:Theorem. ([1]) Let ii be a set of locally finite perimeter whose boundary is Ahlforsregular. Assume that 0 is ô-Reifenberg flat, Ö < ó~ n . Suppose that doj = kda withlogfc G C a (90) (0 0 in O, u| = 0,Au = 0 in O and h = ||. Thus, knowledge of the regularity of the Cauchy data ofv (v\ dQ , §§| ö Q) yields regularity of 90 (or of n, the normal).The first connection between the above Theorem and the work of Pommerenkewas made by Jerison [8], who was also the first to formulate the higher-dimensionalanalogues of Pommerenke's theorem as end-point estimates as a —¥ 0 in the Alt-Caffarelli theorem. Jerison's theorem in [8] states that, if O is a Lipschitz domainand doj = kda with logfc continuous, then n £ VAIO. There is an error in Lemma4 of Jerison's paper. Nonetheless, in [16] we made strong use of the ideas in [8]. Inthe more recent version of our results [14], we bypass this approach.