International Congress of Mathematicians

706 C. E. KenigBefore stating our result, it is useful to classify the assumptions in the Alt-Caffarelli theorem. For this, we recall some examples:Examples. When n = 1, Keldysh-Lavrentiev [12] (see also [5]) constructed domainsin R 1+1 with locally rectifiable boundaries which ([5]) can be taken to beReifenberg vanishing and for which doj = da, i.e., k = 1, but which are not verysmooth.For instance, they fail to be chord-arc. These domains do not, of course,have Ahlfors regular boundaries. When n = 2, Alt-Caffarelli constructed a doublecone F in R 3 such that, for 0 the domain outside the cone, doj°° = da, i.e., k = 1.This is of course not smooth near the origin, the problem being that, while 0 isNTA and 90 is Ahlfors regular, 0 is not oJ-Reifenberg flat for small 8. When n = 3,the Preiss cone we saw before exhibits the same behavior.Our first result was:Theorem 4. ([16]) Assume that 0 Ç R" +1 is ô-chord-arc, 8 < 8 n , that OJ (OJ°°)is asymptotically optimal doubling and that logfc £ VAIO (log h £ VAIOJ. Thenn £ VAIO and 0 is vanishing chord-arcNotice, however, that, when comparing the hypothesis of Theorem 4 to theAlt-Caffarelli theorem two things are apparent : first, we are making the additionalassumption that OJ is asymptotically optimal doubling, and hence, in lightof Theorem 2, 0 is Reifenberg vanishing. Next, the "flatness" assumption in theAlt-Caffarelli theorem is $-Reifenberg flatness, while in Theorem 4 we make the apriori assumption that, in addition, the BAIO norm of n is smaller than 8. R...".This does not make much sense, ecently we have developed a new approach whichhas removed these objections. We have:Theorem 5. ([14]) Let ii be a set of locally finite perimeter whose boundary isAhlfors regular. Assume that 0 is 8-Reifenberg flat, 8 < 8 n . Suppose that doj = kda(doj°° = h da) with logfc G VMO(da) flog h £ VMO(da)). Then ft £ VMO(da)and ii is a vanishing chord-arc domain.Note that Theorems 3 and 5 together give a complete characterization of thevanishing chord-arc domains in terms of their harmonic measure, in analogy withPommerenke's 2-dimensional result, thus answering a question posed by Semmes[21].Our technique for the proof of Theorem 5 is to use a suitable "blow-up" toreduce matters to the following version of the "Liouville theorem" of Alt-Caffarelli(W, [13]):Theorem 6. ([1], [13]) Let ii be a set of locally finite perimeter whose boundaryis (unboundedly) Ahlfors regular. Assume that 0 is an unbounded 8-Reifenberg flatdomain, 8 < 8 n . Suppose that u and h satisfy:andI A« = 0 in 0u > 0 inii u\ ar ,=0. \auuAp = ph da, for p £ C 0 °° (R" +1 ).n Jan