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Doubling and Flatness: Geometry of Measures Tatiana Toro
he goal of this article is to explore the locally, the graph of a function which is required relationship between the geometry of a to be at least Lipschitz continuous. The other point domain in Euclidean space and the of view is suitable in the more general setting properties of a canonical measure sup- where the boundary is only assumed to be non- ported onΩ ∂ , which arises in potential tangentially accessible (i.e., points in the bound- Ttheory. This measure is called the harmonic mea- ary can be reached by twisted cones from inside sure ω. In some instancesΩ the ω-measure of a set and outside the domain, which satisfies some ad- E ∂ can be understood as the probability that ⊂ ditional connectivity hypothesis). In this case the the Brownian motion starting at a fixed point in- harmonic measure is doubling (see [9] and also [5, side theΩ domain will first hit the boundary at a point 23] for related results). A measure is doubling if of E. Our project is twofold. Initially we study how the measure of the ball of radius r controls the the geometry of the boundary of a domain deter- measure of the ball of radius 2r. Our results par- mines the regularity of the harmonic measure (see tially reconcile the large disparity that exists be- the section “Boundary Regularity” and [12]). Then tween these two points of view. we look at the converse problem, namely we es- The converse problem is a free boundary regu- tablish how the regularity of the harmonic measure larity problem. The regularity of the boundary of prescribes the geometry of the boundary (see the the set = υ>0 , where υ is a solution of the section “A Free Boundary Regularity Problem” and { } Laplace equation, is deduced from some informa- [13]). tion aboutΩ the “regularity of the normal derivative” The first problem is that of boundary regular- of υ on ∂ (see “A Free Boundary Regularity Prob- ity for the solution of a partial differential equa- lem”). The main idea behind most results in this tion. This question has been extensively analyzed Ω from two different points of view. One point of view direction is that the oscillation (in a Hölder norm) is that the oscillation of the unit normal vector to of the “derivative” of harmonic measure controls the boundary controls the behavior of the “deriv- the oscillation of the unit normal vector (also in ative” of harmonic measure (see [4, 10, 11, 18]). Al- the Hölder norm) (see [1, 2, 3]). most all of these results are based on the as- In the first two sections we introduce the no- sumption that the boundary of the domain is, tions of flatness and doubling. Our notion of flat- ness generalizes properties of C1 hypersurfaces to much more general sets. Roughly speaking, a set is flat if it is well approximated by hyperplanes at Tatiana Toro is assistant professor of mathematics at the University of Washington, Seattle. Her e-mail address is every point and at every scale. In our work the dou- [email protected]. bling character of a measure and the flatness of a This article is based on a talk given at the Julia Robinson set replace conventional notions of regularity. In Celebration at MSRI, July 1996. The results discussed here the context of the problems sketched above, the are joint work with Carlos Kenig. regularity of the harmonic measure is most often
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described in terms of its behavior with respect to dorff distances between the boundary and n-planes the surface measure of the boundary of the domain. through Q, namely, by The notion of doubling enables us to talk about the regularity of the harmonic measure for domains θ(r,Q)= whose boundary is so rough that the surface mea- 1 inf D[∂ B(r,Q),L B(r,Q)] , sure is not well defined. Similarly the regularity of L r ∩ ∩ the domain has almost always been described in terms of the oscillation of the unit normal vector where the infimumΩ is taken over all n-planes L con- to the boundary. Unfortunately this unit normal taining Q. vector does not exist for many interesting domains Our work requires uniform control of several (e.g., domains with fractal boundary or pseudo- quantities on compact sets, thus for each com- pact set K Rn+1 we define spheres). The notion of flatness allows us to speak ⊂ about the regularity of some of these domains. θK(r) = sup θ(r,Q) and For the sake of exposition we restrict our dis- Q ∂ K cussion to the case of unbounded domains. Thus ∈ ∩ θ(r) = sup Ωθ(r,Q), is always assumed to be an open connected un- Q ∂ bounded domain in Rn+1 whose boundary sepa- ∈ n+1 n+1 Ω ratesΩ R . Namely, R ∂ has exactly two non- The quantity θK(r) provides a uniform measure- \ c empty connected components and int . The ment over K of how far ∂ is from being an affine canonical example to keep inΩ mind is the upper half plane at scale r>0. It also gives an upper bound n+1 space R+ . Similar results to theΩ ones statedΩ below for the oscillation of theΩ approximating affine hold for bounded domains satisfying appropriate spaces at scale r. In this sense it is a good re- separation and connectivity conditions. placement for the oscillation of the unit normal to the boundary, which measures the oscillation of Flatness the tangent planes. If Rn+1 is a smooth domain (i.e., ∂ is locally ⊂ Definition 1.1. Let Rn+1. We say that is a representable as the graph of a smooth function), ⊂ itsΩ boundary is well approximated byΩ n-dimen- Reifenberg flat domain if there exists δ (0, 1/8) n+1 ∈ sional affine spaces. For Q Rn+1 and r>0, so that for each compactΩ set K R there Ωexists ∈ ⊂ B(r,Q) denotes the (n +1)-dimensional ball of ra- R>0 such that dius r and center Q. The fact that ∂ can be well 1 approximated by n-dimensional planes can be ex- (1) sup θK(r) δ, and sup θ(r) . 0
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snowflake, π/3. Recall that the Koch snowflake Summarizing, the fact that is a Reifenberg flat S is constructed from the following generating curve domain guarantees that at small scales ∂ can be 0 approximated by n-planes. ThisΩ approximation is C uniform on compact sets. The deviationΩ of ∂ from being an n-dimensional affine space only depends on the parameter δ (0, 1/8). If is Ωa ∈ Reifenberg vanishing domain, the approximation improves as the scale diminishes. On the Ωother hand, one should not be misled to believe that if by iteration of the similitude that maps 0 onto C is a Reifenberg vanishing domain, then ∂ ad- 1. C mits tangent planes. In fact, let ϕ : R R be de- → finedΩ by Ω
∞ cos(2kx) ϕ(x)= . 2k√k kX=1 The function ϕ can be shown to belong to λ , the After 9 iterations we have: ∗ little-o Zygmund class (see [27], page 47). This im- plies that ϕ is well approximated by affine func- tions whose graphs are affine spaces. Using this information, one can show without difficulty that = (x, t) R2 : t>ϕ(x) is a Reifenberg vanish- { ∈ } ing domain. On the other hand, ϕ is a continuous functionΩ which is nowhere differentiable (in par- is a flat version of π/3, where the angle of ticular it is a variant of the Weierstrass function). Sβ S the spike with respect to the horizontal is β instead Thus ∂ is not rectifiable (for a precise definition of π/3. is a fractal set, which admits a Hölder of rectifiability see [24]), which in particular implies Sβ continuous parameterization that is nowhere dif- that ∂ Ωdoes not have a tangent line almost every- ferentiable. This implies that both the unit normal where; i.e., there is not even a weak notion of the vector and the surface measure of β ∂ are not Ω S ⊂ unit normal vector to ∂ . Furthermore, the surface well defined. On the other hand, it is not difficult to measure to ∂ is not well defined (see [26]). see that since 0 < sin β<1/8, is a ReifenbergΩ flat A way to understandΩ the pathologies present in domain. In fact, sup θ(r) = sin β, where θ(r)= r>0 Reifenberg vanishingΩ domains is to think about sup θ(r,Q). Ω Q ∂ them as domains which admit C0,α parameteri- Reifenberg∈ flat domains should not be seen as Ω zations for every α (0, 1) but which might not exotic objects; they exhibit minimal geometric con- 0,1 ∈ ditions necessary for some natural properties in admit C parameterizations. As a matter of fact, analysis and potential theory to hold. As we shall Reifenberg’s theorem guarantees that the bound- see there is a correspondence between the flatness ary of a Reifenberg vanishing domain is locally of a domain and the doubling properties of the har- representable as the image (not the graph!) via a monic measure. We now introduce Reifenberg van- homeomorphism of an open subset of Rn (see [17, ishing domains which are the appropriate gener- 20]). Moreover, the proof also shows that this alization of smooth domains in this context. Note homeomorphism yields a local Hölder parameter- that if Rn+1 is smooth, then the approxima- ization of the boundary. On the other hand, the ex- ⊂ tion of ∂ by affine spaces improves as r tends ample above shows that the boundary of a Reifen- to 0. ThisΩ translates into the following statement: berg vanishing domain might not contain any for each compactΩ set K Rn+1 we have ⊂ Lipschitz piece. Some refinements of the Reifen- berg vanishing condition guarantee the existence lim θK(r)=0. 0,1 r 0 of C (i.e., bilipschitz) parameterizations for the → corresponding class of domains (see [25, 26]).
Definition 1.2. Let Rn+1. We say that is a Remark. Reifenberg introduced this notion of flat- Reifenberg vanishing domain⊂ if is a Reifenberg ness in 1960 (see [20]). He was interested in the ex- flat domain and if forΩ each compact set K ΩRn+1 istence and regularity of solutions for the Plateau ⊂ Ω problem in higher dimensions. The result men- lim θK(r)=0. tioned above enabled him to show that an n-di- r 0 → mensional minimal surface with prescribed bound- ary is a topological manifold except for a set of n-dimensional Hausdorff measure zero.
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n+1 Doubling In particular, if = R+ , υ(x1, ..., xn+1)=xn+1, n+1 n+1 A measure ω, supported in a subset of R , is then the Poisson kernel of R+ with pole at infinity n+1 doubling if the ω-measure of the ball of radius 2r is identically 1 and Ωthe harmonic measure of R+ and center Q can be controlled byΣ the ω-mea- with pole at infinity is the Lebesgue measure of Rn. ∈ sure of the ball of radius r and center Q for r>0 Thus (2) becomes small enough. IfΣ is a smooth domain, the sur- face measure of its boundary is a doubling mea- xn+1 φdx1...dxn+1 = φdx1...dxn. Rn+1 Rn sure. On the otherΩ hand, since the surface measure Z + Z of the boundary of a general Reifenberg flat domain The Hopf boundary∆ lemma combined with clas- is not well defined, we are compelled to use a dif- sical boundary regularity results for the solution ferent measure which makes sense in a more gen- of the Laplace equation on a smooth domain guar- eral context. We motivate the definition of this antees that the harmonic measure with pole at in- measure by looking at smooth domains. Let finity ω is asymptotically optimal doubling. This Rn+1 be a smooth (unbounded) domain which ⊂ means that ω is a Radon measure; i.e., the ω- is flat at infinity in the sense that measure of compact sets is finite, and for each com- Ωsupr>0 θ(r) 1/8; then there exists a harmonic pact set K Rn+1 such that K ∂ = and for ≤ ⊂ ∩ 6 ∅ function υ defined in satisfying each τ>0, υ =0 in ω(B(rτ,Q)) Ω Ω lim inf = τn υ>0in r 0 Q K ∂ ω(B(r,Q)) (3) → ∈ ∩ ∆ Ω ω(B(rτ,Q)) υ =0 on∂ . Ω n Ω lim sup = τ . r 0 Q K ∂ ω(B(r,Q)) → ∈ ∩ n+1 ∂2 Ω Here = i=1 2 denotes the Laplacian. The func- Ω ∂xi On the one hand, (3) states that ω is a doubling tion υ is uniquelyP determined up to multiplication measure. On the other hand, it claims that as r 0 by a ∆positive constant and is called the Green’s → the ratio ω(B(2r,Q)) behaves more and more like the function of with pole at infinity (see [13]). In gen- ω(B(r,Q)) eral the Green’s function of a domain with pole corresponding ratio for the Lebesgue measure at X0 Ωis a positive harmonic function in (resp. Hausdorff measure) in n-dimensional Eu- ∈ X0 which vanishes on ∂ . The functionΩ υ can clidean space (resp. n-dimensional smooth hy- \{ } be constructedΩ as the limit of scaled Green’s func- persurface). Nevertheless, (3) does not imply any- tionsΩ whose poles convergeΩ to infinity. If ω(B(r,Q)) thing about the behavior of the ratio r n for φ C (Rn+1) , integration by parts gives us ∈ c∞ Q ∂ as r tends to 0. In fact, if = Green’s formula ∈ 2 cos(2kx) (x, t) R : t>ϕ(x) for ϕ(x)= ∞ , Ω k=1 2k√Ωk ∂φ ∂υ { ∈ } (υ φ φ υ)dx = (υ φ )dσ, then ω is asymptotically optimally doubling,P but Z − Z∂ ∂ν − ∂ν 2 ω(B(r,Q)) for each compact set K R , supK ∂ r ∆ ∆ ∂ ⊂ ∩ whereΩ σ denotes the surfaceΩ measure and ∂ν de- tends to infinity as r tends to 0. notes the normal derivative at the boundary (i.e., In summary, a smooth domain is ReifenbergΩ ∂ = ν where ν denotes the outward unit nor- vanishing and its harmonic measure with pole at ∂ν ·∇ mal and denotes the gradient). Since υ is har- infinity is asymptotically optimal doubling. We ∇ monic in and vanishes on the boundary, the in- now show that these notions of flatness and dou- tegration by parts formula above becomes bling are deeply intertwined and are independent Ω of the smoothness assumption, as suggested by the ∂υ example above. They provide some weak notions (2) υ φdx = φ dσ. Z − Z∂ ∂ν of regularity which suffice to answer several ques- tions in analysis and geometric measure theory. Be- ∆ By analogy withΩ the conventionalΩ Poisson kernel, fore stating any results we need to guarantee that ∂υ is called the Poisson kernel of with pole it makes sense to talk about the harmonic measure − ∂ν at infinity. The measure ω supported in ∂ and with pole at infinity for a general Reifenberg flat defined by Ω domain. This is the content of the next proposi- Ω tion. ∂υ ω(A)= hdσ , where h = , Proposition [13]. Let Rn+1 be a Reifenberg ∂ A −∂ν ⊂ Z ∩ flat domain; then there exist a unique (up to a n+1 for any Borel setΩ A R , is called the harmonic positive constant multiple)Ω function υ, continuous ⊂ measure of with pole at infinity. Both the Pois- on the closure of , and a unique (up to a posi- son kernel and the harmonic measure are deter- tive constant multiple) doubling Radon measure ω mined up to Ωmultiplication by a positive constant. such that Ω
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1 n+1 C function), then log h VMO(dσ) (see [10]). υ φdx = φdω φ Cc∞(R ) ∈ Z Z∂ ∀ ∈ This insures that log h can be well approximated ∆ by uniformly continuous functions in an integral where Ω υΩ =0 in sense (namely, in the mean oscillation norm). Nev- υ>0in ertheless, it does not guarantee that log h is con- ∆ Ω tinuous. In fact, it is easy to construct examples υ =0 on∂ . 1 Ω of C domains for which the logarithm of the Pois- son kernel is not continuous. Ω Along these lines we prove that if the unit nor- Here ω denotes the harmonic measure of with mal vector ν to the boundary of has small in- pole at infinity, and υ denotes the Green’s func- tegral oscillation, then so does the logarithm of the tion of with pole at infinity. Ω Poisson kernel. Our results concernΩ domains where a suitable version of the divergence theorem holds, BoundaryΩ Regularity In this section we discuss how the regularity of the i.e., sets of locally finite perimeter (see [7]) whose boundary of Rn+1 (a domain as in the propo- boundary is Ahlfors regular. ∂ is Ahlfors regu- ⊂ sition above) determines the regularity of its har- lar if there exists a constant C>1 such that, for Q ∂ and r>0, Ω monic measureΩ ω. If at each scale ∂ has big Lip- ∈ schitz pieces—more precisely, if ∂ is uniformly n(∂ B(r,Q)) σ (B(r,Q)) rectifiable (see [6] for a precise definition)—thenΩ Ω1 C− H n = n C. the surface measure σ, given by theΩ restriction of ≤ rT r ≤ Ω the n-dimensional Hausdorff measure to ∂ , i.e. We show that if is a set of locally finite perime- σ = n ∂ , is a Radon measure. Under the ap- H ter whose boundary is Ahlfors regular and propriate geometric conditions for , resultsΩ in [4, ν VMO(dσ), thenΩ log h VMO(dσ) (see [12]). ∈ ∈ 5], and [23] Ωguarantee that ω is a doubling mea- A domain satisfying these hypotheses is called a sure and that σ and ω are mutually Ωabsolutely con- chord arc domain with vanishing constant. In par- tinuous (i.e., a subset of ∂ has σ-measure zero ticular this result applies to a domain whose if and only if it has ω-measure zero). In this case boundary is locally representable as the graph of the Radon-Nikodym theoremΩ insures that the Pois- a function whose gradient is in VMO. NoteΩ that we son kernel h = dω = ∂υ(the Radon-Nikodym de- dσ − ∂ν establish the same result as Jerison and Kenig rivative of ω with respect to σ) exists. The Pois- under much weaker assumptions. In some sense son kernel is the density of ω with respect to σ. this shows that chord arc domains with vanishing In this setting the regularity of ω is described in constant provide a good generalization of C1 do- terms of the behavior of h. If ∂ is not uniformly mains from a potential theoretic point of view. rectifiable, we concentrate on the doubling prop- The first results of this type for nonsmooth do- erties of ω rather than on propertiesΩ of the Pois- mains were proved by Lavrentiev (n =1; see [15]) son kernel which might not even exist. The proofs and Dahlberg (n 2). In particular Dahlberg ≥ of all the results discussed in this section mainly showed that if is a Lipschitz domain, then σ and use techniques from partial differential equations. ω are mutually absolutely continuous. In this case, A classical boundary regularity result states while the Radon-NikodymΩ theorem guarantees only that if the oscillation of the unit normal vector to that h L1 (dσ) , he proved that in fact the boundary is small in some Hölder norm, then ∈ loc h L2 (dσ) (see [4]). This result has played a cen- so is the oscillation of the Poisson kernel and its ∈ loc inverse. Controlling the oscillation of h and of tral role in the study of boundary regularity for so- 1/h amounts to controlling the oscillation of log h. lutions of elliptic partial differential equations in More precisely, if is a C1,α domain for α (0, 1) nonsmooth domains. ∈ (i.e., ∂ is locally representable as the graph of a We saw in the example above that the surface function whose gradientΩ is Hölder continuous with measure of the boundary of a Reifenberg vanish- exponentΩ α), then log h C0,α. In particular h ing domain need not be well defined. In this set- ∈ and 1/h are Hölder continuous (see [11]). Jerison ting the regularity of the harmonic measure needs and Kenig proved that if the oscillation of the unit to be expressed in terms of its doubling proper- normal vector to the boundary is small in the C0 ties, not in terms of the regularity of its density with norm, then log h has small oscillation in an inte- respect to surface measure. The appropriate reg- gral sense. Namely, the L2 averages on balls of log h ularity statement is given by the following result. minus its average (i.e. the mean oscillation) con- Theorem 1 [12]. The harmonic measure of a Reifen- verge to zero as the radius of the balls tends to berg vanishing domain is asymptotically optimal zero. In this case log h is said to have vanishing doubling. mean oscillation, which we denote by log h VMO(dσ). Explicitly, if is a C1 domain This theorem shows that if the boundary of a ∈ (i.e., ∂ is locally representable as the graph of a domain is well approximated by affine spaces in Ω Ω OCTOBER 1997 NOTICES OF THE AMS 1091 toro.qxp 4/15/98 4:00 PM Page 1092
the Hausdorff distance sense, then, from a doubling then is a C1,α domain for some α (0, 1) which ∈ point of view, its harmonic measure behaves like depends on β. Moreover, if h is identically equal the harmonic measure of the half spaces bounded to 1, thenΩ is a half-space. by these affine spaces. Theorem 2, discussed in the Note that these results combined with those next section, establishes the converse of this state- Ω ment. This provides a complete characterization mentioned above reinforce the idea that the regu- of Reifenberg vanishing domains in terms of the larity of the boundary of a domain (described in doubling properties of the harmonic measure. The terms of the oscillation of the unit normal vector) main ingredients in the proof of Theorem 1 are the and the regularity of its harmonic measure (de- maximum principle, the comparison principle for scribed in terms of the regularity of the logarithm nontangentially accessible (NTA) domains (of which of its density) are “equivalent”. Along these lines Reifenberg flat domains are an example), and the we prove that this “equivalence” prevails even boundary regularity for nonnegative harmonic when the notions of smoothness involved are functions on NTA domains (see [9, 12]). weaker than the ones above. We show that on a do- main which is flat enough, the behavior of the log- A Free Boundary Regularity Problem arithm of the Poisson kernel together with the In this section we discuss the converse problem. doubling properties of the harmonic measure de- Namely, we explain that either the regularity of the termine the regularity and the geometry of the Poisson kernel of a domain or the doubling prop- boundary. We say that is a chord arc domain with erties of its harmonic measure determine the reg- small constant if is a set of locally finite perime- ularity of its boundary. This problem should be un- ter whose boundary isΩ Ahlfors regular for which derstood as a free boundary regularity problem in the unit normal vectorΩ ν to the boundary has small the sense that we are trying to deduce the regu- mean oscillation (i.e., the L2 averages of ν minus larity of the boundary of the set = υ>0 , where its average are small). In [13] (see also [8]) we prove { } υ denotes the Green’s function with pole at infin- that if is a chord arc domain with small constant, ity from some information aboutΩ the “regularity of ω is asymptotically optimal doubling, and the normal derivative” of υ on ∂ . In the case log h ΩVMO(dσ), then ν VMO(dσ). This re- ∈ ∈ where the boundary of the domain is rectifiable we sult should be contrasted with the results dis- have a unit normal vector to ∂ . Moreover,Ω the nor- cussed in “Boundary Regularity” concerning chord mal derivative of υ at the boundary is a well de- arc domains with vanishing constant. This yields fined function h, the PoissonΩ kernel with pole at the following theorem: is a chord arc domain infinity. In the case where the boundary is not rec- with vanishing constant if and only if is a chord tifiable, we do not have a unit normal vector to the arc domain with small constantΩ whose harmonic boundary and therefore the “normal derivative” measure ω is asymptotically optimalΩ doubling should be understood not as a function but as a and whose Poisson kernel h satisfies measure, the harmonic measure with pole at in- log h VMO(dσ). finity. This interpretation is validated by the inte- ∈ Note that all of these results convey the idea that gral equality that appears in the proposition at the the oscillation of the unit normal to the boundary end of “Doubling”. The proofs of these results re- of a domain and the oscillation of the logarithm quire some tools from geometric measure theory of its Poisson kernel are “equivalent” quantities on that are developed in [14] and [13], and are inspired flat domains. For chord arc domains in R2 this by techniques discussed in [16] and [19]. equivalence was explicitly proved by Pommerenke. Roughly speaking, Alt and Caffarelli (see [1, 2, He showed, using complex analytic methods, that 3]) showed that on a domain which is sufficiently flat the behavior of the logarithm of the Poisson the unit normal to the boundary belongs to VMO(dσ) if and only if log h VMO(dσ) (see kernel determines the regularity and the geome- ∈ try of the boundary. The key idea behind their [18]). It is the subject of current investigation proof is the following: in a domain whose bound- whether this result is also true in higher dimen- ary is a priori well approximated by affine spaces sions. More precisely, we would like to show that if Rn+1 is a chord arc domain with small con- the uniform continuity of the logarithm of the ⊂ stant and log h VMO(dσ), then ω is an asymp- Poisson kernel insures that as the scale decreases ∈ the approximation by affine spaces improves. toticallyΩ optimal doubling measure. This would allow us to apply the result stated above to con- Theorem [1]. Assume that: clude that is a chord arc domain with vanish- 1. Rn+1 is a set of locally finite perimeter ing constant, i.e., that ν VMO(dσ). ⊂ ∈ whose boundary is Ahlfors regular, For generalΩ Reifenberg flat domains the notion of surface measure of the boundary is not well de- 2. Ω Rn+1 is a Reifenberg flat domain and (1) ⊂ fined. Therefore the regularity of the free bound- holds for some δ>0 small enough, ary in this case depends solely on the doubling Ω 3. log h C0,β for some β (0, 1); properties of the harmonic measure. ∈ ∈
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Theorem 2 [13]. Let Rn+1 be a Reifenberg volved. In particular they are very well adapted to ⊂ flat domain whose harmonic measure is asymp- sets which are not rectifiable, whose Hausdorff totically optimal doubling;Ω then is a Reifenberg dimension might be larger than their topological vanishing domain. dimension, or which are not locally graphs. The no- Ω tions of doubling and flatness presented here are Theorem 2 is a corollary of a more general re- powerful tools to study potential theory on con- sult: A Reifenberg flat set which supports an as- tinuous domains which lack differentiability. ymptotically optimal doubling measure is Reifen- berg vanishing. Theorems 1 and 2 provide a Acknowledgments complete characterization of Reifenberg vanishing I would like to thank Richard Pollack and Hugo domains in terms of the doubling properties of Rossi for their comments on a preliminary version their harmonic measure. The proof of Theorem 2 of this paper and Don Marshall who created the pic- uses tools from [16, 19]. It relies heavily on the tures that appear in “Flatness”. Kowalski-Preiss classification of n-uniform mea- sures (see [14] and Remark 1 below) as well as on References the notions of tangent and pseudo-tangent mea- [1] H. W. Alt and L. A. Caffarelli, Existence and regu- sure (see [13]). larity for a minimum problem with free boundary, J. Reine Angew. Math. 325 (1981), 105–144. Remarks [2] L. A. Caffarelli, A Harnack inequality approach to 1. In Alt and Caffarelli’s result as well as in The- the regularity of free boundaries. Part I: Lipschitz free 1,α orem 2 the assumption that is a Reifenberg flat boundaries are C , Rev. Mat. Iberoamericana 3 domain is crucial. An example by Kowalksi and (1987), 139–162. [3] , A Harnack inequality approach to the regu- Preiss (see [14]) combined withΩ a calculation of the ——— larity of free boundaries. Part II: Flat free boundaries harmonic measure carried out in [13] shows that are Lipschitz, Comm. Pure Appl. Math. 42 (1989), in dimensions greater than 4 there exist unbounded 55–78. domains whose Poisson kernel at infinity is iden- [4] B. Dahlberg, Estimates for harmonic measure, Arch. tically 1, whose harmonic measure with pole at in- Rational Mech. Anal. 65 (1977), 275–288. finity is asymptotically optimal doubling, and [5] G. David and D. Jerison, Lipschitz approximation to which are very far from being Reifenberg vanish- hypersurfaces, harmonic measure, and singular in- ing. If n 3, let tegrals, Indiana Univ. Math. J. 39 (1990), 831–845. ≥ [6] G. David and S. Semmes, Analysis of and on uni- formly rectifiable sets, Math. Surveys Monographs, n+1 = (x1, ..., xn+1) R : x4 ⊂ | | vol. 38, Amer. Math. Soc., Providence, RI, 1993. n [7] L. C. Evans and R. F. Gariepy, Measure theory and 2 2 2 Ω < x1 + x2 + x3 . fine properties of functions, Studies in Advanced q Mathematics, CRC Press, 1992. is an unbounded nontangentially accessible do- [8] D. Jerison, Regularity of the Poisson kernel and free main whose harmonic measure ω with pole at in- boundary problems, Colloquium Mathematicum, finityΩ appropriately normalized satisfies 60–61 (1990), 547–567. [9] D. Jerison and C. Kenig, Boundary behavior of har- monic functions in nontangentially accessible do- mains, Adv. in Math., 46 (1982), 80–147. n dω ω = σ = ∂ which implies h = =1. [10] , The logarithm of the Poisson kernel of a C1 H dσ ——— domain has vanishing mean oscillation, Trans. Amer. Moreover, ω is n-uniform; i.e., for Q ∂ , r>0, Ω ∈ Math. Soc., 273 (1982), 781–794. and τ>0 [11] O. D. Kellogg, Foundations of potential theory, Ω Springer-Verlag, 1929. Reprinted 1967, Dover Pub- ω(B(r,Q)) = n(∂ B(r,Q)) = r n H lications. ω(B\(rτ,Q)) [12] C. Kenig and T. Toro, Harmonic measure on locally which implies Ω = τn. ω(B(r,Q)) flat domains, Duke Math. J. 87 (1997), 509–551. [13] ———, Free boundary regularity for harmonic mea- On the other hand, it is easy to see that is not sures and Poisson kernels, preprint. Reifenberg vanishing. In fact, 0 ∂ and for r>0 [14] O. Kowalski and D. Preiss, Besicovitch-type proper- ∈ the Hausdorff distance between ∂ B(r,Ω0) and ties of measures and submanifolds, J. Reine Angew. ∩ L B(r,0) for any n-dimensional planeΩ containing Math. 379 (1987), 115–151. ∩ the origin is at least r/√2, i.e., θ(r,Ω0) 1/√2. [15] M. Lavrentiev, Boundary problems in the theory of ≥ univalent functions, Mat. Sb. (N.S.) 1 (1936), 815–844. 2. The previous characterization of Reifenberg [English Trans.]: Amer. Math. Soc., Trans., Ser. 2, 32 vanishing domains is quite natural if looked at (1963), 1–36. under the appropriate light. The Reifenberg flat- [16] P. Mattila, Geometry of sets and measures in Eu- ness of a set and the doubling character of a mea- clidean spaces, Cambridge University Press, 1995. sure extend some classical notions of regularity [17] C. B. Morrey, Multiple integrals in the calculus of vari- that require differentiability of the quantities in- ations, Springer-Verlag 1966.
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[18] Ch. Pommerenke, On univalent functions, Bloch func- tions and VMOA, Math. Ann. 236 (1978), 199–208. [19] D. Preiss, Geometry of measures in Rn: Distribution, rectifiability, and densities, Ann. of Math. 125 (1987), 537–643. [20] E. Reifenberg, Solution of the plateau problem for m-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. [21] S. Semmes, Chord-arc surfaces with small constant, I, Adv. in Math., 85 (1991), 198–223. [22] ———, Chord-arc surfaces with small constant, II: Good parameterizations, Adv. in Math., 88 (1991), 170–199. [23] ———, Analysis vs. geometry on a class of rectifiable hypersurfaces, Indiana Univ. J. 39 (1990) 1005–1035. [24] L. Simon, Lectures on geometric measure theory, Australian National University 1983. [25] T. Toro, Surfaces with generalized second funda- mental form in L2 are Lipschitz manifolds, J. Dif- ferential Geometry, 39 (1994), 65–101. [26] ———, Geometric conditions and existence of bilipschitz parameterizations, Duke Math. J. 77 (1995), 193–227. [27] A. Zygmung, Trigonometric Series, Vol 1, Cambridge University Press, 1959.
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