Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 459–505

ON THE DIMENSION OF p-HARMONIC MEASURE

Björn Bennewitz and John L. Lewis

University of Kentucky, Department of Mathematics Lexington, Kentucky, 40506-0027, U.S.A. bbennewi@ms.uky.edu; john@ms.uky.edu

Abstract. In this paper we study the dimension of a measure associated with a positive p-harmonic function which vanishes on the boundary of a certain domain.

Introduction

Denote points in Euclidean 2-space R^{2} by x = (x_{1}, x_{2}). Let ⟨ · , · ⟩ be the standard inner product on R^{2} and let |x| = ⟨x, x⟩^{1/2} be the Euclidean norm of x. Set B(x, r) = {y ∈ R^{2} : |x − y| < r} whenever x ∈ R^{2} and r > 0. Let dx denote Lebesgue measure on R^{2} and define k dimensional Hausdorff measure, in R^{2} , 0 < k ≤ 2, as follows: For fixed δ > 0 and E ⊆ R^{2} , let L(δ) = {B(x_{i}, r_{i})} be such that E ⊆ ⋃ B(x_{i}, r_{i}) and 0 < r_{i} < δ, i = 1, 2, …. Set

ϕ_{δ}^{k}(E) = inf_{L(δ)}(∑ α(k) r_{i}^{k}),

where α(k) denotes the volume of the unit ball in R^{k}. Then

H^{k}(E) = lim_{δ→0} ϕ_{δ}^{k}(E), 0 < k ≤ 2.

If O is open and 1 ≤ q ≤ ∞, let W^{1,q}(O) be the space of equivalence classes of functions u with distributional gradient ∇u = (u_{x_{1}}, u_{x_{2}}), both of which are q th power integrable on O. Let

∥u∥_{1,q} = ∥u∥_{q} + ∥∇u∥_{q}

be the norm in W^{1,q}(O) where ∥ · ∥_{q} denotes the usual Lebesgue q -norm in O. Let C_{0}^{∞}(O) be infinitely differentiable functions with compact support in O and let W_{0}^{1,q}(O) be the closure of C_{0}^{∞}(O) in the norm of W^{1,q}(O). Let Ω be a domain (i.e. an open connected set) and suppose that the boundary of Ω (denoted ∂Ω) is bounded and non-empty. Let N be a neighborhood of ∂Ω, p fixed, 1 < p < ∞, and u a positive weak solution to the p Laplace partial differential equation in Ω ∩ N. That is, u ∈ W^{1,p}(Ω ∩ N) and

2000 Mathematics Subject Classification: Primary 35J65; Secondary 31A15.

Both authors were partially supported by NSF grant 0139748.

(1.1)  ∫ |∇u|^{p−2}⟨∇u, ∇θ⟩ dx = 0

whenever θ ∈ W_{0}^{1,p}(Ω ∩ N). Observe that if u is smooth and ∇u ≠ 0 in Ω ∩ N, then ∇ · (|∇u|^{p−2} ∇u) ≡ 0, in the classical sense, where ∇· denotes divergence. We assume that u has zero boundary values on ∂Ω in the Sobolev sense. More specifically if ζ ∈ C_{0}^{∞}(N), then u ζ ∈ W_{0}^{1,p}(Ω∩N). Extend u to N \Ω by putting u ≡ 0 on N \ Ω. Then u ∈ W^{1,p}(N) and it follows from (1.1), as in [HKM], that there exists a positive finite Borel measure μ on R^{2} with support contained in ∂Ω and the property that

(1.2)  ∫ |∇u|^{p−2} ⟨∇u, ∇ϕ⟩ dx = − ∫ ϕ dμ

whenever ϕ ∈ C_{0}^{∞}(N). For the reader’s convenience we outline another proof of existence for μ under the assumption that u is continuous in N. We claim that it suffices to show

∫ |∇u|^{p−2} ⟨∇u, ∇ϕ⟩ dx ≤ 0

whenever ϕ ≥ 0 and ϕ ∈ C_{0}^{∞}(N). Indeed once this claim is established, we get existence of μ as in (1.2) from basic Caccioppoli estimates (see Lemma 2.6) and the same argument as in the proof of the Riesz representation theorem for positive linear functionals on the space of continuous functions. To prove our claim we note that θ = [(η + max[u − ε, 0])^{ε} − η^{ε}]ϕ can be shown to be an admissible test function in (1.1) for small η > 0. From (1.1) we see that

∫_{{u≥ε}∩N}[(η + max[u − ε, 0])^{ε} − η^{ε}]|∇u|^{p−2}⟨∇u, ∇ϕ⟩ dx ≤ 0.

Using dominated convergence, letting first η and then ε → 0 we get our claim. We note that if ∂Ω is smooth enough, then

(1.3)  dμ = |∇u|^{p−1} dH^{1}|_{∂Ω}.

In this paper we study for given p, 1 < p < ∞, the Hausdorff dimension of μ (denoted H-dim μ) defined as follows:

H-dim μ = inf{k : there exists E Borel ⊂ ∂Ω with H^{k}(E) = 0 and μ(E) = μ(∂Ω)}.

On the dimension of p-harmonic measure 461

To outline previous work, let p = 2 and u be the Green’s function for Ω with pole at some point x_{0} ∈ Ω. Then μ is called harmonic measure for Ω relative to x_{0} . Carleson [C] showed H-dim μ = 1 when ∂Ω is a snowflake and that H-dim μ ≤ 1 for any scale invariant Cantor set. He was also the first to recognize the importance of ∫_{∂Ω_{n}} |∇g_{n}| log |∇g_{n}| dH^{1} where g_{n} is the Green’s function for Ω_{n} with pole at x_{0} ∈ Ω_{0} and (Ω_{n}) is an increasing sequence of smooth domains whose union is Ω. Later Makarov [M] proved for any simply connected domain Ω ⊂ R^{2} , that H-dim μ = 1. Jones and Wolff [JW] proved that H-dim μ ≤ 1 when Ω ⊂ R^{2} and μ exists. Wolff [W1] strengthened [JW] by showing that harmonic measure is concentrated on a set of σ -finite H^{1} measure. We also mention results of Batakis [Ba], Kaufmann–Wu [KW], and Volberg [V] who showed for certain fractal domains and domains whose complements are Cantor sets that

(1.4)  Hausdorff dimension of ∂Ω = inf{k : H^{k}(∂Ω) = 0} > H-dim μ.

Finally we note that higher-dimensional results for the dimension of harmonic measure can be found in [B], [W], and [LVV]. In this paper we prove the following theorems.

Theorem 1. Let u, μ be as in (1.1), (1.2). If ∂Ω is a certain snowflake and 1 < p < 2, then H-dim μ > 1 while if 2 < p < ∞, then H-dim μ < 1.

Theorem 2. Let u, μ be as in (1.1), (1.2). If ∂Ω is a certain self-similar Cantor set and 2 < p < ∞, then H-dim μ < 1.

Theorem 3. Let u, μ be as in (1.1), (1.2). If ∂Ω is a quasicircle, then H-dim μ ≤ 1 for 2 ≤ p < ∞, while H-dim μ ≥ 1 for 1 < p ≤ 2.

The snowflakes and Cantor sets we consider are defined in Sections 3 and 5, respectively. The definition of a k -quasicircle, 0 < k < 1, is given in Section 2. As motivation for our theorems we note that Wolff in [W] was able to estimate the dimension of certain snowflakes in R^{3} . The first step in his construction was to determine the sign of

(1.5)  ∫_{∂_{D}(ε)} |∇g̃|( · , ε) ln |∇g̃( · , ε)| dH^{2}

where D̃ (ε) is a certain domain with smooth boundary and g̃( · , ε) is Green’s function for the Laplacian in D̃ (ε) with pole at ∞. In analogy with Wolff and for p fixed, 1 < p < ∞, put D(ε) = {x ∈ R^{2} : x_{2} > ε ψ(x_{1})} where ψ ∈ C_{0}^{∞}(R) with ψ ≡ 0 in R \ (−1, 1). Let g( · , ε) be a positive weak solution to the p-Laplace equation in D(ε) with g( · , ε) = 0 continuously on ∂D(ε) and g(x, ε) = x_{2} + ω(x, ε) where ω( · , ·) is infinitely differentiable in \overline{D} (ε) × (−ε_{0}, ε_{0}) and

|∇ω| ≤ k(1 + |x|)^{−2}, x ∈ D(ε),

for some constant k independent of ε ∈ (−ε_{0}, ε_{0}). Existence of g( · , ε) and ω( · , ε) can be deduced by Picard iteration and Schauder type estimates for ε_{0} > 0, sufficiently small. In fact proceeding operationally, assume g(x, ε) = x_{2} + ω(x, ε) exists. Assuming that ∇g ≠ 0 in D(ε) and writing out the p-Laplace equation for g, one deduces that if w(x_{1}, x_{2}) = ω(x_{1}, \sqrt{p − 1} x_{2}), then w satisfies

Δw(x) = F (w_{x_{1}}, \frac{w_{x_{2}}}{\sqrt{p − 1}}, w_{x_{1}x_{1}}, \frac{w_{x_{1}x_{2}}}{\sqrt{p − 1}} , \frac{w_{x_{2}x_{1}}}{\sqrt{p − 1}} , \frac{w_{x_{2}x_{2}}}{p − 1} ) = H(w, x),

x ∈ D(ε), where

\begin{aligned}F (q_{1}, q_{2}, q_{11}, q_{12}, q_{21}, q_{22}) & = −(p − 2) ∑_{i,j=1}^{2}q_{i}q_{j}q_{ij} \\ & − (2q_{2} + ∑_{i,j=1}^{2} q_{i}^{2})(∑_{i=1}^{2} q_{ii}) − 2(p − 2) ∑_{i,j=1}^{2} q_{i}q_{ij}. \end{aligned}

Also, w = −\sqrt{p − 1} x_{2} on ∂D(ε). Let w_{0} be the bounded solution to Laplace’s equation in D(ε) with w_{0} = −\sqrt{p − 1} x_{2} on ∂D(ε). Proceeding by induction, let w_{n+1} for n = 1, 2, …, be the bounded solution to

Δw_{n+1}(x) = H(w_{n}, x) in D(ε) with w_{n+1} = −\sqrt{p − 1} x_{2} continuously on ∂D(ε).

Using Schauder type estimates one can show for ε_{0} > 0, sufficiently small that lim_{n→∞} w_{n} = w exists with the desired smoothness properties. Thus g( · , ε) exists.

From (1.3) we get that the analogue of (1.5) for p fixed, 1 < p < ∞, is

(1.6)  I(ε) = ∫_{∂D(ε)} |∇g( · , ε)|^{p−1} ln |∇g( · , ε)| dH^{1}.

Following Wolff one calculates that I(0) = I′(0) = 0 and

I′′(0) = \frac{p − 2}{p − 1} ∫_{R}(\frac{dψ}{dx_{1}} )^{2} dx_{1}.

Now if ε_{0} is small enough, then I has three continuous derivatives on (−ε_{0}, ε_{0}) and so by Taylor’s theorem, (1.7) I(ε) > 0 when p > 2 and I(ε) < 0 for 1 < p < 2, when ε is sufficiently small.

Initially we found it quite surprising that (1.7) held for fixed p, independently of ψ, especially in view of the examples in [W], [LVV]. Our first attempt at explaining (l.7) was to observe that if ∇g( · , ε) ≠ 0 in D(ε) and v = log |∇g| satisfies

(1.8)  ∇ · (|∇g|^{p−2}∇v) < 0 (> 0) when 1 < p < 2 (p > 2)

then (1.7) follows from the divergence theorem applied to |∇g|^{p−2}[g∇v − v∇g]. However a direct calculation shows no reason for (1.8) to hold. Next we tried to imitate Wolff’s construction in order to produce examples of snowflakes where the conclusion of Theorem 1 held. To indicate the difficulties involved we note that Wolff shows Carleson’s integral over ∂Ω_{n} can be estimated at the nth step in the construction of certain snowflakes ⊂ R^{3} , provided the integral in (1.5) has a sign for small ε > 0. His calculations make key use of a boundary Harnack inequality for positive harmonic functions vanishing on a portion of the boundary in a non-tangentially accessible domain (see [JK]). Thus we prove a ‘rate theorem’ for the ratio of two positive p harmonic functions u_{1} , u_{2} which are defined in B(z, r) ∩ Ω and vanish continuously on B(z, r) ∩ ∂Ω, whenever z ∈ ∂Ω and ∂Ω is a quasicircle. We show that u_{1}/u_{2} is bounded in B(z, r/2) ∩ Ω, by a constant that depends only on p, Ω, when 1 < p < ∞ (see Lemma 2.16), but are not able to show u_{1}/u_{2} is Hölder continuous in Ω ∩ B(z, r/2) when 1 < p < ∞, p ≠ 2, as is the case when p = 2 (see [JK]). The p = 2 argument for Hölder continuity uses linearity of the Laplacian which is clearly not available for the p-Laplacian. Using just boundedness in the boundary Harnack inequality, we are still able to deduce that μ has a certain weak mixing property. An argument of Carleson–Wolff can then be applied to obtain an invariant ergodic measure ν on ∂Ω (with respect to a certain shift) satisfying (see Section 3),

(1.9)  μ, ν are mutually absolutely continuous.

From ergodicity and (1.9) it follows that the ergodic theorem of Birkhoff and entropy theorem of Shannon–McMillan–Breiman can be used to get that

(1.10)  lim_{r→0} \frac{log μ[B(x, r)]}{log r} = H-dim μ for μ almost every x ∈ ∂Ω.

In [W], Wolff uses Hölder continuity of the ratio in order to make effective use of (1.10) in his estimates of H-dim μ. We first tried to avoid the use of Hölder continuity in our estimates by a finess type argument which was supposed to take advantage of the constant sign in (1.7) when p ≠ 2 is fixed. However, later this argument was shown to be incorrect because of a calculus type mistake. Finally in desperation we returned to our original idea of using the divergence theorem and finding a partial differential equation for which u is a solution and v = log |∇u| is a subsolution (super solution) when p > 2 (1 < p < 2). To describe our efforts we note for u as in Theorem 1, that if η ∈ R^{2} with |η| = 1, while ∇u is nonzero and sufficiently smooth in Ω ∩ N, then ζ = ⟨ ∇u, η ⟩, is a strong solution in Ω ∩ N to

(1.11)  Lζ = ∇ · [(p − 2)|∇u|^{p−4}⟨∇u, ∇ζ⟩∇u + |∇u|^{p−2}∇ζ] = 0.

Clearly,

(1.12)  Lu = (p − 1)∇ · [|∇u|^{p−2}∇u] = 0 in Ω ∩ N.

464 Björn Bennewitz and John L. Lewis (1.11) can be rewritten in the form

(1.13)  Lζ = ∑_{i,k=1}^{2} \frac{∂}{∂x_{i}} [b_{ik}(x)ζ_{x_{k}}(x)] = 0,

where at x ∈ Ω ∩ N,

(1.14)  b_{ik}(x) = |∇u|^{p−4}[(p − 2)u_{x_{i}}u_{x_{k}} + δ_{ik}|∇u|^{2}](x), 1 ≤ i, k ≤ 2,

and δ_{ij} is the Kronecker δ.

Next we assume at x that

(1.15)  ∇u(x) = (1, 0)|∇u(x)|

which is permissible since (1.11) is rotationally invariant. Then

v_{x_{k}} = |∇u|^{−2} ∑_{l=1}^{2} u_{x_{l}}u_{x_{l}x_{k}}

and so

Lv = ∑_{i,k=1}^{2} \frac{∂(b_{ik}v_{x_{k}})}{∂x_{i}} = ∑_{i=1}^{2} \frac{∂}{∂x_{i}} (|∇u|^{−2} ∑_{k,l=1}^{2} b_{ik}u_{x_{l}} u_{x_{l}x_{k}}).

Using (1.13) on the right-hand side of this display with η = (1, 0), (0, 1), we get

(1.16)  \begin{aligned}Lv & = −2|∇u|^{−4} ∑_{i,k,l,m=1}^{2} b_{ik}(u_{x_{l}}u_{x_{l}x_{k}}u_{x_{m}}u_{x_{m}x_{i}}) \\ & + |∇u|^{−2} ∑_{i,k,l=1}^{2}b_{ik}u_{x_{l}x_{i}}u_{x_{l}x_{k}} = T_{1} + T_{2}. \end{aligned}

From (1.14), (1.15) we see at x that

(1.17)  b_{11} = (p − 1)|∇u|^{p−2}, b_{22} = |∇u|^{p−2}, and b_{12} = b_{21} = 0.

Also from (1.12), (1.15) we find that

(1.18)  (p − 1)u_{x_{1}x_{1}} + u_{x_{2}x_{2}} = 0.

Using (1.17), (1.18) in the definitions of T_{1} , T_{2} we obtain at x, T_{1} = −2|∇u|^{p−4}[(p − 1)(u_{x_{1}x_{1}})^{2} + (u_{x_{1}x_{2}})^{2}] and

T_{2} = p|∇u|^{p−4}[(p − 1)(u_{x_{1}x_{1}})^{2} + (u_{x_{1}x_{2}})^{2}].

Putting these equalities for T_{1} , T_{2} in (1.16) we deduce

(1.19)  Lv = (p − 2)|∇u|^{p−4}[(p − 1)(u_{x_{1}x_{1}})^{2} + (u_{x_{1}x_{2}})^{2}].

From (1.19) we conclude that

(1.20)  Lv ≥ 0 when p > 2 and Lv ≤ 0 when p < 2.

The sign in (1.7) can now be explained by using (1.20) and applying the divergence theorem to the vector field whose ith component (i = 1, 2) is

u ∑_{k=1}^{2} b_{ik}v_{x_{k}} − v ∑_{k=1}^{2} b_{ik}u_{x_{k}}.

Hence we use (1.19), (1.20), and the divergence theorem in place of Hölder continuity of the ratio to make estimates and finally get Theorem 1. Our examples are more general than in [W] as our estimates do not require a smallness assumption on the Lipschitz norm of the piecewise linear function defining a snowflake. Theorem 2 is proved similarly. That is, we follow the general scheme of Carleson–Wolff and use (1.19)–(1.20) to make final estimates. To prove Theorem 3 we follow [M] and first prove an integral inequality involving log |∇u| (see Lemma 6.1). Theorem 3 follows easily from this inequality and the estimates in Section 2 for u, μ. The theorems are proved in the order they were conceived.

As for the plan of this paper, in Section 2 we list smoothness results for the p-Laplacian. We also list some results for quasiregular mappings and discuss their relationship to ∇ũ, where ũ is a certain p-harmonic function satisfying the hypotheses of Theorem 1 and/or Theorem 2. In Section 2 we also prove the ‘rate theorem’ discussed above and then use it in Section 3 to set up the ergodic apparatus necessary to prove Theorem 1. Finally in Section 4, we obtain Theorem 1. In Section 5, we prove Theorem 2. In Section 6 we prove Theorem 3 and make concluding remarks.

Finally we remark that the term p-harmonic measure was first used by Martio in [Mar]. His definition and our definition agree for p = 2 provided u is chosen to be the Green’s function corresponding to the Laplacian with pole at some point in D. However, for p ≠ 2, the two definitions are quite different due to the nonlinearity of the p-Laplace equation. In fact Martio’s measure need not even be subadditive (see [LMW] for references and examples).

2. Basic estimates

We begin with some definitions. A Jordan curve J is said to be a k - quasicircle, 0 < k < 1, if J = f(∂B(0, 1)) where f ∈ W^{1,2}(R^{2}) is a homeomorphism of R^{2} and

(2.1)  |fz̄| ≤ k|f_{z}|, H^{2} almost everywhere in R^{2}.

Here we are using complex notation, i = \sqrt{−1}, z = x_{1} + ix_{2} , 2fz̄ = f_{x_{1}} + if_{x_{2}} , 2f_{z} = f_{x_{1}} − if_{x_{2}} . We say that J is a quasicircle if J is a k -quasicircle for some 0 < k < 1. Let w_{1} , w_{2} be distinct points on the Jordan curve J and let J_{1} , J_{2} be the arcs on J with endpoints w_{1} , w_{2} . Then J is said to satisfy the Ahlfors threepoint condition provided there exists 1 ≤ M < ∞ such that whenever w_{1}, w_{2} ∈ J, we have

(2.2)  min{diam J_{1}, diam J_{2}} ≤ M|w_{1} − w_{2}|.

Ω is said to be a uniform domain provided there exists M̂ , 1 ≤ M̂ < ∞, such that if w_{1}, w_{2} ∈ Ω, then there is a rectifiable curve γ: [0, 1] → Ω with γ(0) = w_{1} , γ(1) = w_{2} , and

(2.3)  \begin{matrix}(a) H^{1}(γ) ≤ M̂ |w_{1} − w_{2}|, \\ (b) min\{H^{1}(γ([0, t])), H^{1}(γ([t, 1]))\} ≤ M̂ d(γ(t), ∂Ω). \end{matrix}

Here as in the sequel, d(E, F) denotes the distance between the non-empty sets E and F. If 1 ≤ M̃ < ∞ and Ω is a domain, then a ball B(w, r) ⊂ Ω is said to be M̃ non-tangential provided

M̃ r > d(B(w, r), ∂Ω) > M̃^{−1}r.

If w_{1}, w_{2} ∈ Ω, then a Harnack-chain from w_{1} to w_{2} in Ω is a sequence of M̃ - non-tangential balls such that the first ball contains w_{1} , the last ball contains w_{2} , and consecutive balls intersect. A domain Ω is called non-tangentially accessible (NTA) if there exist M̃ (as above) such that:

(2.4) (α) Corkscrew condition. For any w ∈ ∂Ω, 0 < r ≤ diam Ω, there exists a = a_{r}(w) ∈ Ω such that M̃^{−1}r < |a − w| < r and d(a, ∂Ω) > M̃^{−1}r,

(2.4) (β) R^{2} \ Ω satisfies the corkscrew condition,

(2.4) (γ) Harnack chain condition. Given ε > 0, w_{1}, w_{2} ∈ Ω, d(w_{j}, ∂Ω) > ε, and |w_{1} − w_{2}| < Cε, there is a Harnack chain from w_{1} to w_{2} whose length depends on C but not on ε.

In (2.4)(α), diam Ω denotes the diameter of Ω. For use in the sequel we list the following equivalences.

Lemma 2.5. Ω is a uniform domain if and only if (2.4)(α), (2.4)(γ) hold. If ∂Ω = J is a Jordan curve, then the conditions:

J is a quasicircle,

J satisfies the Ahlfors three point condition, Ω is a uniform domain, Ω is non-tangentially accessible,

all imply each other and constants in one definition can be determined from the constants in another definition.

Proof. See [G].

Note that Ω in Lemma 2.5 can be either of the two components of R^{2} \ J. In the sequel c will denote a positive constant ≥ 1 (not necessarily the same at each occurrence), which may depend only on p, unless otherwise stated. In general, c(a_{1}, …, a_{n}) denotes a positive constant ≥ 1, which may depend only on p, a_{1}, …, a_{n}, not necessarily the same at each occurrence. We also assume that Ω is a domain and 0 < r < diam ∂Ω < ∞. Next we state some interior and boundary estimates for ũ a positive weak solution to the p-Laplacian in B(w, 4r) ∩ Ω with ũ ≡ 0 in the Sobolev sense on ∂Ω ∩ B(w, 4r) when this set is non-empty. More specifically, ũ ∈ W^{1,p}(B(w, 4r) ∩ Ω) and (1.1) holds whenever θ ∈ W_{0}^{1,p}(B(w, 4r) ∩ Ω). Also ζũ ∈ W_{0}^{1,p}(B(w, 4r) ∩ Ω) whenever ζ ∈ C_{0}^{∞}(B(w, 4r)). Extend ũ to B(w, 4r) by putting ũ ≡ 0 on B(w, 4r) \ Ω. Then there exists a locally finite positive Borel measure μ̃ with support ⊂ B(w, 4r)∩∂Ω and for which (1.2) holds with u replaced by ũ and ϕ ∈ C_{0}^{∞}(B(w, 4r)). Let max_{B(z,s)} ũ, min_{B(z,s)} ũ be the essential supremum and infimum of ũ on B(z, s) whenever B(z, s) ⊂ B(w, 4r).

Lemma 2.6. Let ũ be as above. Then

c^{−1}r^{p−2} ∫_{B(w,r/2)} |∇ũ|^{p} dx ≤ max_{B(w,r)} ũ^{p} ≤ cr^{−2} ∫_{B(w,2r)} ũ^{p} dx.

If B(w, 2r) ⊂ Ω, then

max_{B(w,r)} ũ ≤ c min_{B(w,r)} ũ.

Proof. The first display in Lemma 2.6 is a standard subsolution estimate while the second display is a standard weak Harnack estimate for positive weak solutions to nonlinear partial differential equations of p-Laplacian type (see [S]).

Lemma 2.7. Let ũ be as in Lemma 2.6. Then ũ has a representative in W^{1,p}(B(w, 4r) ∩ Ω) (also denoted ũ) with Hölder-continuous partial derivatives in B(w, 4r) ∩ Ω. That is, for some σ = σ(p) ∈ (0, 1) we have

c^{−1}|∇ũ(w_{1}) − ∇ũ(w_{2})| ≤ (|w_{1} − w_{2}|/s)^{σ} max_{B(z,s)} |∇ũ| ≤ cs^{−1}(|w_{1} − w_{2}|/s)^{σ} max_{B(z,2s)} ũ

whenever w_{1}, w_{2} ∈ B(z, s) and B(z, 4s) ⊂ B(w, 4r) ∩ Ω.

Proof. The proof of Lemma 2.7 can be found in [D], [L1] or [T] and in fact is true when B(w, 4r) ∩ Ω ⊂ R^{n}. In R^{2} the best Hölder exponent in Lemma 2.7 is known when p > 2 while for 1 < p ≤ 2 a solution has continuous second partials (see [IM]).

In order to describe further (R^{2} ) results for solutions to the p-Laplacian we note that h: B(w, 4r)∩Ω → R^{2} is said to be quasiregular in B(w, 4r) ∩ Ω provided h ∈ W^{1,2}(B(w, 4r) ∩ Ω) and (2.1) holds with f replaced by h in B(w, 4r) ∩ Ω. From a factorization theorem for quasiregular mappings, it then follows that h = τ ∘ f where f is quasiconformal in R^{2} and τ is an analytic function on f(B(w, 4r) ∩ Ω). We have

Lemma 2.8. If ũ is as in Lemma 2.6 and z = x_{1} +ix_{2} , then ũ_{z} is quasiregular in B(w, 4r) ∩ Ω for some 0 < k < 1 (depending only on p) and consequently ∇ũ has only isolated zeros in B(w, 4r) ∩ Ω.

Proof. For a proof of quasiregularity (see [ALR], [L]). The fact that the zeros of ∇ũ are isolated follow from the above factorization theorem and the corresponding theorem for analytic functions.

From Lemma 2.8 we deduce

Lemma 2.9. If ũ is as in Lemma 2.6, then ũ is real analytic in B(w, 4r) ∩ [Ω \ {x : ∇ũ(x) ≠ 0}]. Moreover if B(w, 4r) ⊂ Ω, ∇ũ ≠ 0 in B(w, 4r), and max_{B(w,2r)} |∇ũ| ≤ λ max_{B(w,r)} |∇ũ| then

max_{B(w,2r)} |∇ũ| ≤ c(λ) min_{B(w,r)} |∇ũ|, (+)

max_{x∈B(w,r)} ∑_{i,j=1}^{2} |ũ_{x_{i}x_{j}}|(x) ≤ c(λ)r^{−1} max_{x∈B(w,2r)} |∇ũ|(x), (++)

max_{x,y∈B(w,r/2)} ∑_{i,j=1}^{2} |ũ_{x_{i}x_{j}}(x) − ũ_{y_{i}y_{j}}(y)| (+ + +)

≤ c(λ)(|x − y|/r) max_{x∈B(w,r)} ∑_{i,j=1}^{2} |ũ_{x_{i}x_{j}}|(x).

Proof. To prove Lemma 2.9, we first observe from (1.1) with u replaced by ũ and Lemmas 2.7, 2.8 that

(2.10)  ũ ∈ W^{2,2}(B(w, 4r) ∩ Ω) and ∑_{i,k=1}^{2} a_{ik}(x)ũ_{x_{i}x_{k}}(x) = 0

for H^{2} almost every x ∈ B(w, 4r) ∩ Ω. Here a_{ik} = |∇u|^{2−p}b_{ik}, where b_{ik}, 1 ≤ i, k ≤ 2, are as in (1.14). It is easily checked that (a_{ik}) are measurable, L^{∞} bounded, and uniformly elliptic in B(w, 4r) ∩ Ω with L^{∞} norm and ellipticity constant depending only on p. Second, using this observation, Lemma 2.7, Schauder theory, and a bootstrap argument (see [GT, Section 9.6]), we get that ũ ∈ C^{∞}(Ω ∩ B(w, 4r)). Real analyticity of ũ (i.e. a power series expansion in x_{1}, x_{2} ) at points of B(w, 4r) ∩ Ω follows from a theorem of Hopf (see [H], [F], [F1]). To prove (+) we note from Lemma 2.8 that v = log |∇ũ| is a weak solution in B(w, 4r) to the divergence form partial differential equation (see [Re, Chapter III]),

∑_{i,j=1}^{2} \frac{∂}{∂x_{i}} (A_{ij}(x)v_{x_{j}}) = 0

where if Dũ_{z} denotes the Jacobian matrix of ũ_{z}, det Dũ_{z} the Jacobian of ũ_{z} and D^{t}ũ_{z} the transpose of Dũ_{z}, then A = (A_{ij}) is defined by

A(x) = \{ \begin{matrix}[det Dũ_{z}](D^{t}ũ_{z} · Dũ_{z})^{−1} if Dũ_{z} is invertible, \\ Identity matrix, otherwise. \end{matrix}

From quasiregularity of ũ_{z} and/or (2.10) we find that (A_{ij}) are L^{∞} -bounded and uniformly elliptic (again with constants depending only on p). Using Harnack’s inequality for positive solutions to partial differential equations of this type (see [S]), applied to max_{B(w,2r)}v − v in B(w, r) we get (+). (++) and (+ + +) follow from (+), (2.10), and once again Schauder estimates.

Next we consider the behaviour of ũ near B(w, 4r) ∩ ∂Ω and the relation between ũ, μ̃. By a simply connected domain Ω we shall always mean that R^{2} \ Ω is a connected set of more than one point.

Lemma 2.11. Let ũ be as in Lemma 2.6 and w ∈ ∂Ω. If p > 2 and ∂Ω is bounded, then there exists α = α(p) ∈ (0, 1) such that ũ has a Hölder α continuous representative in B(w, r) (also denoted ũ). Moreover if x, y ∈ B(w, r) then

|ũ(x) − ũ(y)| ≤ c(|x − y|/r)^{α} max_{B(w,2r)} ũ.

If 1 < p ≤ 2, and Ω is simply connected, then this inequality is also valid when 1 < p ≤ 2, with α = α(p).

Proof. For p > 2, Lemma 2.11 is a consequence of Lemma 2.6 and Morrey’s theorem (see [E, Chapter 5]). If 1 < p ≤ 2 and Ω is simply connected we deduce from the interior estimates in Lemma 2.7 that it suffices to consider only the case when y ∈ B(w, r) ∩ ∂Ω. We then show for some θ = θ(p, k), 0 < θ < 1, that

(2.12)  max_{B(z,ϱ/4)} ũ ≤ θ max_{B(z,ϱ/2)} ũ whenever 0 < ϱ < r and z ∈ ∂Ω ∩ B(w, r).

(2.12) can then be iterated to get Lemma 2.11 for x, y as above. To prove (2.12) we use the fact that B(z, ϱ/4) ∩ ∂Ω and B(z, ϱ/4) have comparable p-capacities (see [HKM]) and estimates for subsolutions to elliptic partial differential equations of p-Laplacian type (see [GZ], [L]).

Lemma 2.13. Let ũ, Ω, w be as in Lemma 2.11. Assume also that Ω is a uniform domain. Then there exists c = c(M̂) with

max_{B(w,2r)} ũ ≤ cũ(a_{r}(w))

where M̂ is as in (2.3) and a_{r}(w) is as in (2.4)(α). Hence,

|ũ(x) − ũ(y)| ≤ c(|x − y|/r)^{α}ũ(a_{r}(w)) for x, y ∈ B(w, r).

Proof. The first display in Lemma 2.13 follows from Harnack’s principle in Lemma 2.6, Hölder continuity of ũ in Lemma 2.11, the fact that Ω is a uniform domain and a general argument which can be found in [CFMS]. The second display in Lemma 2.13 is a consequence of the first display and Lemma 2.11.

Lemma 2.14. Let ũ, Ω, w, r be as in Lemma 2.11 and μ̃ as in (1.2) relative to ũ. Then there exists c such that

c^{−1}r^{p−2}μ̃[B(w, r/2)] ≤ max_{B(w,r)} ũ^{p−1} ≤ cr^{p−2}μ̃[B(w, 2r)].

Proof. The proof of Lemma 2.14 when ũ is a subsolution to a class of partial differential equations that includes the p-Laplacian can be found in [KZ, Lemma 3.1]. Their proof of the above inequality essentially just uses Harnack’s inequality—Lemma 2.11 and is modeled on a previous proof for solutions to partial differential equations of p-Laplacian type in [EL, Lemma 1].

From Lemmas 2.13, 2.14 and Harnack’s principle, we note that if Ω is a uniform domain, x ∈ ∂Ω ∩ B(w, 4r), and B(x, 4ϱ) ⊂ B(w, 4r), then

(2.15)  ϱ^{p−2}μ̃(B(x, ϱ)) ≈ ũ(a_{ϱ}(x))^{p−1} ≈ ( max_{B(x,ϱ)} ũ)^{p−1},

where e ≈ f means e/f is bounded above and below by positive constants. In this case the constants depend only on p, M̂ .

Next we prove the rate theorem referred to in Section 1.

Lemma 2.16. Let ∂Ω be a quasicircle and ũ, μ̃, w as in Lemma 2.11. Let v > 0, ν be as defined above Lemma 2.6, (1.2), respectively, with ũ, μ̃ replaced by v, ν. If ũ(a_{r}(w)) = v(a_{r}(w)), then there exists c_{+} = c_{+}(M̂), M_{+} = M_{+}(M̂) ≥ 1, such that

c−1_{+} < \frac{ũ(x)}{v(x)} < c_{+} for all x ∈ B(w, M −1_{+}r) ∩ Ω.

Proof. Let γ: ] − 1, 1[→ R^{2} be a parametrization of ∂Ω ∩ B(w, 2r) such that γ(0) = w. Let r_{1} = r/c_{1} where c_{1} = c_{1}(M̂) ≥ 1 will be chosen later. Let t_{1} = sup{t < 0 : |γ(t) − w| = r_{1}}, t_{2} = inf{t > 0 : |γ(t) − w| = r_{1}}, z_{1} = γ(t_{1}), and z_{2} = γ(t_{2}). Then |w − z_{1}| = |w − z_{2}| = r_{1} and the part of ∂Ω between z_{1} and z_{2} is contained in B(w, r_{1}). If r_{2} = r_{1}/c_{1} , then from (2.2) we see for c_{1} large enough that B(z_{1}, r_{2}) ∩ B(z_{2}, r_{2}) = ∅. For any point ζ_{1} ∈ B(z_{1}, r_{2}) ∩ ∂Ω and ζ_{2} ∈ B(z_{2}, r_{2}) ∩ ∂Ω we can use (2.3) to construct a curve with endpoints ζ_{1} , ζ_{2} , in the following way: Take ϱ such that B(ζ_{i}, ϱ) ⊂ B(z_{i}, r_{2}) for i = 1, 2. Draw the curve from a_{ϱ}(ζ_{1}) to a_{ϱ}(ζ_{2}) guaranteed by (2.3). Similarly, connect a_{ϱ}(ζ_{1}) to a_{ϱ/2}(ζ_{1}) and then a_{ϱ/2}(ζ_{1}) to a_{ϱ/4}(ζ_{1}), etc. Since a_{ϱ/2^{n}}(ζ_{1}) → ζ_{1} as n → ∞ this curve ends up at ζ_{1} . We go from a_{ϱ}(ζ_{2}) to ζ_{2} in the same way. The total curve from ζ_{1} to ζ_{2} is denoted by Γ.

From our construction and (2.3) we note for c_{1} large enough that

(2.17)

(a) Γ \ {ζ_{1}, ζ_{2}} ⊂ B(w, r) ∩ Ω,
(b) H^{1}(Γ) ≤ c_{1}r,
(c) min{H^{1}(Γ([0, t])), H^{1}(Γ([t, 1]))} ≤ c_{1}d(Γ(t), ∂Ω).

Fix c_{1} satisfying the above requirements. Now suppose that u/v ≥ λ at some point in B(z, M −1_{+}r) ∩ Ω where M_{+}(M̂) is chosen so large that Γ ∩ B(w, M −1_{+}r_{1}) = ∅ independently of ζ_{1} ∈ B(z_{1}, r_{2}) ∩ ∂Ω, ζ_{2} ∈ B(z_{2}, r_{2}) ∩ ∂Ω. Existence of M_{+} follows from (2.17). Note from Lemma 2.11 that ũ ≡ v ≡ 0 on ∂Ω ∩ B(w, r) and that ũ, v are continuous in \overline{Ω} ∩ B(w, r). Using this note and the weak maximum principle for solutions to the p-Laplacian, we see that ũ/v ≥ λ at some point ξ on Γ. Then from (2.17), (2.15), and Harnack’s inequality we deduce for some s, 0 < s < r/2, that

(2.18)  \frac{μ[B(ζ, s)]}{ν[B(ζ, s)]} ≥ c^{−1}[\frac{u(ξ)}{v(ξ)}]^{p−1} ≥ c^{−1}λ^{p−1} = λ′

where ζ = ζ_{1} or ζ = ζ_{2} . Allowing ζ_{i} to vary in B(z_{i}, r_{2}) ∩ ∂Ω, i = 1, 2, we get a covering of either B(z_{1}, r_{2}) ∩ ∂Ω or B(z_{2}, r_{2}) ∩ ∂Ω by balls of the form B_{ζ} = B(ζ, s). Assume for example that G = B(z_{1}, r_{2}) ∩ ∂Ω is covered by balls of this type. Then using a standard covering argument we get a subcovering {B_{ζ_{n}}} of G such that the balls with one-fifth the diameter of the original balls but the same centers, call them {B_{ζ_{n}}^{∗}}, are disjoint. From (2.17), (2.18), and Lemma 2.14 we deduce for some c = c(M̂) that

(2.19) λ′ν(G) ≤ λ′ν(⋃_{n} B_{ζ_{n}}) ≤ ∑_{n}μ(B_{ζ_{n}}) ≤ c∑_{n} μ(B_{ζ_{n}}^{∗}) ≤ c^{2}μ(B(w, 2r)).

From (2.15) with u, μ replaced by v, ν, (2.3) and Harnack’s principle we see for some c = c(M̂) that

(2.20)  ν(B(w, 2r)) ≤ cν(G)

which together with (2.19), (2.15) and u(a_{r}(w)) = v(a_{r}(w)) implies for some c_{∗} = c_{∗}(M̂),

(2.21)  c_{∗}^{−1}λ′ ≤ \frac{μ(B(w, 2r))}{ν(B(w, 2r))} ≤ c_{∗}.

From (2.21) we conclude the validity of Lemma 2.16.

We observe for later use that Lemma 2.16 remains valid for suitable c_{+} if B(w, M −1_{+}r) is replaced by B(w, r/2), as follows easily from applying Lemma 2.16 at points in ∂Ω∩B(w, r/2) and using Harnack’s inequality. Let HΔK = (H \K) ∪(K \ H) be the symmetric difference of H and K. For use in Section 3 we need

Lemma 2.22. Let ∂Ω̃_{1}, ∂Ω̃_{2} be quasicircles, w ∈ ∂Ω̃_{1} ∩ ∂Ω̃_{2} and ũ_{1} , ũ_{2} as in Lemma 2.6 with ũ, Ω replaced by ũ_{i}, Ω̃_{i}, i = 1, 2. Let μ̃_{i} be the corresponding measures for ũ_{i}, i = 1, 2, and suppose E ⊂ ∂Ω̃_{1} ∩ ∂Ω̃_{2} is Borel. Also suppose that C_{1} ≥ 1, B(w, C_{1}^{−1}r) ∩ ∂Ω̃_{1} ⊂ E, diam E ≤ r and d(E, ∂Ω̃_{1}Δ∂Ω̃_{2}) > C_{1}^{−1}r. If Y ⊂ E, then for some C = C(M̂ , C_{1}),

C^{−1} \frac{μ̃_{2}(Y)}{μ̃_{1}(Y)} ≤ \frac{μ̃_{2}(E)}{μ̃_{1}(E)} ≤ C \frac{μ̃_{2}(Y)}{μ̃_{1}(Y)}

whenever Y ⊂ E is Borel and μ̃_{1}(Y) ≠ 0.

Proof. In Lemma 2.22, M̂ is a constant for which (2.3) is valid with Ω replaced by Ω̃_{i}, i = 1, 2. The assumptions in the lemma imply that we can cover E with a bounded number of balls of radius \frac{1}{2}C_{1}^{−1}r whose doubles do not intersect ∂Ω̃_{1}Δ∂Ω̃_{2} . In each of these balls we can apply the rate theorem and use Harnack’s inequality to get

C^{−1} \frac{ũ_{1}(a_{r}(w))}{ũ_{2}(a_{r}(w))} < \frac{ũ_{1}(z)}{ũ_{2}(z)} < C \frac{ũ_{1}(a_{r}(w))}{ũ_{2}(a_{r}(w))}

provided d(z, E) < \frac{1}{4}C_{1}^{−1}r and z ∈ Ω̃_{1} ∩ Ω̃_{2} . Furthermore using (2.15) we get

(2.23)  C^{−1} \frac{μ̃_{1}(E)}{μ̃_{2}(E)} < (\frac{ũ_{1}(z)}{ũ_{2}(z)})^{p−1} < C \frac{μ̃_{1}(E)}{μ̃_{2}(E)}

where C in (2.23) has the same dependence as in Lemma 2.22. Using (2.23) and (2.15) once again we conclude for y ∈ E, 0 < ϱ < r,

\frac{μ̃_{1}(B(y, ϱ))}{μ̃_{2}(B(y, ϱ))} ≈ \frac{μ̃_{1}(E)}{μ̃_{2}(E)}

so if

\frac{dμ̃_{1}}{dμ̃_{2}} (y) = lim_{ϱ→0} \frac{μ̃_{1}(B(y, ϱ))}{μ̃_{2}(B(y, ϱ))}

then

\frac{dμ̃_{1}}{dμ̃_{2}} (y) ≈ \frac{μ̃_{1}(E)}{μ̃_{2}(E)}.

From Theorem 7.15 in [R] it follows that μ̃_{1} has no singular part with respect to μ̃_{2} and vice versa. Thus by the Radon–Nikodym theorem

\frac{μ̃_{1}(Y)}{μ̃_{2}(Y)} = \frac{∫_{Y} \frac{dμ̃_{1}}{dμ̃_{2}} dμ̃_{2}}{μ̃_{2}(Y)} ≈ \frac{μ̃_{1}(E)}{μ̃_{2}(E)}

where the proportionality constant depends on C_{1} , M̂ , p.

Next suppose that Ω is simply connected and ũ_{i}, μ̃_{i}, i = 1, 2, are as in (1.1), (1.2) with u, μ replaced by ũ_{i}, μ̃_{i}. Then from the maximum principle for p-harmonic functions we first see for some c = c(ũ_{1}, ũ_{2}, N) that ũ_{i} ≤ cũ_{j}, i, j ∈ {1, 2} and thereupon from Lemma 2.14 that

c^{−1}μ̃_{1}(B(w, r/2)) ≤ μ̃_{2}(B(w, r)) ≤ cμ̃_{1}(B(w, 2r))

whenever w ∈ ∂Ω and 0 < r < diam ∂Ω. This inequality and basic measure theoretic arguments, similar to the above, imply that

(2.24)  H-dim μ̃_{1} = H-dim μ̃_{2}.

Thus to estimate H-dim μ where u is as in (1.1), (1.2) and Ω is simply connected it suffices to find the Hausdorff dimension of either μ_{1} or μ_{2} , corresponding to u_{1} , u_{2} , respectively, where u_{1} , u_{2} are defined as follows. If Ω is bounded we write Ω = Ω_{1} , while if Ω is unbounded we put Ω = Ω_{2} . In the bounded case choose x̂ in Ω_{1} and r̂ > 0 so that B(x̂, 4r̂) ⊂ Ω_{1} . In the unbounded case, choose x̂ ∈ R^{2} \ Ω and R̂ > 0 so that ∂B(x̂, R̂/4) ⊂ Ω_{2} . Also,

(2.25)  either r̂ ≈ d(x̂, ∂Ω) or R̂ ≈ diam Ω.

In the bounded case, let u_{1} be a weak solution to the p-Laplacian in D_{1} = Ω_{1} \ B(x̂, r̂) with u_{1} = 0 on ∂Ω_{1} and u_{1} = 1 on ∂B(x̂, r̂) in the Sobolev sense. In the unbounded case, let u_{2} be a weak solution to the p-Laplacian in D_{2} = Ω_{2} ∩ B(x̂, R̂) with u_{2} = 0 on ∂Ω_{2} and u_{2} = 1 on ∂B(x̂, R̂) in the Sobolev sense. Existence of u_{1} , u_{2} follow from the usual variational argument (see [GT]). Let u′, D′, r′ = u_{1}, D_{1}, r̂, when Ω is bounded and u′, D′, r′ = u_{2}, D_{2}, R̂, otherwise.

Lemma 2.26. We have ∇u′ ≠ 0 in D′ and u′ is real analytic in an open set containing D′ ∪ ∂B(x̂, r′). Also if x ∈ D′ and ∂Ω is a quasicircle, then for some c = c(M̂) ≥ 1,

c^{−1}d(x, ∂Ω)^{−1}u′(x) ≤ |∇u′(x)| ≤ cd(x, ∂Ω)^{−1}u′(x)

and

∑_{l,m=1}^{2} |u′_{x_{l}x_{m}}|(x) ≤ cd(x, ∂Ω)^{−2}u′(x).

Proof. ∇u′ ≠ 0 in D′, is proved in [L, (1.4)]. Real analyticity of u′ in D′ then follows from Lemma 2.9. To outline the proof of real analyticity on ∂B(x̂, r′) suppose x̂ = 0 and r′ = r̂. Let f(y_{1}, y_{2}) = (y_{1}, y_{2} + \sqrt{r̂^{2} − y_{1}^{2}} ) when (y_{1}, y_{2}) ∈ H = B(0, r̂/2) ∩ {y : y_{2} > 0}. Then f(H) ⊂ D_{1} and f(∂H ∩ {y : y_{2} = 0}) ⊂ ∂B(0, r̂). Moreover w = 1 − u_{1} ∘ f is a weak solution to

(2.27)  ∇ · (⟨A∇w, ∇w⟩^{(p/2−1)}A∇w) = 0 in H,

where

A(y) = ( \begin{matrix}1 & \frac{−y_{1}}{\sqrt{r̂^{2} − y_{1}^{2}}} \\ \frac{−y_{1}}{\sqrt{r̂^{2} − y_{1}^{2}}} & \frac{r̂^{2}}{r̂^{2} − y_{1}^{2}} \end{matrix} )

and ∇w is regarded as a column matrix. Also, w = 0 on ∂H ∩ {y : y_{2} = 0} in the Sobolev sense. Note that A has real analytic coefficients which depend only on y_{1} . We now apply our previous program to w. That is, arguing as in [D], [L1], [T], we first show that w has Hölder continuous derivatives locally in H. Next using a boundary Harnack inequality for nondivergence form equations of Krylov and a Campanato type argument as in [Li], we see that w extends to a function with Hölder continuous derivatives in \overline{H} . Moreover partial derivatives of w satisfy the inequality in Lemma 2.7 with u replaced by w and B(z, s), B(z, 2s) replaced by \overline{H} . Before proceeding further we note that these arguments also give boundary regularity in higher dimensions. A somewhat simpler proof of the above results, valid only in two dimensions, will be given in the first author’s thesis. Second, using Schauder estimates and a bootstrap argument we get that w ∈ C^{∞}(\overline{H}). Third, real analyticity of w follows (once again) from a theorem of Hopf (see [H], [F], [F1]). Finally from w ≤ 1 we deduce that |∇w| ≤ cr̂^{−1} . Transforming back by f we get results for u_{1} in a neighborhood of (0, r̂). Covering ∂B(0, r̂) by such neighborhoods we conclude that u_{1} is real analytic on ∂B(0, r̂). Moreover, for some positive c,

c^{−1}r̂^{−1} ≤ |∇u_{1}| ≤ cr̂^{−1}, x ∈ B(x̂, [1 + c^{−1}]r̂) \ B(x̂, [1 − c^{−1}]r̂),

as follows for example from the analogue of Lemma 2.7 for the extension of u_{1} and barrier type estimates. As in Lemma 2.9, this inequality implies

∑_{l,m=1}^{2} |(u_{1})_{x_{l}x_{m}}| ≤ c^{2}r̂^{−2}, x ∈ B(x̂, [1 + c^{−2}]r̂) \ B(x̂, [1 − c^{−2}]r̂)

provided c is large enough. From these inequalities and (2.25) we see that to complete the proof of Lemma 2.26 when u′ = u_{1} and ∂Ω is a quasicircle, it remains to consider the case when x ∈ Ω_{1} \ B(x̂, [1 + c^{−2}]r̂). First suppose that x ∈ Ω_{1} and B(x, d(x)) ∩ B(x̂, [1 + c^{−2}]r̂) = ∅. Choose y ∈ B(x, d(x)) with u(y) = u(x)/2. We apply the mean value theorem of elementary calculus to u restricted to the line segment with endpoints, x, y. Then from this theorem and Lemma 2.13 we deduce the existence of c̃ = c̃(M̂) ≥ 4 and z such that y, z ∈ B(x, (1 − c̃^{−1})d(x)) and

u_{1}(x)/2 = |u_{1}(x) − u_{1}(y)| ≤ |∇u_{1}(z)| |x − y|.

Using this equality and Lemma 2.7 we see for some positive ĉ that if

t_{2} = [1 − (2c̃)^{−1}] d(x, ∂Ω), t_{1} = (1 − c̃^{−1})d(x, ∂Ω),

then

(2.28) ĉ^{−1}u_{1}(x)/d(x, ∂Ω) ≤ max_{B(x,t_{1})} |∇u_{1}| ≤ max_{B(x,t_{2})} |∇u_{1}| ≤ ĉu_{1}(x)/d(x, ∂Ω).

From (2.28), Lemma 2.9, and an iteration type argument, we conclude that Lemma 2.26 is valid for u_{1} at x. Lemma 2.26 for x ∈ Ω_{1} \ B(x̂, [1 + c^{−2}]r̂) follows from the previous special case, simple connectivity of Ω_{1} , and once again an iteration argument using Lemma 2.9. Thus in all cases Lemma 2.26 is valid when u′ = u_{1} . The proof of Lemma 2.26 for u′ = u_{2} is similar. We omit the details.

3. Construction of snowflakes and a shift invariant ergodic measure

We begin this section by constructing “snowflake”-type regions and showing that they are quasi circles. We follow the construction in [W]. Let ϕ: R → R be a piecewise linear function with supp ϕ ⊂] − 1, 1[. Let 0 < b, 0 < b̂ < b̃, and Q a line segment (relatively open) with center a_{Q}, length l(Q). If e is a given unit normal to Q define

T_{Q} = cch(Q ∪ {a_{Q} + bl(Q)e}),
T̃_{Q} = int cch(Q ∪ {a_{Q} − b̃l(Q)e}),
T̂_{Q} = int cch(Q ∪ {a_{Q} − b̂l(Q)e})

where cch E, int E, denote the closed convex hull and interior of the set E. Let e = −e_{2} and define

\begin{aligned}Λ & = \{x ∈ T_{]−1,1[} ∪ T̃_{]−1,1[} : x_{2} > ϕ(x_{1})\}, \\ ∂ & = \{x ∈ R^{2} : x_{1} ∈] − 1, 1[, x_{2} = ϕ(x_{1})\}. \end{aligned}

We assume that

(3.1)  ∂ ⊂ int (T_{]−1,1[} ∪ T̂_{]−1,1[}).

Suppose Ω is a domain with a piecewise linear boundary and Q ⊂ ∂Ω is a line segment with T_{Q} ∩ Ω = ∅, T̃_{Q} ⊂ Ω. Let e be the normal to Q pointing into T_{Q}. Form a new domain Ω̃ as follows: Let S be the affine map with S(] − 1, 1[) = Q, S(0) = a_{Q}, S(−e_{2}) = e. Let Λ_{Q} = S(Λ) and ∂_{Q} = S(∂). Then Ω̃∩(T_{Q}∪ T̃_{Q}) = Λ_{Q} and Ω̃ \ (T_{Q} ∪ T̃_{Q}) = Ω \ (T_{Q} ∪ T̃_{Q}). We call this process ‘adding a blip to Ω along Q’. Let Ω_{0} be one of the following domains: (a) the interior of the unit square with center at the origin, (b) the equilateral triangle with sidelength one and center at the origin, (c) the exterior of the set in (a) and (d) the exterior of the set in (b). We assume that b, b̂, b̃, ϕ are such that inductively we can define Ω_{n} as follows. If n ≥ 1 and Ω_{n−1} has been defined, then its boundary is given as ∂Ω_{n−1} = E_{n−1} ∪ (⋃_{\underline{G}_{n−1}} Q) where E_{n−1} is a discrete set of points and

(3.2)

(α) Each Q ∈ \underline{G}_{n−1} is a line segment with T_{Q} ∩ Ω_{n−1} = ∅, T̃_{Q} ⊂ Ω_{n−1}.

(β) If Q_{1}, Q_{2} ∈ \underline{G}_{n−1} then int T_{Q_{1}} ∩ int T_{Q_{2}} = ∅ and T̃_{Q_{1}} ∩ T̃_{Q_{2}} = ∅.
(γ) If Q_{1}, Q_{2} ∈ \underline{G}_{n−1} have a common endpoint, z, then there exists

an open half space P with z ∈ ∂P and

⋃_{i=1}^{2} T̃(Q_{i}) ∪ T (Q_{i}) ⊂ P ∪ {z}.

To form Ω_{n} add a blip along each Q ∈ \underline{G}_{n−1} . Then ∂Ω_{n} = E_{n−1} ∪ (⋃_{Q∈\underline{G}_{n−1}} ∂_{Q})

is decomposed as E_{n} ∪ (⋃_{Q∈\underline{G}_{n}} Q′) where Q′ are the line segments which make up ∂_{Q}. In this case we say that Q′ is directly descended from Q. Assume that (3.2) holds with Q, n−1 replaced by Q′, n. Also assume for Q′ directly descended from Q ∈ \underline{G}_{n−1} that

(3.3)  \begin{matrix}(+) T_{Q′} ∪ T̃_{Q′} ⊂ T_{Q} ∪ T̃_{Q}, \\ (++) T_{Q′} ∪ T̂_{Q′} ⊂ T_{Q} ∪ T̂_{Q}, \\ (+ + +) T_{Q′} ⊂ R^{2} \ Λ_{Q}, T̃_{Q′} ⊂ Λ_{Q}. \end{matrix}

So by assumption, the induction hypothesis is satisfied and the construction can continue. By induction, we obtain Ω_{n}, n = 1, 2, …. Assuming there exists (Ω_{n}) satisfying (3.2), (3.3) for n = 1, 2, …, we claim that Ω = lim_{n→∞}Ω_{n}, where convergence is in the sense of Hausdorff distance. In fact from (3.1)–(3.3) we deduce the existence of c ≥ 1 and θ, 0 < θ < 1, such that for m ≥ n ≥ 0,

(3.4)  sup_{x∈Ω_{m}} d(x, Ω_{n}) + sup_{x∈Ω_{n}} d(x, Ω_{m}) ≤ cθ^{n}.

Let

Ω = ⋂_{n=1}^{∞}( ⋃_{m=n}^{∞} Ω_{m}).

Using (3.4) and taking limits we deduce that (3.4) holds with Ω_{m} replaced by Ω. Hence our claim is true. We note that (3.1)–(3.3) are akin to the ‘open set condition’ for existence of a self similar set (see [Ma, Chapter 4]).

Next let \underline{G} = ⋃_{n=1}^{∞} \underline{G}_{n}. If Q, Q′ ∈ \underline{G} we say Q′ is descended from Q in n stages if there are Q_{0} = Q′, Q_{1}, …, Q_{n} = Q with Q_{j} directly descended from Q_{j+1} for j = 0, …, n − 1. If Q′ is descended from Q we write Q′ ≺ Q. Also if Q ∈ \underline{G}, let Γ_{Q} = ∂Ω ∩ [T_{Q} ∪ T̃_{Q}]. Note from (3.1)–(3.3) that if Q′ ∈ \underline{G}_{m}, Q ∈ \underline{G}_{n}, and m > n, then Γ_{Q′} ⊂ Γ_{Q} provided Q′ ≺ Q while otherwise Γ_{Q} ∩ Γ_{Q′} = ∅. We prove

Lemma 3.5. ∂Ω is a quasicircle.

Proof. To prove Lemma 3.5 it suffices to prove that

(3.6) ∂Ω_{m} satisfies the Ahlfors three-point condition in (2.2) for m = 1, 2, …,

with constant M independent of m. Indeed, once (3.6) is proved we can use Lemma 2.5 to get (f_{m}) a sequence of quasiconformal mappings of R^{2} and 0 < k < 1, independent of m, with

(3.7)  f_{m}(∂B(0, 1)) = ∂Ω_{m} and |(f_{m})z̄| ≤ k|(f_{m})_{z}|

for all z = x_{1} + ix_{2} in R^{2} . Using the fact that a subsequence of (f_{m}) converges uniformly on compact subsets of R^{2} to a quasiconformal f: R^{2} → R^{2} (see [A] or [Re]) and (3.4), we deduce that (3.7) holds with f_{m}, Ω_{m}, replaced by f, Ω. Thus Lemma 3.5 is true once we show that (3.6) holds. For this purpose note that by construction Ω_{m} is a piecewise linear Jordan curve with a finite number of segments. First assume z_{1} , z_{2} , z_{3} lie on line segments in ∂Ω_{m} which are descendants of the same side in \underline{G}_{0} and z_{3} lies between z_{1} , z_{2} . Then there exists a line segment Q_{0} ∈ \underline{G} with the property that z_{1} , z_{2} , z_{3} all are descendants of Q_{0} and this property is not shared by any descendant of Q_{0} . Suppose z_{i}, i = 1, 2, 3 belong to line segments descended from Q_{i} where Q_{i} ⊂ ∂_{Q_{0}} and Q_{1} ∩ Q_{2} = ∅. Let z_{i}^{∗} denote the projection of z_{i} on Q_{i}. Using (3.1)–(3.3) we see for b, b̃, small enough that

|z_{1} − z_{2}| ≥ |z_{1}^{∗} − z_{2}^{∗}|/2 and |z_{i} − z_{3}| ≤ 2|z_{i}^{∗} − z_{3}^{∗}| for i = 1, 2, ….

These inequalities and Lipschitzness of ϕ now imply the existence of M, 1 ≤ M < ∞, depending only on b, b̂, b̃, and the Lipschitz norm of ϕ with

(3.8)  \frac{max\{|z_{1} − z_{3}|, |z_{2} − z_{3}|\}}{|z_{1} − z_{2}|} ≤ M.

Choosing J_{1} , J_{2} to be arcs on ∂Ω_{m} connecting z_{1} to z_{2} and taking the supremum over all z_{3} between z_{1} , z_{2} , we deduce from (3.8) that (2.2) is valid. The situation when z_{1} , z_{2} lie on different sides in \underline{G}_{0} is handled similarly using (3.1)–(3.3). We omit the details. From our earlier reductions we conclude first (3.6) and second Lemma 3.5.

As for existence of Ω, Ω_{n} satisfying (3.2)–(3.4), Lemma 3.5, (3.6), and (3.7) we note that if ψ is any piecewise linear Lipschitz function with support contained in ] − 1, 1[, then the above construction can be carried out for ϕ(x) = A^{−1}ψ(Ax), x ∈ R, provided A is suitably large and b, b̃ small enough. To give some explicit examples, fix θ, 0 < θ < π/2, and let ϱ = (1 + cos θ)^{−1}. Define ϕ = ϕ( · , θ) on R as follows.

ϕ(x) = \{ \begin{matrix}0 & when |x| ≥ 1 − ϱ, \\ ϱ sin θ − |x| tan θ & when 0 ≤ |x| < 1 − ϱ. \end{matrix}

Then the above construction can be carried out for this ϕ and suitable b, b̃, b̂. If for example Ω_{0} is the exterior of the unit square with center at the origin, one can show from some basic trigonometry that it suffices to choose b = δ, b̂ = \frac{1}{2} sin(\frac{1}{2}θ + δ), b̃ = \frac{1}{2} sin(\frac{1}{2}θ + 2δ) provided δ > 0 is small enough (any δ with 0 < 4δ < π/2 − θ works for b̂, b̃). The choice θ = π/3 yields the Van Koch snowflake for ∂Ω.

Next we relate our snowflake, Ω, constructed as in (3.1)–(3.8), to a certain Bernoulli shift. Following [C], [W], we number the line segements on the graph of the Lipschitz function ϕ in (3.1) over ] − 1, 1[. Number these segments from left to right (so the first and last segments are subsets of ] − 1, 1[). Suppose there are l > 1 segments in the graph of ϕ over ] − 1, 1[. If Σ ∈ \underline{G}_{0} , then there is a one to one correspondence between the set \{1, …, l\} and the line segments Q ∈ \underline{G}_{1} , Q ≺ Σ. Using the same numbering of the graph of ϕ on the second generation of line segments we get a correspondence between Q ∈ \underline{G}_{2} , Q ≺ Σ and \{(ω_{1}, ω_{2}) : ω_{i} ∈ {1, …, l}}. If we continue in this way it is clear that we get a correspondence between all but a countable set of points on the snowflake (corresponding to the endpoints of intervals in \underline{G}) and the set Θ of infinite sequences:

Θ = {ω = (ω_{1}, ω_{2}, …) : ω_{i} ∈ {1, …, l}}.

Define the shift S on Θ by

(3.9)  S(ω) = (ω_{2}, ω_{3}, …) whenever ω = (ω_{1}, ω_{2}, …) ∈ Θ.

Hence for ω as in (3.9),

(3.10)  S^{−1}(ω) = {ω^{1}, ω^{2}, …, ω^{l}} where ω^{i} = (i, ω_{1}, …), 1 ≤ i ≤ l.

The geometric meaning of the shift S on Γ_{Σ} is as follows. If Q ∈ \underline{G}_{1}, Q ≺ Σ, the restriction of S to Γ_{Q} is a composition of a rotation, translation, and dilation (i.e. a conformal affine map) which takes Q to Σ and maps the normal into T_{Q} to the normal into T_{Σ} . Thus if n ≥ 1, Q ∈ \underline{G}_{n}, Q ≺ Σ, then the restriction of S^{n} to Γ_{Q} takes Q to Σ and maps the normal into T_{Q} to the normal into T_{Σ} . Therefore, we can regard S as either a function defined on Θ or on ∂Ω minus a countable set. For any x ∈ Γ_{Σ} which is not an endpoint of a Q ∈ \underline{G}_{n}, let Q_{n}(x) be the line segment with x ∈ Γ_{Q_{n}(x)} and Q_{n}(x) ∈ \underline{G}_{n}. Let u′ be as in Lemma 2.26 defined relative to ∂Ω with x̂ = 0, r̂ = 1/200, R̂ = 200. Then either Ω = Ω_{1} or Ω = Ω_{2} . We write u, D for u′, D′ in Lemma 2.26. Let μ be the measure corresponding to u as in (1.2). We shall show later (see (3.17) or (3.28)) that μ has no point masses. Thus we can regard μ as being defined on Θ.

Lemma 3.11. The measure μ is mutually absolutely continuous with respect to a Borel measure ν which is invariant under S.

Proof. By ‘ν invariant under S ’, we mean that

(3.12)  ν(S^{−1}E) = ν(E) whenever E ⊂ ∂Ω is Borel.

Let

ν^{(n)}(Y) = \frac{1}{n} ∑_{j=0}^{n−1} μ(S^{−j}Y)

and let ν be a weak* limit point of {ν^{(n)}}. Then if Y is a Borel subset of Γ_{Σ}, we have

ν^{(n)}(S^{−1}Y) − ν^{(n)}(Y) = \frac{1}{n}(μ(S^{−n}Y) − μ(Y)) → 0 as n → ∞,

since μ is finite. Thus ν is invariant under S. To prove mutual absolute continuity, given ε > 0, let E_{ε} = {x ∈ Γ_{Σ} : d(x, {a, b}) < ε} where a, b are the endpoints of Γ_{Σ} . We claim for some c, c_{ε} = c(ε), that

(3.13) (∗) If Y ⊂ Γ_{Σ} and Y ∩ E_{ε} = ∅, then c_{ε}^{−1}μ(Y) ≤ ν(Y) ≤ c_{ε}μ(Y), (∗∗) ν(E_{ε}) ≤ cε^{α} for some α, 0 < α < 1.

To prove (3.13)(∗) let Q ∈ \underline{G}_{n} and let V be the restriction of S^{n} to Γ_{Q}. Then as noted below (3.10), V is a conformal affine map of R^{2} onto R^{2} with V (Γ_{Q}) = Γ_{Σ} . Let ũ_{2}(x) = u(V^{−1}x), x ∈ V (\overline{D}). We note that the p-Laplacian partial differential equation is rotation, translation and dilation invariant. Thus u_{2} is p-harmonic and we can apply Lemma 2.22 with ũ_{1}, Ω̃_{1} = u, Ω, respectively, while Ω̃_{2} = V (Ω) and E = Γ_{Σ} \ E_{ε}. If Y ⊂ E is Borel, we get for some c̃_{ε} = c(ε) ≥ 1 that

(3.14)  c̃_{ε}^{−1} \frac{μ̃_{2}(E)}{μ(E)} ≤ \frac{μ̃_{2}(Y)}{μ(Y)} ≤ c̃_{ε} \frac{μ̃_{2}(E)}{μ(E)}.

Next we note for some λ > 0 that μ̃_{2}(X) = λμ(S^{−n}X ∩ Γ_{Q}) whenever X is a Borel subset of Γ_{Σ} . Using this fact and (2.15) we deduce that

(3.15)  μ̃_{2}(E) ≈ λμ(Γ_{Q}) and μ(E) ≈ 1

From (3.14), (3.15) it follows that

(3.16)  c_{ε}^{−1}μ(Y)μ(Γ_{Q}) ≤ μ(S^{−n}Y ∩ Γ_{Q}) ≤ c_{ε}μ(Y)μ(Γ_{Q}).

Summing (3.16) over Q ∈ \underline{G}_{n}, we get

c_{ε}^{−1}μ(Y) ≤ μ(S^{−n}Y) ≤ c_{ε}μ(Y)

which in view of the definiton of ν, implies (3.13)(∗). To prove (3.13)(∗∗) observe from (2.15) and Harnack’s inequality that for some c = c(M̂) ≥ 2,

(3.17) μ(B(z, 2r)) ≤ cμ(B(z, r)) and μ(B(z, r)) ≤ (1 − c^{−1})μ(B(z, 2r))

whenever z ∈ ∂Ω and 0 < r ≤ diam ∂Ω. If Q ∈ \underline{G}_{n} then (S^{−n}E_{ε}) ∩ Γ_{Q} is contained in two disks of radius εl(Q) whose centers are in ∂Ω. Using this fact and iterating (3.17) with r = l(Q), we conclude for some 0 < α < 1 that

(3.18)  μ[(S^{−n}E_{ε}) ∩ Γ_{Q}] ≤ cε^{α}μ(Γ_{Q}).

Summing (3.18) over n and Q ∈ \underline{G}_{n}, as well as, using the definition of ν we obtain (3.13)(∗∗). To complete the proof of Lemma 3.11 note from (3.13)(∗) that μ, ν are proportional on Γ_{Σ} \ E_{ε}, Σ ∈ G_{0} , while (3.13)(∗∗) shows lim_{ε→0} ν(E_{ε}) = lim_{ε→0} μ(E_{ε}) = 0. These facts clearly imply that μ, ν are mutually absolutely continous on Γ_{Σ} . Since Σ ∈ \underline{G}_{0} is arbitrary we conclude the validity of Lemma 3.11.

Next if Σ ∈ \underline{G}_{0} we say that ν is ergodic with respect to S on Γ_{Σ} , provided that whenever H ⊂ Γ_{Σ} is Borel and S^{−1}(H) = H, it is true that either ν(Γ_{Σ} \ H) = 0 or ν(H) = 0.

Lemma 3.19. S is ergodic with respect to ν on Σ, whenever Σ ∈ \underline{G}_{0} .

Proof. Let H ⊂ Γ_{Σ} be as above, F = Γ_{Σ} \ H, and E_{ε} as in (3.13). If ν(H) > 0, we choose ε > 0 so small that ν(E_{ε}) < ν(H)/4. Let H′ = H \ E_{ε} and suppose x ∈ F \ E_{2ε}. Then for n large enough, Γ_{Q_{n}(x)} ∩ E_{ε} = ∅. By (3.16), (3.13) with Y = H′, S^{−n}H′ ∩ Γ_{Q_{n}(x)} , Γ_{Q_{n}(x)} , we have

(3.20)  \begin{aligned}\frac{ν(H ∩ Γ_{Q_{n}(x)})}{ν(Γ_{Q_{n}(x)})} & = \frac{ν(S^{−n}H ∩ Γ_{Q_{n}(x)})}{ν(Γ_{Q_{n}(x)})} \\ & ≥ \frac{ν(S^{−n}H′ ∩ Γ_{Q_{n}(x)})}{ν(Γ_{Q_{n}(x)})} ≥ c_{ε}^{−3}ν(H′) ≥ 3c_{ε}^{−3}ν(H)/4. \end{aligned}

From (3.17), (3.13), and differentiation theory (see [R]), we deduce

\frac{ν(H ∩ Γ_{Q_{n}(x)})}{ν(Γ_{Q_{n}(x)})} → 0

for ν almost every x ∈ F \ E_{2ε} (i.e., ν almost every x ∈ F \ E_{2ε} is a point of ν density one). (3.20) and the above display imply that ν(F \ E_{2ε}) = 0. Since ε can be arbitrarily small and (3.13)(∗∗) holds, we conclude that ν(F) = 0. Thus ν is ergodic on each Σ ∈ G_{0} .

Let f(x) = − log l(Q) if x ∈ Γ_{Q} and Q ∈ \underline{G}_{1} . Then
(3.21)  ∥f∥_{L^{∞}(ν)} ≤ c < ∞,

where c depends only on the length of the segments on the graph of ϕ over ]−1, 1[. Armed with Lemma 3.19 we now are in a position to apply the ergodic theorem of Birkhoff and entropy theorem of Shannon–McMillan–Breiman (see [Bi] or [Sh]).

Lemma 3.22. If σ_{n}(x) = l(Q_{n}(x)) and h_{n}(x) = μ(Γ_{Q_{n}(x)}), then

σ = lim_{n→∞} \frac{1}{n} log \frac{1}{σ_{n}(x)}, h = lim_{n→∞} \frac{1}{n} log \frac{1}{h_{n}(x)},

exist almost everywhere ν and are constant.

Proof. First we note that

\frac{1}{n} log \frac{1}{σ_{n}(x)} = \frac{1}{n} ∑_{j=0}^{n−1} f(S^{j}x)

so (3.21) and the ergodic theorem applied to f yields the limit involving σ_{n}. The other part of Lemma 3.22 follows from the entropy theorem applied to ν and (3.13)(∗). It should be noted though that in [Bi], [Sh], the proof of the entropy theorem is given for the shift S on the space, Ψ, of sequences: {(x_{i})_{−∞}^{∞} : x_{i} ∈ {1, 2, …, l}}. However, the proofs of these authors is easily adapted to our situation. For example the proof in [Sh] uses Ψ to conclude that if g(x) = lim inf_{n→∞}[n^{−1} log ν(Γ_{Q_{n}(x)})], then g(Sx) = g(x) for ν almost every x ∈ Ψ. In fact if Q_{n}(x) ≺ Q ∈ \underline{G}_{1} , then this equality follows from (3.17), (3.13), and (3.16) with Y = Q_{n}(Sx), n = 1. The rest of the proof in [Sh] is essentially unchanged if Ψ is replaced by Θ.

As a consequence of Lemma 3.22 we have

Lemma 3.23. For μ almost every x ∈ ∂Ω, we have

lim_{t→0} \frac{log μ[B(x, t)]}{log t} = h/σ = H-dim μ.

Proof. Let x ∈ ∂Ω and suppose x is not an endpoint of any Γ_{Q} with Q ∈ G. Given t, 0 < t < 1, choose n a nonnegative integer so that l(Q_{n+1}(x)) ≤ t < l(Q_{n}(x)). Then

(3.24)  t ≈ l(Q_{n}(x)) since l(Q_{n+1}(x)) ≈ l(Q_{n}(x))

where the proportionality constants are independent of n. Also from the doubling property of μ in (3.17) we have for some c = c(M̂) that

(3.25)  |ln \frac{μ[Γ_{Q_{n}(x)}]}{μ[B(x, t)]}| ≤ c.

Using (3.24), (3.25) and Lemmas 3.11, 3.22 we deduce for μ almost every x ∈ ∂Ω, that

h/σ = lim_{n→∞} \frac{log h_{n}(x)}{log σ_{n}(x)} = lim_{t→0} \frac{log μ[B(x, t)]}{log t}.

Finally h/σ = H-dim μ follows from the definition of H-dim μ and basic measure theoretic type arguments.

Next given Q ∈ \underline{G}, let T̂_{Q} be as defined above (3.1). Given n a positive integer, put

(3.26)  O_{n} = ⋃_{Q∈\underline{G}_{n}} T̂_{Q} and put Ω̂_{n} = Ω_{n} \ \overline{O}_{n} when n = 1, 2, ….

For use in Section 4 we note that

(3.27)

(α) Ω̂_{n} ⊂ Ω_{m} for m ≥ n so Ω̂_{n} ⊂ Ω,

(β) ∂Ω̂_{n} is a k-quasicircle with k independent of n.

(3.27)(α) follows from (3.1)–(3.3). (3.27)(β) is proved in the same way as the corresponding statement for Ω_{n} (see (3.6)–(3.8)). Recall that x̂ = 0, r̂ = 1/200, R̂ = 1/200 in the definition of u, D. Define D̂_{n}, û_{n} relative to Ω̂_{n} in the same way as D, u were defined relative to Ω. Then û_{n}, n = 1, 2, …, is the p-harmonic function in D̂_{n} with boundary values 1 on either ∂B(0, 1/200) or ∂B(0, 200) and 0 on ∂Ω̂_{n} in the continuous sense. Continuity of û_{n}, n = 1, 2, …, on the closure of D̂_{n} follows from Lemma 2.11.

Let μ̂_{n}, n = 1, 2, …, be the measure corresponding to û_{n}, as in (1.2). From (3.17) we see for some 0 < λ < 1 < λ′, that if x̃ ∈ ∂Ω̂_{m}, 0 < t < ϱ < 1/100, and ỹ ∈ ∂Ω̂_{m} ∩ B(x̃, ϱ), then

(3.28)  μ̂_{n}[B(x̃, ϱ)](\frac{t}{ϱ})^{λ′} ≤ μ̂_{n}[B(ỹ, t)] ≤ cμ̂_{n}[B(x̃, ϱ)](\frac{t}{ϱ})^{λ}.

Observe that (3.28) also holds with û_{n}, μ̂_{n} replaced by u, μ. From this observation with ϱ = 1/200, we deduce that

(3.29)  h > 0

and since l(Q_{n+1}(x) ≈ l(Q_{n}(x)) (with constants depending on the definition of ϕ above (3.1)), we also have

(3.30)  σ > 0.

4. Proof of Theorem 1

In this section we complete the program outlined in the introduction and get Theorem 1. Let T̂_{Q} be as defined above (3.1) and let τ(Q) be the union of the two closed line segments joining the endpoints of Q to a_{Q} − b̂l(Q)e. From (3.2), (3.3) and the definition of Ω̂_{n} in (3.26) we see that τ(Q), τ(Q′) have disjoint interiors whenever Q, Q′ are distinct intervals in \underline{G}_{n} and

(4.1)  ∂Ω̂_{n} = ⋃_{Q∈\underline{G}_{n}} τ(Q), n = 1, 2, ….

Also let μ̂_{n}, û_{n}, be as defined after (3.27). From the maximum principle for p-harmonic functions and (3.27) we see that û_{n} ≤ u in D̂_{n}. Put

\begin{aligned}μ_{n} & = \frac{μ̂_{n}}{μ̂_{n}(∂Ω̂_{n})}, \\ u_{n} & = \frac{û_{n}}{μ̂_{n}(∂Ω̂_{n})} for n = 1, 2, …. \end{aligned}

From (2.15) and the above facts we see for some c = c(M̂) ≥ 1 that

(4.2)  u_{n} ≤ cu in D̂_{n}.

From the definition of μ_{n}, (4.2), (3.27), (3.28) and once again (2.15) we conclude that

(4.3)  μ_{n}(∂ D̂_{n}) = 1 and μ_{n}(τ(Q)) ≤ cμ(Γ_{Q})

whenever Q ∈ \underline{G}_{n}, n = 1, 2, …. We first prove Lemma 4.4. If h is as in Lemma 3.22, then

lim_{n→∞}[\frac{1}{n} ∑_{Q∈\underline{G}_{n}} μ_{n}(τ(Q)) log \frac{1}{μ_{n}(τ(Q))}] = h.

Proof. From Lemma 3.22, (3.29), and Egoroff’s theorem we see for given ε, 0 < ε ≤ 1/4, that there exists N = N(ε) such that

(4.5)  (1 − ε)h < \frac{1}{n} log \frac{1}{μ(Γ_{Q_{n}(x)})} < (1 + ε)h

when n ≥ N for all Q = Q_{n}(x) ∈ \underline{G}_{n}, except for a set ⋃_{Q∈A_{n}} Γ_{Q} of μ-measure < ε where A_{n} ⊂ \underline{G}_{n}. It follows from (4.3), (4.5) that

(4.6) \begin{aligned}\frac{1}{n} ∑_{Q∈\underline{G}_{n}\A_{n}} μ_{n}(τ(Q)) log \frac{1}{μ_{n}(τ(Q))} & ≥ \frac{1}{n} ∑_{Q∈\underline{G}_{n}\A_{n}} μ_{n}(τ(Q)) log \frac{1}{cμ(Γ_{Q})} \\ & ≥ −c/n + (1 − ε)h ∑_{Q∈\underline{G}_{n}\A_{n}}μ_{n}(τ(Q)) \\ & ≥ −c′/n + (1 − c′ε)h \end{aligned}

for c′ = c′(k), large enough. From (4.6) and arbitrariness of ε we obtain

(4.7)  lim inf_{n→∞}[\frac{1}{n} ∑_{Q∈\underline{G}_{n}} μ_{n}(τ(Q)) log \frac{1}{μ_{n}(τ(Q))}] ≥ h.

To obtain an estimate from above for the sum in (4.7) we note from (3.28) for μ, μ_{n} with ϱ = 1/200 and (4.3) that for some c = c(k) ≥ 1,

(4.8)  c^{−1}n ≤ log \frac{1}{cμ(Γ(Q))} ≤ log \frac{1}{μ_{n}(τ(Q))} ≤ cn

whenever n = 2, 3, …, and Q ∈ \underline{G}_{n}. Also note for 0 < λ ≤ 1/4 that λμ(Γ_{Q}) < μ_{n}(τ(Q)) except for a set E = ⋃_{Q∈B_{n}} τ(Q) with μ_{n}(E) ≤ λμ(∂Ω) ≤ cλ. Using this weak type estimate, (4.5), and (4.8) we deduce

(4.9)

\frac{1}{n} ∑_{Q∈\underline{G}_{n}\B_{n}} μ_{n}(τ(Q)) log \frac{1}{μ_{n}(τ(Q))} < \frac{1}{n} ∑_{Q∈\underline{G}_{n}\B_{n}} μ_{n}(τ(Q)) log \frac{1}{λμ(Γ_{Q})}

< (1 + ε)h − (1/n) log λ

+ n^{−1} ∑_{Q∈A_{n}} μ_{n}(τ(Q)) log(1/μ(Γ_{Q}))

≤ (1 + ε)h − (1/n) log λ + cε.

Also from (4.8) and our weak type estimate we have

(4.10)  \frac{1}{n} ∑_{Q∈B_{n}} μ_{n}(τ(Q)) log \frac{1}{μ_{n}(τ(Q))} ≤ cλ.

Choosing λ = 1/n in (4.9), (4.10) and using the arbitrariness of ε we see that

(4.11)  lim sup_{n→∞}[\frac{1}{n} ∑_{Q∈\underline{G}_{n}} μ_{n}(τ(Q)) log \frac{1}{μ_{n}(τ(Q))}] ≤ h.

Combining (4.7), (4.11) we conclude that Lemma 4.4 is valid.

Next we note from a Schwarz reflection type argument that if Q ∈ \underline{G}_{n} and x ∈ τ(Q) is not an endpoint of either of the two line segments making up τ(Q), then u_{n} has a p-harmonic extension to a neighborhood of x. From p-harmonicity, it follows as in Lemma 2.7 that this extension has Hölder continuous partial derivatives in a neighborhood of x. This fact, (1.1), (1.2) for u_{n}, μ_{n}, and an easy limiting argument imply that at x

(4.12)  dμ_{n} = |∇u_{n}|^{p−1}(x) dH^{1}x.

Moreover from the doubling property of μ_{n}, n = 2, …, (see (3.28)) and (4.8) it follows that μ_{n} has no point masses at the endpoints of τ(Q) whenever Q ∈ \underline{G}_{n}. In view of (4.1), we see that (4.12) holds for μ_{n} almost every x ∈ ∂Ω̂_{n}. Next let l(τ(Q)) be the length of τ(Q) whenever Q ∈ \underline{G}_{n}.

Lemma 4.13. If Q ∈ \underline{G}_{n}, then for some c = c(k, ϕ) ≥ 1,

∑_{Q∈\underline{G}_{n}} ∫_{τ(Q)}|log(\frac{l(τ(Q))|∇u_{n}|^{p−1}(x)}{μ_{n}(τ(Q))})||∇u_{n}|^{p−1} dH^{1}x ≤ c.

Proof. To prove this lemma we use results from [K] where examples are constructed of p-harmonic functions w = w( · , p) in the region S(α), 0 < α < π, where if x_{1} = ϱ cos θ, x_{2} = ϱ sin θ, in polar coordinates ϱ, θ, |θ| < π, then

S(α) = {(ϱ, θ) : ϱ > 0 and |θ| < α},
w(ϱ, θ) = ϱ^{λ}f(θ) in S(α),
w(1, 0) = 1 and w(ϱ, ±α) = 0.

Moreover f ∈ C_{0}^{∞}[−α, α] and for some c ≥ 1,

(4.14)  c^{−1}λ^{2}θ ≤ |f′(θ)| ≤ cλ^{2}θ.

The relation between λ, α is given by (see the remark in [K]),

(4.15)  1 − \frac{(λ − 1)\sqrt{p − 1}}{\sqrt{λ^{2}(p − 1) − λ(p − 2)}} = 2α/π.

Let Q ∈ \underline{G}_{n} and suppose that Q̂, Q′ are the distinct closed line segments in τ(Q). If Q̂ ∩ Q′ = {z}, let Ŝ be the open sectorial region with vertex z,

Ŝ ∩ D̂_{n} ∩ B(z, l(Q̂)/2) ≠ ∅ and Q̂, Q′ ⊂ ∂Ŝ.

After a rotation and translation we assume, as we may, that z = 0 and Ŝ = S(θ_{0}) for some π/2 < θ_{0} < π. For fixed p, Q̂, Q′, let w be the corresponding p harmonic function in S(θ_{0}). Given ε > 0, let t = 3l(Q̂)/4 and put

b_{n}^{p−1} = t^{p−2}μ_{n}(Q̂) = t^{p−2} ∫_{Q} |∇u_{n}|^{p−1} dH^{1}x.

From (2.15) with μ̃, ũ replaced by μ_{n}, u_{n}, and Harnack’s inequality, we see that u_{n}(t, 0) ≈ b_{n}. This fact, (3.1)–(3.3), and Lemma 2.16 with Ω, ũ, v, replaced by S(θ_{0}), u_{n}, w, imply that

(4.16)  t^{−λ}w ≈ u_{n}/b_{n} in S(θ_{0}) ∩ \overline{B}(0, t).

From smoothness of u_{n}, w in Q̂ ∪ S(θ_{0}) ∩ B(0, t) and (4.16) it follows that

(4.17)  t^{−λ}|∇w(x)| ≈ |∇u_{n}(x)|/b_{n} for x ∈ Q̂ ∩ \overline{B}(0, t).

Proportionality constants may depend on k, ϕ but are independent of n = 1, 2, …. From (4.17), (4.14), (4.15), we see for some c = c(k, ϕ) that

\begin{aligned}∫_{_{Q}∩\overline{B}(0,t)}| & log[(t|∇u_{n}|/b_{n})^{p−1}]||∇u_{n}|^{p−1} dH^{1}x \\ & ≤ c(b_{n}/t^{λ})^{p−1}∫_{0}^{t} log(\frac{ct}{ϱ})ϱ^{(λ−1)(p−1)} dϱ ≤ c^{2}t^{2−p}b_{n}^{p−1} ≤ c^{3}μ_{n}(Q̂). \end{aligned}

Here we have also used λ > (p − 2)/(p − 1) when p > 2, as we see from (4.15). Thus,

(4.18)  ∫_{_{Q}∩B(0,t)}|log(\frac{l(Q̂)|∇u_{n}|^{p−1}}{μ_{n}(Q̂)} )| |∇u_{n}|^{p−1} dH^{1}x ≤ cμ_{n}(Q̂)

for some c = c(k, ϕ). Let q denote the other endpoint of Q̂. If x ∈ Q̂ \ B(0, t), observe from (3.1)–(3.3) and Lemma 2.16 that we can apply the rate theorem to the above functions in Ω̂_{n} ∩ B(x, |x − q|/c) for some c = c(k, ϕ). Doing this we get

|∇u_{n}(x)| ≈ u_{n}(x^{∗})|x − q|^{−1}

where x^{∗} ∈ B(x, |x − q|/c) and d(x^{∗}, ∂Ω̂_{n}) ≥ |x − q|/c^{2} . Using this fact, (2.15), and (3.28), we see that

∫_{Q}\B(0,t)|log(\frac{l(Q̂)|∇u_{n}|^{p−1}}{μ_{n}(Q̂)} )| |∇u_{n}|^{p−1} dH^{1}x

≤ ∫_{_{Q}\B(0,t)}|log(\frac{cl(Q̂)μ_{n}[B(q, 2|x − q|)]}{|x − q|μ_{n}(Q̂)})| |∇u_{n}|^{p−1} dH^{1}x

≤ cμ_{n}(Q̂) + c ∫_{_{Q}\B(0,t)} log(\frac{2t}{|x − q|})|∇u_{n}|^{p−1} dH^{1}x

≤ cμ_{n}(Q̂),

where we have written the last integral as a sum over Q̂ ∩ [B(q, 2^{−m}l(Q̂)) \ B(q, 2^{−m−1}l(Q̂))], m = 2, 3, …, and then used (3.28) to make estimates. Combining this display with (4.18) we have

∫_{Q}|log(\frac{l(Q̂)|∇u_{n}|^{p−1}}{μ_{n}(Q̂)} )||∇u_{n}|^{p−1} dH^{1}x ≤ cμ_{n}(Q̂).

A similar inequality holds for Q′. Since by (3.28)

μ_{n}(τ(Q)) ≈ μ_{n}(Q̂) ≈ μ_{n}(Q′)

with proportionality constants depending on k, ϕ, we can sum the above for Q̂, Q′ over Q ∈ G_{n} to get Lemma 4.13.

Our final lemma before the proof of Theorem 1 is

Lemma 4.19. For fixed p, 1 < p < ∞,

η = lim_{n→∞} n^{−1} ∫_{∂_{Ω}_{n}} |∇u_{n}|^{p−1} log |∇u_{n}| dH^{1}x

exists. If η > 0 then H-dim μ < 1 while if η < 0, then H-dim μ > 1.

Proof. First from Lemma 3.22, (3.30), and an argument similar to the one used in Lemma 4.4, we see that

(4.20)  lim_{n→∞}[\frac{1}{n} ∑_{Q∈\underline{G}_{n}} μ_{n}(τ(Q)) log l(τ(Q))] = −σ.

From (4.1) and Lemma 4.13 we have for n = 1, 2, …,

(4.21) |(p−1) ∫_{∂_{Ω}_{n}} |∇u_{n}|^{p−1} log |∇u_{n}| dH^{1}x− ∑_{Q∈\underline{G_{n}}}μ_{n}(τ(Q)) log \frac{μ_{n}(τ(Q))}{l(τ(Q))}| ≤ c.

Dividing this inequality by n, letting n → ∞, and using Lemma 4.4, (4.20), we conclude that the limit exists in Lemma 4.19. In fact,

(p − 1)η = σ − h = σ[1 − h/σ] = σ(1 − H-dim μ).

Clearly this equality gives the last sentence in Lemma 4.19.

We now prove Theorem 1. Note from Lemma 2.26 and the arguments leading to (4.12), (4.17) that u_{n} is infinitely differentiable and ∇u_{n} ≠ 0 in the closure of D̂_{n}, except at endpoints of segments in ∂Ω̂_{n}. Let Z_{n} denote the finite set of endpoints of segments on ∂Ω̂_{n}. Let t_{0} be 1/2 of the length of the smallest segment contained in ∂Ω̂_{n}. If z ∈ Z_{n}, x ∈ ∂B(z, t)∩∂Ω̂_{n} and 0 < t < t_{0} , then as in (4.17), we get an estimate for |∇u_{n}(x)| in terms of t_{0}^{−λ}, u(x_{0}), where x_{0} ∈ ∂B(z, t_{0})∩Ω_{n} with d(x_{0}, ∂Ω_{n}) ≈ t_{0} . Using the Schwarz reflection argument once again to extend u_{n}, and C^{1,σ} results for the p-Laplacian (see Lemma 2.7), we deduce that this estimate also holds in the closure of B(x, t/c) ∩ Ω̂_{n} provided c is large enough. This fact and Lemma 2.26 for the extension of u_{n}, imply that on the closure of ∂B(z, t)∩Ω̂_{n} we have for sufficiently small t > 0, some λ > max{0, (p−2)/(p−1)}, and c = c(k, ϕ, t_{0}, u(x_{0})),

(4.22)

tu_{n}|∇u_{n}|^{p−3} ∑_{i,j=1,}^{2} |(u_{n})_{x_{i}x_{j}}| + t|∇u_{n}|^{p−1}|log |∇u_{n}||
≤ ct ln(1/t)|∇u_{n}|^{p−1}
≤ ct ln(1/t)t^{(λ−1)(p−1)} → 0 as t → 0.

Let b_{ik}, 1 ≤ i, k ≤ 2, be as in (1.14). Let

v = log |∇u_{n}| in Õ_{n} = D̂_{n} \ ⋃_{z∈Z_{n}} \overline{B}(z, t).

Let L be the divergence form operator in (1.11). As in the discussion following (1.20) we apply the divergence theorem in Õ_{n} to the vector field whose ith component, i = 1, 2, is

u_{n} ∑_{k=1}^{2} b_{ik}v_{x_{k}} − v ∑_{k=1}^{2} b_{ik}(u_{n})_{x_{k}}.

Such an application is permissible, since u_{n}, v are smooth in the closure of Õ_{n} (see Lemma 2.26). We get

(4.23)  ∫Õ_{n} u_{n}Lv dx = ∫_{∂Õ_{n}} ∑_{i,k=1}^{2} b_{ik}ξ_{i}[u_{n}v_{x_{k}} − v(u_{n})_{x_{k}}] dH^{1}x

where ξ denotes the outer unit normal to ∂Õ_{n}. Next observe from (1.14), (4.22) that for some c = c(k, ϕ),

(4.24)  \begin{aligned}∑_{z∈Z_{n}} & ∫_{∂B(z,t)∩∂Õ_{n}} ∑_{i,k=1}^{2} |b_{ik}|[|v| |∇u_{n}| + u_{n}|∇v|] dH^{1}x \\ & ≤ ∑_{z∈Z_{n}} c ∫_{∂B(z,t)∩∂Õ_{n}}[| log | |∇u_{n}| |∇u_{n}|^{p−1} \\ & + u_{n}|∇u_{n}|^{p−3} ∑_{i,j=1}^{2} |(u_{n})_{x_{i}x_{j}}|] dH^{1}x \\ & → 0 as t → 0. \end{aligned}

Letting t → 0 in (4.23), we see from (4.24) that

(4.25)  ∫_{_{D}_{n}} u_{n}Lv dx = ∫_{∂ _{D}_{n}} ∑_{i,k=1}^{2} b_{ik}ξ_{i}[u_{n}v_{x_{k}} − v(u_{n})_{x_{k}}] dH^{1}x.

Observe from (1.18), (1.19) that if p ≠ 2, 1 < p < ∞, then at x ∈ D̂_{n}

(4.26)  (p − 2)^{−1}Lv ≈ |∇u_{n}|^{p−4} ∑_{i,j=1}^{2}[(u_{n})_{x_{i}x_{j}}]^{2}

where the proportionality constant depends only on p. Also from Lemma 2.26, the fact that u_{n} = 0, ξ = −∇u_{n}|∇u_{n}|^{−1} on ∂Ω̂_{n} \ Z_{n}, and the definition of (b_{ik}), we see for some c = c(M̂) that

(4.27)  \begin{matrix}∫_{∂ _{D}_{n}} ∑_{i,k=1}^{2} b_{ik}ξ_{i}[u_{n}v_{x_{k}} − v(u_{n})_{x_{k}}] dH^{1}x \\ = (p − 1) ∫_{∂_{Ω}_{n}} |∇u_{n}|^{p−1} log |∇u_{n}| dH^{1}x + E \end{matrix}

where |E| ≤ c. From (4.25)–(4.27) and Lemma 4.19 we see that in order to prove Theorem 1 it suffices to show

(4.28)  lim inf_{n→∞}(n^{−1} ∫_{_{D}_{n}} u_{n}|∇u_{n}|^{p−4} ∑_{i,j=1}^{2}[(u_{n})_{x_{i}x_{j}}]^{2}(x) dx) > 0.

To prove (4.28) let 1 ≤ m < n and suppose that Q ∈ \underline{G}_{m}. Choose Q′ ∈ \underline{G}_{m} with the same endpoint (say z) as Q and the property that Q, Q′ are distinct direct descendants of the same Q′′ ∈ \underline{G}_{m−1} . Let β′ be the angle between the rays drawn from z through Q, Q′, as measured from within Ω_{m}. If s = \frac{1}{100} min\{l(Q), l(Q′)\}, note from (3.1)–(3.3) that there exists a sectorial region S̃ with vertex z, angle opening β, 0 < β < 2π, and S̃∩ B(z, 10s) ∩ Ω_{m} ≠ ∅. Also,

(4.29)

(a) β, β′ lie in the same component of (0, 2π) \ {π}.

(b) If β′ < π, then β′ < β and ∂S̃∩ B(z, 10s) ∩ Ω_{l} = ∅ for l ≥ m.

(c) If β′ > π, then β < β′ and ∂S̃∩ B(z, 10s) ⊂ Ω_{l} for l ≥ m.

We assume as we may that z = 0 and S̃ = {(ϱ, θ) : 0 < ϱ, |θ| < β/2}. If (b) in (4.29) is valid, then from (3.1)–(3.3) we deduce for some θ_{0}, δ > 0, that

S′ = {(ϱ, θ) : 0 < ϱ ≤ 5s, |θ − θ_{0}| < δ}

satisfies

(4.30)  S′ ⊂ S ∩ [ T̃_{Q} \ T̂_{Q}].

Here δ depends on our choice of constants in (3.1)–(3.3) and ϕ but is independent of m = 1, 2, …. Let w, λ, be as defined above (4.14) relative to S(β/2) and write u_{n}(ϱ, θ) for u_{n}(x) when x ∈ Ω̂_{n} has polar coordinates ϱ, θ. If (b) holds then from the maximum principle for p-harmonic functions and Lemma 2.13, we get upon comparing boundary values of u_{n}/u_{n}(s, θ_{0}) and w/w_{n}(s, θ_{0}),

(4.31)  \frac{u_{n}(ϱ, θ_{0})}{u_{n}(s, θ_{0})} ≤ c(\frac{ϱ}{s})^{λ}, 0 < ϱ ≤ s,

where λ > 1, as follows from (4.15). From (4.31) and Lemma 2.26 we deduce that

(4.32)  \frac{|∇u_{n}|(ϱ, θ_{0})}{|∇u_{n}|(s, θ_{0})} ≤ c^{∗}(\frac{ϱ}{s})^{λ−1}

where c^{∗} is independent of m. Taking logarithms, we see that there exists η < 1/2 such that if ϱ = ηs, then

(4.33)  log |∇u_{n}|(ϱ, θ_{0}) − log |∇u_{n}|(s, θ_{0}) ≤ −1.

Applying the mean value theorem from calculus and using Lemmas 2.7, 2.26, we find from (4.33) for some x̃ with polar coordinates ϱ, θ_{0} , ηs < ϱ < s, that

(4.34)  cs ∑_{i,j=1}^{2} \frac{|(u_{n})_{x_{i}x_{j}}|(x̃)}{|∇u_{n}|(x̃)} ≥ 1

where c = c(k, ϕ). From (4.34), Lemma 2.26, and Lemma 2.9, we see first that (4.34) is valid in B(x̃, s/c) ⊂ S′ provided c = c(k, ϕ) is large enough and thereupon from (2.15), Lemma 2.26, that

(4.35)  c ∫_{B_{Q}} u_{n}|∇u_{n}|^{p−4} ∑_{i,j=1}^{2} |(u_{n})_{x_{i}x_{j}}|^{2} dx ≥ μ_{n}(B(z, 10l(Q)).

Here B_{Q} = B(x̃, (s/2c)) and we have used the fact that l(Q) ≈ l(Q′) (since Q, Q′, are direct descendants of Q′′). If (c) holds a similar argument applies only now λ < 1 and

(4.36)  S′ ∩ S = ∅, S′ ⊂ T̃_{Q} \ T̂_{Q}.

From (4.36) and the maximum principle for p-harmonic functions we get as in (4.31),

c\frac{u_{n}(ϱ, 0)}{u_{n}(s, 0)} ≥ (\frac{ϱ}{s})^{λ}, 0 < ϱ ≤ s,

for some λ < 1, c ≥ 1, and thereupon from Harnack’s inequality that

(4.37)  c^{2} \frac{u_{n}(ϱ, θ_{0})}{u_{n}(s, θ_{0})} ≥ (\frac{ϱ}{s})^{λ} for 0 < ϱ ≤ s and c large.

Arguing as following (4.31), we still get (4.35) for a ball with a radius and center having the above properties. Carrying out this procedure for each Q ∈ \underline{G}_{m} and 1 ≤ m < n, we obtain \{B_{Q} : Q ∈ \underline{G}_{m}, 1 ≤ m < n\}, satisfying (4.35). We claim that each point in Ω_{n} lies in at most c = c(k, ϕ) of the balls in {B_{Q} : Q ∈ Ω_{m}, 1 ≤ m < n}. To prove this claim we note from (3.1)–(3.3) that if 1 ≤ m < j < n, Q ∈ \underline{G}_{m}, Q′ ∈ \underline{G}_{j}, then B_{Q} ∩ B_{Q′} = ∅ unless Q′ ≺ Q. If Q′ ≺ Q, then from (3.1)–(3.3) we see that B_{Q′} has radius, r_{Q′}, with

r_{Q′} ≤ c^{m−j+1}l(Q).

Since

c_{+}d(B_{Q}, T̂_{Q}) ≥ l(Q) and d(B_{Q′}, T̂_{Q}) ≤ c_{+}r_{Q′}

where c_{+} = c_{+}(k, ϕ), we see that our claim is true. From our claim, (4.35), (3.28), and (3.1)–(3.3) we conclude that

c^{3} ∫_{Ω}_{n} u_{m}|∇u_{m}|^{p−4} ∑_{i,j=1}^{2} |u_{x_{i}x_{j}}|^{2} dx ≥ c^{2} ∑_{m=1}^{n−1} ∑_{Q∈\underline{G}_{m}}∫_{B_{Q}}u_{m}|∇u_{m}|^{p−4} ∑_{i,j=1}^{2} |u_{x_{i}x_{j}}|^{2} dx

≥ c ∑_{m=1}^{n−1} μ_{n}(∂Ω̂_{n}) ≥ n. (4.38)

Thus (4.28) is true and the proof of Theorem 1 is now complete.

5. Proof of Theorem 2

To begin the proof of Theorem 2, let

J = \{(x, y) : |x| = \frac{1}{2}, −\frac{1}{2} ≤ y ≤ \frac{1}{2}, and |y| = \frac{1}{2}, −\frac{1}{2} ≤ x ≤ \frac{1}{2}\}

be the unit square with center at 0 and side length one. Let O_{J} be the open set bounded by J and suppose that J_{1}, J_{2}, …, J_{l} are translated and dilated copies of J with

(5.1)  J_{i} ∩ J_{j} = ∅, i ≠ j, and J_{i} ⊂ O_{J}

for 1 ≤ i ≤ l. Let O_{J_{i}}, 1 ≤ i ≤ l, be the open set bounded by J_{i} and put \underline{G}_{0} = \{J\}, \underline{G}_{1} = \{J_{i}, 1 ≤ i ≤ l\}. By induction, if \underline{G}_{n} has been defined and Q ∈ \underline{G}_{n} let Q_{i}, 1 ≤ i ≤ l, be a translation and dilation of J_{i}, chosen so that if Q is mapped onto J by a composition of a translation and a magnification, then this mapping also sends Q_{i} to J_{i}, 1 ≤ i ≤ l. As in Section 3 we write Q_{i} ≺ Q, 1 ≤ i ≤ l, and put

\begin{aligned}\underline{G}_{n+1} & = \{Q_{i} : Q_{i} ≺ Q for some Q ∈ \underline{G}_{n}\}, \\ \underline{G} & = ⋃_{n=0}^{∞} \underline{G}_{n} \end{aligned}

Let O_{Q} be the open set bounded by Q ∈ \underline{G}_{n} and set

\begin{aligned}E_{n} & = ⋃_{Q∈G_{n}} \overline{O}_{Q}, n = 1, …, \\ E & = ⋂_{n=1}^{∞} E_{n}, \\ Ω_{n} & = B(0, 2) \ E_{n}, n = 1, …, \\ Ω & = B(0, 2) \ E. \end{aligned}

Then Ω is the Cantor set referred to in Theorem 2. We claim that

(5.2) Ω, Ω_{n} are uniform domains with constant M̂ in (2.3), independent of n.

We prove this claim only for Ω. If x, y ∈ Ω let n be the smallest integer such that x, y ∉ ⋃_{Q∈\underline{G}_{n}} \overline{O}_{Q}. Draw the line l′ connecting x to y. If l′ intersects Q ∈ \underline{G}_{n}, let w_{1} , w_{2} be the first and last points of intersection of l′ with Q. We can replace the line segment connecting w_{1} to w_{2} by line segments ⊂ Q, in such a way that the resulting curve connects w_{1} to w_{2} and satisfies (2.3). Continuing in this manner we get σ ⊂ \overline{Ω}_{n} satisfying (2.3) with w_{1} , w_{2} replaced by x, y. Thus claim (5.2) is true and consequently Lemma 2.13, (2.15) are valid since p > 2. To prove Lemma 2.16 for p > 2, let ũ, v be the p-harmonic functions in this lemma and w ∈ ∂Ω. Given r, 0 < r < 1, we see from (5.1) and the definition of Ω that there exists Q ∈ \underline{G}_{n} for some positive integer n with w ∈ O_{Q} ⊂ B(w, r) and

(5.3)  r ≈ l(Q) ≈ d(∂Ω, Q).

Here l(Q) denotes the length of Q and the proportionality constants are independent of w ∈ ∂Ω as well as r, 0 < r < 1. From (5.2), Harnack’s inequality, we deduce first that

(5.4)  c^{−1} ≤ \frac{ũ}{v} ≤ c on Q.

Second from (5.3), (5.4) and the weak maximum principle for p harmonic functions, we get Lemma 2.16.

Next given p > 2, let u ∈ W^{1,p}(Ω) be the p-harmonic function in Ω = B(0, 2) \ E with continuous boundary values 0 on E and 1 on ∂B(0, 2). As in Section 2 we see that existence of u follows from the usual variational argument and Lemma 2.11. Let μ be the measure associated with u as in (1.2). Also suppose ũ is another positive p-harmonic function in Ω∩N (N = a neighborhood of ∂Ω) with boundary value 0 on ∂Ω in the Sobolev sense. Let μ̃ be the measure associated with ũ as in (1.2). Using (2.15) and arguing as in (2.24) we deduce first that μ ≈ μ̃ and thereupon that it suffices to prove Theorem 2 for fixed p > 2 and u, μ as above. If Q ∈ \underline{G}, let Γ_{Q} = O_{Q} ∩ ∂Ω. We can now repeat the arguments in Sections 3 and 4, essentially verbatim. In fact the notation was chosen so that we can define the shift S as in (3.9), (3.10), and identify it with the following mapping: If Q ∈ \underline{G}_{1} , then S restricted to Q is the translation-magnification which takes Γ_{Q} onto E = Γ_{J}. Using Lemma 2.16 and arguing as in (3.12)–(3.18) we get Lemma 3.11. Arguing as in (3.20)–(3.21) we obtain first Lemma 3.19 and then Lemma 3.22. Finally arguing as in (3.24)–(3.25) we get Lemma 3.23.

For fixed p > 2 and n a nonnegative integer, let û_{n} be the p harmonic function in Ω_{n} with continuous boundary values 0 on ∂E_{n} and 1 on ∂B(0, 2). Let μ̂_{n} be the measure corresponding to û_{n} as in (1.2) with u replaced by û_{n}. We observe first that (3.28) is valid and second that (3.29), (3.30) hold. As in Section 4 put

\begin{aligned}μ_{n} & = \frac{μ̂_{n}}{μ̂_{n}(E_{n})}, \\ u_{n} & = \frac{û_{n}}{μ̂_{n}(E_{n})}. \end{aligned}

Arguing as in Section 4 we obtain (4.2), (4.3) with τ(Q), Ω̂_{n} replaced by Q, Ω_{n}. Also arguing as in (4.5)–(4.11) we get Lemma 4.4 with τ(Q) replaced by Q. Moreover, arguing as in (4.14)–(4.18) we get first Lemma 4.13 (Q replacing τ(Q)) and then Lemma 4.19. Armed with Lemma 4.19 we can apply the divergence theorem to the vector field below (4.22). In this case there is an additional complication because ∇u_{n} can vanish at points of Ω_{n}. To handle this complication we first show that

(5.5)  ∇u_{n} vanishes at a finite number of points in Ω_{n}.

Indeed if Q ∈ \underline{G}_{n}, then for small t > 0 we see from the maximum principle for p-harmonic functions that there is a simply connected component U of \{x : u_{n}(x) < t\} with \overline{O}_{Q} ⊂ U. Then D_{Q} = U \ \overline{O}_{Q} is doubly connected and as in Lemma 2.26, it follows that ∇u_{n} ≠ 0 in D_{Q}. Since \underline{G}_{n} has finite cardinality, we conclude that there exists a neighborhood N containing E_{n} with ∇u_{n} ≠ 0 in N ∩ Ω_{n}. Also as in Lemma 2.26, we see that u_{n} has a real analytic extension to a neighborhood of ∂B(0, 2) with |∇u_{n}| ≈ 1 on ∂B(0, 2). From these facts and Lemma 2.8 we conclude that (5.5) is true.

Now suppose w ∈ Ω_{n} and ∇u_{n}(w) = 0. Choose r_{0} > 0 so small that ∇u_{n} ≠ 0 in B(w, r_{0}) \ {w}. Then as noted in the display below (2.10), v = log |∇u_{n}| satisfies a uniformly elliptic partial differential equation in divergence form when ∇u_{n} ≠ 0, so if 0 < 6r < r_{0} , then by the usual Caccioppoli type estimate,

(5.6)  ∫_{B(w,4r)\B(w,3r)} |∇v|^{2} dx ≤ cr^{−2} ∫_{B(w,5r)\B(w,2r)} v^{2} dx ≤ c^{2}A(r)^{2}

where A(r) denotes the essential supremum of |v| in B(w, 6r) \ B(w, r). Using Hölder’s inequality we deduce from (5.6) that

(5.7)  ∫_{B(w,4r)\B(w,3r)} |∇v| dx ≤ c_{−}rA(r)

where c_{−} = c_{−}(w, u_{n}) is independent of r. From (5.7) and a weak type estimate we conclude for some r̂ ∈ (3r, 4r) that

(5.8)  ∫_{∂B(w,r̂)} |∇v| dH^{1} ≤ c2_{−}A(r).

Furthermore using the factorization theorem mentioned after Lemma 2.7 and wellknown properties of analytic functions, quasiconformal mappings, we see that

(5.9)  A(r) ≈ log r as r → 0.

From Lemma 2.7 we also see that

(5.10)  |∇u_{n}| ≤ cr^{σ} on ∂B(w, r̂).

Let {w_{i}}_{1}^{m} denote the zeros of ∇u_{n} in Ω_{n}. Let r = r(w_{i}), r̂ = r̂(w_{i}), 1 ≤ i ≤ m, be as above and put Ω′_{n} = Ω_{n} \ ⋃_{i=1}^{m} \overline{B}(w_{i}, r̂(w_{i})). We repeat the argument following (4.22) to obtain (4.25) with D̂_{n} replaced by Ω′_{n}. Let ξ be the outer unit normal to ∂Ω′_{n}. Using (5.8)–(5.10) and arguing as in (4.27) we get

(5.11)  \begin{matrix}∫_{∂Ω′_{n}} ∑_{i,k=1}^{2} b_{ik}ξ_{i}[u_{n}v_{x_{k}} − v(u_{n})_{x_{k}}] dH^{1}x \\ = (p − 1) ∫_{∂E_{n}} |∇u_{n}|^{p−1} log |∇u_{n}| dH^{1}x + F \end{matrix}

where

(5.12)  \begin{aligned}|F | & ≤ c + ∑_{i=1}^{m} ∫_{∂B(w_{i},r̂(w_{i}))}[|∇u_{n}|^{p−1}|v| + u_{n}|∇u_{n}|^{p−2}|∇v|] dH^{1}x \\ & ≤ c + r^{σ(p−2)} log r. \end{aligned}

Since p > 2 we can let r → 0 in (5.11) and conclude in view of (5.12), as in (4.28), that it suffices to show

(5.13)  lim inf_{n→∞}(n^{−1} ∫_{Ω_{n}} u_{n}|∇u_{n}|^{p−4} ∑_{i,j=1}^{2}[(u_{n})_{x_{i}x_{j}}]^{2}(x) dx) > 0

in order to prove Theorem 2. To do this we first claim there exists c̃, a positive integer, depending only on p and the initial choice of J_{i}, 1 ≤ i ≤ l, such that if m < n − 3c̃, j = m + 2c̃, we have,

(5.14)  ∇u_{n}(w) = 0 for some w ∈ O_{Q} \ E_{j} whenever Q ∈ \underline{G}_{m}.

To prove claim (5.14) recall that u_{n} ≡ 0 on E_{n}. Given Q ∈ \underline{G}_{m}, let q = m + c̃, and choose Q′ ∈ \underline{G}_{q}, with Q′ ≺ Q. Note from Lemma 2.13 that for c̃ large enough,

(5.15)  t_{1} = 2 max_{\overline{O}_{Q′}} u_{n} < min_{Q} u_{n}

and

(5.16)  t_{2} = 2 max_{E_{j}∩O_{Q′}} u_{n} < min_{∂E_{q+1}∩O_{Q′}} u_{n}.

Let D(s) = {x : u_{n}(x) < s}. From (5.16) and the maximum principle for pharmonic functions we see that D(t_{2}) ∩ O_{Q′} splits into β ≥ l > 1 components having a non-empty intersection with E_{j} ∩O_{Q′}. If ∇u_{n} = 0 on ∂D(t_{2})∩O_{Q′}, then (5.14) is valid since ∂D(t_{2})∩O_{Q′} ⊂ O_{Q}\E_{j}. Otherwise, it follows from the implicit function theorem that there exists ε = ε(t_{2}) > 0 for which D(s), s ∈ (t_{2}−ε, t_{2}+ε), has exactly β components which have a non-empty intersection with E_{j} ∩ O_{Q′}. Let τ be the least upper bound of the set of all t such that D(s) has exactly β components which intersect E_{j} ∩O_{Q′} whenever s ∈ (t_{2}, t). Note that τ ≤ t_{1} since O_{Q′} ⊂ D(t_{1}) (so D(t_{1}) has only one component containing points in O_{Q′}). From this note and (5.15) we conclude that ∂K ⊂ O_{Q} \ E_{j} whenever K is a component of D(τ) with K ∩ E_{j} ∩ O_{Q′} ≠ ∅. Also there must exist K̂ a component of D(τ) with K̂ ∩E_{j} ∩O_{Q′} ≠ ∅ and ∇u_{n}(w) = 0 for some w ∈ ∂K̂ . In fact, otherwise, we could repeat the argument given for D(t_{2}) and contradict the maximality of D(τ). Thus claim (5.14) is true. We also observe from (5.15) and the mean value theorem that

(5.17)  ĉl(Q)|∇u_{n}|(x) ≥ max_{Q} u_{n}

for some x ∈ O_{Q} \ E_{q}. Let σ be a smooth curve connecting x to w in (5.14) with length ≤ c′l(Q) and σ ⊂ O_{Q} \ E_{j} (ĉ, c′ have the same dependence as c̃ above (5.14)). Then v(y) → −∞ as y → w on σ. Thus starting from x there exists a first point z such that |v(x) − v(z)| = 1. From Lemma 2.7, (5.17), and our choice of σ we deduce for some c_{+} ≥ 1, having the same dependence as c̃,

(5.18)  c−1_{+}l(Q) ≤ |z − x| ≤ c_{+}l(Q).

Applying the mean value theorem to an arclength parametrization of σ we get y = y_{Q} on the arc of σ connecting x to z with

(5.19)  l(Q)^{−1}u_{n}(x) ≈ |∇u_{n}| and cl(Q)|∇u_{n}|^{−1} ∑_{i,j=1}^{2} |(u_{n})_{y_{i}y_{j}}| ≥ 1

at y. From (5.19), Lemma 2.7, we see that

\frac{1}{2}|∇u_{n}|(y) ≤ |∇u_{n}|(ŷ) ≤ 2|∇u_{n}|(y)

when ŷ ∈ B(y, l(Q)/c) provided c is large enough. Lemma 2.9 can now be applied to conclude that (5.19) is valid on B_{Q} = B(y_{Q}, r_{Q}) ⊂ Q \ E_{j+1} with

(5.20)  r_{Q} ≈ l(Q).

Again all constants in (5.19), (5.20) have the same dependence as c̃. From (5.19), (5.20), Harnack’s inequality and (2.15), we conclude as in (4.35) that

(5.21)  c ∫_{B_{Q}} u_{n}|∇u_{n}|^{p−4} ∑_{i,j=1}^{2} |(u_{n})_{x_{i}x_{j}}|^{2} dx ≥ μ_{n}(O_{Q}).

We do this for each Q ∈ \underline{G}_{m} and 1 ≤ m < n − 3c̃. We obtain, Θ = \{B_{Q} : Q ∈ \underline{G}_{m}, 1 ≤ m < n − 3c̃\}, satisfying (5.21). Since B_{Q} ⊂ O_{Q} \ E_{j+1} when Q ∈ \underline{G}_{m}, we see (as in the argument following (4.37)) that each point in Ω_{n} lies in at most c members of Θ. Using this fact and summing (5.21), we get (5.13). The proof of Theorem 2 is complete.

6. Proof of Theorem 3

To begin the proof of Theorem 3, fix p ≠ 2, 1 < p < ∞. We note from (2.24) that it suffices to prove Theorem 3 for u_{1} , u_{2} as defined below (2.25). In fact, we just prove Theorem 3 for u_{1} , since the proof for u_{2} is unchanged. We assume, as we may, that x̂ = 0, r̂ = 1 in (2.25), since otherwise we translate and dilate Ω. Thus if u = u_{1} , then u is continuous in the closure of D = Ω \ \overline{B}(0, 1) with u ≡ 1 on ∂B(0, 1) and u ≡ 0 on ∂Ω. Also B(0, 4) ⊂ Ω, ∂Ω is a k - quasicircle, and u is a solution to the p-Laplacian partial differential equation in D. If v(x) = log |∇u(x)|, x ∈ D, let w(x) = max(v, 0) when 1 < p < 2 and w(x) = max(−v, 0) when p > 2. Following Makarov (see [M], [P, Chapter 8, Section 5]), we first prove

Lemma 6.1. Let m be a nonnegative integer. There exists c_{+} = c_{+}(k) ≥ 1 such that for 0 < t < 1,

∫_{{x:u(x)=t}} |∇u|^{p−1}w^{2m} dH^{1}x ≤ cm+1_{+}m![log(2/t)]^{m}.

Proof. Put h(x) = max(w(x) − c′, 0), x ∈ D. Here c′ is chosen so large that h ≡ 0 in B(0, 2) ∩ D. Existence of c′ follows from Lemma 2.26. Extend h continuously to Ω by putting h ≡ 0 in \overline{B}(0, 1). Let Ω(t) = Ω \ \{x : u(x) ≤ t\}, whenever 0 < t < 1 and let L, (b_{ik}), be as in (1.13), (1.14). We note that h^{2} ∈ W^{2,∞}(Ω(t)). Also from (1.14), (1.20) we deduce for H^{2} almost every x ∈ Ω(t),

(6.2)  \begin{aligned}L(h^{2m})(x) & = ∑_{i,j=1}^{2} \frac{∂}{∂x_{i}} [b_{ij}(x)(h^{2m})_{x_{j}}(x)] \\ & ≤ 2m(2m − 1)h^{2m−2}(x) ∑_{i,j=1}^{2} b_{ij}(x)h_{x_{i}}(x)h_{x_{j}}(x) \\ & ≤ 2m(2m − 1)p|∇u|^{p−2}(x)h^{2m−2}(x)|∇h|^{2}(x). \end{aligned}

Next we apply the divergence theorem in Ω(t) to the vector field whose ith component for i = 1, 2 is

(u − t) ∑_{j=1}^{2} b_{ij}(h^{2m})_{x_{j}} − h^{2m} ∑_{j=1}^{2} b_{ij}u_{x_{j}}.

Using (6.2) and our choice of c′ we deduce from Lemma 2.26, as in (4.27), that (6.3)

(p−1) ∫_{{x:u(x)=t}} |∇u|^{p−1}h^{2m} dH^{1}x ≤ 2m(2m−1)p ∫_{Ω(t)} u|∇u|^{p−2}h^{2m−2}|∇h|^{2} dx.

We note from Lemma 2.26 that for some c = c(k) ≥ 1,

(6.4)  |∇h| ≤ cd(x, ∂Ω)^{−1} and d(x, ∂Ω)^{−1}u(x) ≈ |∇u(x)|

for H^{2} almost every x ∈ D. Using (6.4) in (6.3) and the coarea formula (see [Fe, Section 3.2]), we obtain with p′ = p/(p − 1),

(6.5) \begin{aligned}I_{m}(t) & = ∫_{\{x:u(x)=t\}} |∇u|^{p−1}h^{2m} dH^{1}x \\ & ≤ 2m(2m − 1)p′ ∫_{Ω(t)} u|∇u|^{p−2}h^{2m−2}|∇h|^{2} dx \\ & = 2m(2m − 1)p′ ∫_{t}^{1} τ(∫_{\{x:u(x)=τ\}} |∇u|^{p−3}h^{2m−2}|∇h|^{2} dH^{1}x) dτ \\ & ≤ 2m(2m − 1)p′c ∫_{t}^{1}(∫_{\{x:u(x)=τ\}} |∇u|^{p−1}h^{2m−2} dH^{1}x)τ^{−1} dτ \\ & = 2m(2m − 1)p′c ∫_{t}^{1} I_{m−1}(τ)τ^{−1} dτ. \end{aligned}

Observe that

(6.6)  ∫_{{x:u(x)=τ}} |∇u|^{p−1} dH^{1}x = constant

for 0 < τ < 1 as follows easily from ∇ · (|∇u|^{p−2}∇u) = 0 and the divergence theorem. From (6.6) we have trivially, for some c_{∗} = c_{∗}(k),

(6.7)  I_{0}(t) ≤ c_{∗} for 0 < t < 1.

By induction if m ≥ 1 and

(6.8)  I_{m−1}(t) ≤ c_{∗}^{m}(m − 1)! log^{m−1}(2/t), 0 < t < 1,

then from (6.5), (6.8), we obtain upon integrating that (6.8) holds with m − 1 replaced by m provided c_{∗} is suitably large. Thus by induction (6.8) is true for m = 1, 2, …. Since w ≤ h + c′, we conclude from (6.8) and simple estimates that Lemma 6.1 is valid.

Dividing the display in Lemma 6.1 by (2c_{+})^{m}m! log^{m}(2/t) and summing we see for 0 < t < 1 that

(6.9)  ∫_{\{x:u(x)=t\}} |∇u|^{p−1} exp[\frac{w^{2}}{2c_{+} log(2/t)}] dH^{1}x ≤ 2c_{+}.

Let

λ(t) = \sqrt{4c_{+} log(2/t)} \sqrt{log(− log t)} for 0 < t < e^{−2},
F (t) = {x : u(x) = t and w(x) ≥ λ(t)}.

Then from (6.9) and weak type estimates we deduce

(6.10)  ∫_{F (t)} |∇u|^{p−1} dH^{1}x ≤ 2c_{+}[log(1/t)]^{−2}.

Next for a fixed and large, we define Hausdorff measure (denoted σ) with respect to

γ(r) = \{ \begin{matrix}re^{aλ(r)} & when 1 < p < 2, \\ re^{−aλ(r)} & when p > 2. \end{matrix}

That is, for fixed 0 < δ < e^{−2} and E ⊆ R^{2} , let L(δ) = {B(z_{i}, s_{i})} be such that E ⊆ ⋃ B(z_{i}, s_{i}) and 0 < s_{i} < δ, i = 1, 2, …. Set

ϕ_{δ}(E) = inf_{L(δ)}(∑_{i} γ(s_{i})).

Then

σ(E) = lim_{δ→0}ϕ_{δ}(E).

We claim for a = a(k) large enough, that

(6.11)  μ is absolutely continuous with respect to σ when 1 < p < 2.

To prove this claim let K ⊂ ∂Ω be a Borel set and suppose that σ(K) = 0. Let K_{1} be the subset of all x ∈ K with

(6.12)  lim sup_{r→0} \frac{μ(B(x, r))}{γ(r)} ≤ 2.

Then from the definition of σ, it is easily shown that μ(K_{1}) = 0. Thus to prove (6.11) it suffices to show μ(E) = 0 when E is Borel and is equal μ almost everywhere to the set of all points in ∂Ω for which (6.12) is false. For this purpose given r_{0} > 0, small, we use a well-known covering argument to get {B(x_{i}, r_{i})} pairwise disjoint with r_{i} ≤ r_{0} ,

(6.13)  μ[B(x_{i}, 5r_{i})] ≥ γ(5r_{i}) and E ⊂ ⋃ B(x_{i}, 5r_{i}).

Let t_{m} = 2^{−m} for m = 4, 5, …. From Lemmas 2.13, 2.26 and (2.15) we see there exists y_{i} ∈ B(x_{i}, r_{i}/2) with u(y_{i}) = t_{m} for some m = m(i), d(y_{i}, ∂Ω) ≈ r_{i}, and

(6.14)  μ[B(x_{i}, r_{i})]/r_{i} ≈ [u(y_{i})/d(y_{i}, ∂Ω)]^{p−1} ≈ |∇u(y)|^{p−1}

whenever y ∈ B(y_{i}, d(y_{i}, ∂Ω)/2). We conclude from (6.13), (6.14) for some c̃ = c̃(k) ≥ 1 that

(6.15)  w(y) = log |∇u(y)| ≥ aλ(5r_{i})/c̃ on B(y_{i}, d(y_{i}, ∂Ω)/2).

Note that

(6.16)  H^{1}[B(y_{i}, d(y_{i}, ∂Ω)/2) ∩ {x : u(x) = t_{m}}] ≥ d(y_{i}, ∂Ω)/2

as we see from the maximum principle for p-harmonic functions, a connectivity argument and basic geometry. Finally from Lemma 2.13, Harnack’s inequality, (3.28) we find for some β = β(k), 0 < β < 1, c̄ = c̄(k), that

(6.17)  r_{i} ≤ c̄t_{m}^{β} ≤ c̄^{2}r_{i}^{β^{2}}.

Using (6.14)–(6.17) and the doubling property of μ (see (3.28)) we conclude for a large enough that

(6.18)  μ[B(x_{i}, 5r_{i})] ≤ c ∫_{F (t_{m})∩B(x_{i},r_{i})} |∇u|^{p−1} dH^{1}x.

From (6.18), disjointness of {B(x_{i}, r_{i})}, and (6.10) it follows for c large enough that

(6.19)  \begin{aligned}μ(E) & ≤ ⋃_{i} μ[B(x_{i}, 5r_{i})] \\ & ≤ c ∑_{m=m_{0}}^{∞} ∫_{F (t_{m})} |∇u|^{p−1} dH^{1}x ≤ c^{2} ∑_{m=m_{0}}^{∞} m^{−2} ≤ c^{3}m_{0}^{−1}, \end{aligned}

where 2^{−m_{0}β} = c̄r_{0}^{β^{2}} . Since r_{0} can be arbitrarily small we see from (6.19) and the remark after (6.12) that claim (6.11) is true. Theorem 3 follows for 1 < p < 2 from claim (6.11) since σ is absolutely continuous with respect to H^{1−ε} measure for each ε > 0.

Finally to prove Theorem 3 for p > 2, we show there exists a Borel set K̂ ⊂ ∂Ω with

(6.20)  μ(K̂) = μ(∂Ω) and σ(K̂) ≤ 100μ(K̂).

Since H^{1+ε} measure is absolutely continuous with respect to σ for each ε > 0 it follows from (6.20) that Theorem 3 is true when p > 2. The proof of (6.20) is similar to the proof of (6.11). Let K̂ be the set of all x ∈ ∂Ω with

(6.21)  lim sup_{r→0} \frac{μ(B(x, r))}{γ(r)} ≥ 2.

From the definition of σ, (6.21), and a Vitali covering type argument (see [Ma, Chapter 2]) it follows easily that σ(K̂) ≤ 100μ(K̂). Thus to prove (6.20) it suffices to show μ(Ê) = 0 when Ê is Borel and equal σ almost everywhere to the set of all points in ∂Ω for which (6.21) is false. For this purpose given r_{0} > 0, small, we argue as above to first get {B(x_{i}, r_{i})} pairwise disjoint with r_{i} ≤ r_{0} ,

μ[B(x_{i}, 5r_{i})] ≤ γ(5r_{i}) and Ê ⊂ ⋃ B(x_{i}, 5r_{i}).

Next we repeat the argument after (6.13) leading to (6.18). We obtain

μ[B(x_{i}, 5r_{i})] ≤ c ∫_{F (t_{m})∩B(x_{i},t_{i})} |∇u|^{p−1} dH^{1}x

(t_{m} as defined above). Finally as in (6.19) we get

μ(Ê) ≤ cm_{0}^{−1} → 0 as r_{0} → 0.

Hence (6.20) is true and the proof of Theorem 3 is complete.

7. Closing remarks

We now discuss some open problems arising from Theorems 1–3.

(1) We would like to know if the rate theorem in Lemma 2.16 is valid for Ω an NTA domain ⊂ R^{n} and 1 < p < ∞, as is the case when p = 2. The argument in the proof of Lemma 2.16 cannot be used in R^{n}, n ≥ 3. In R^{2} , this argument works for somewhat more general domains than domains bounded by quasicircles. For example the same argument would work for a disk with a slit. However if we only assume that Ω is an NTA-domain our proof no longer works. For example, let x_{i,j} = (i2^{−j}, 2^{−j}) for −2^{j+1} ≤ i ≤ 2^{j+1} , j = 1, 2, …. Put Δ = ⋃_{i,j} B(x_{i,j}, 2^{−j−2}) and Ω = {x ∈ R^{2} : x_{2} > 0, |x| < 4} \ Δ̄. It is easily shown that Ω is an NTA domain. If we proceed as in the proof of Lemma 2.16, we can construct curves between points in I_{1} = B((1, 0), 1/2) ∩ {x : x_{2} = 0} and I_{2} = B((−1, 0), 1/2) ∩ {x : x_{2} = 0} and obtain as in (2.18)–(2.21) that

μ(O) ≥ λ′ν(O)

for some open set O containing either I_{1} or I_{2} . However this inequality does not lead to a contradiction since (2.15) only gives an estimate for

\frac{μ[B((±1, 0), 1/2)]}{ν[B((±1, 0), 1/2)]}

and O ⊂ {x : |x_{2}| < ε} is possible. In fact, for each j we have by doubling that μ[B((i2^{−j}, 0), 2^{−j−2})] < cμ[B(x_{i,j}, 2^{−j−2})]. Since the projections of the balls B((i2^{−j}, 0), 2^{−j−2}), j fixed, cover one fourth of (−2, 2) and are equally spaced, we find (again using doubling) that

μ[\{x : x_{2} = 0\} ∩ (−\frac{3}{2}, \frac{3}{2})] < cμ[⋃_{i} B(x_{i,j}, 2^{−j−2})].

Summing this inequality over 1 ≤ j ≤ n we get

nμ(\{x : x_{2} = 0\} ∩ (−\frac{3}{2}, \frac{3}{2})) < cμ(Δ).

Letting n → ∞ it follows that μ(\{x : x_{2} = 0\}∩(−\frac{3}{2}, \frac{3}{2})) = 0. A similar argument holds for ν. Thus,

μ[I_{1}] ≥ λ′ν[I_{1}]

is valid for any λ′ > 0.

(2) Given boundedness of the ratio in Lemma 2.16, we would like to know if this ratio is also Hölder continuous (as is the case when p = 2).

(3) As for Theorem 1, we would like to know the exact value of H-dim μ when ∂Ω is the Van Koch snowflake and p ≠ 2. One can also ask for a given p, what the supremum (p < 2) or infimum (p > 2) is of H-dim μ taken over the class of quasicircles or even simply connected domains. If this question is too hard, perhaps one can obtain estimates for H-dim μ as p → 1 and p → ∞? At p = 1, the p-Laplacian degenerates into the mean curvature equation, so for a given Ω, a natural conjecture would be that H-dim μ → the Hausdorff dimension of ∂Ω as p → 1. Is H-dim μ continuous and/or decreasing as a function of p when ∂Ω is the Van Koch snowflake? Are the p-harmonic measures defined on each side of a snowflake mutually singular? The answer is yes when p = 2 (see [BCGJ]).

(4) Is it always true for p > 2 that H-dim μ < Hausdorff dimension of ∂Ω when ∂Ω is as in Theorem 2? The p = 2 case was handled in [Ba].

(5) Does Lemma 2.26 generalize to simply connected domains? If so, then our proof scheme in Theorem 3 could be used to get an analogue of Makarov’s theorem for 1 < p < ∞. That is, Theorem 3 would generalize to simply connected domains.

(6) Does the Jones–Wolff theorem alluded to in the introduction hold for p > 2 in all planar domains?

(7) In the proof of Theorem 3 we showed for 1 < p < 2 that μ is absolutely continuous with respect to γ where γ is defined below (6.10). In the case p = 2 Makarov proves absolute continuity with respect to a smaller measure and shows by way of example that his result is essentially best possible. His methods also imply that μ is concentrated on a set of σ finite H^{1} measure (see [P, Chapter 6, Section 5] or [W1]) which is considerably better than our result for μ when 2 < p. Can one, at least for quasicircles, obtain the full strength of the results in [M] when 1 < p < ∞?

(8) If p is fixed, 1 < p < ∞, give a criterion for which the p-harmonic measures defined on both sides of a quasicircle (or more generally a Jordan domain), are mutually absolutely continuous. Necessary and sufficient conditions for p = 2 are given in [BCGJ].

(9) There are numerous related problems for p = 2 which one can try to find analogues of when p ≠ 2 (see [CM]).

(10) Let u be a positive weak solution to some divergence form partial differential equation in a neighborhood of ∂Ω with u ≡ 0 on ∂Ω in the Sobolev sense. Then the existence of a measure μ satisfying an integral equality, similar to (1.2), is essentially equivalent to showing that powers of u satisfy a basic Caccioppoli inequality. Thus μ corresponding to u as above, exists for a large class of divergence form partial differential equations. For which divergence form partial differential equations can one prove analogues of Theorems 1–3? Regarding this question we note that the invariance of the p-Laplacian under rotation, translation, and dilation was of crucial importance in the proof of Theorems 1 and 2. We also made important use in Theorems 1–3 of the fact that u and its derivatives both satisfy (1.11). This property is also somewhat special for the p-Laplacian, as one sees from considering smooth solutions u to

∇ · [f(|∇u|^{2})∇u] = 0

(f smooth on (0, ∞)). As in (1.11) we see that if ζ = ⟨∇u, η⟩ and ∇u(x) ≠ 0, then

L̃ζ = ∇ · [2f′(|∇u|^{2})⟨∇u, ζ⟩∇u + f(|∇u|^{2})∇ζ] = 0

at x. Moreover,

L̃u = ∇ · [2f′(|∇u|^{2})|∇u|^{2}∇u]

at x and this equation is only obviously zero if f(t) = at^{λ} for some real a, λ. Thus our theorems may be special for the p-Laplacian.

504 Björn Bennewitz and John L. Lewis

References

[A] Ahlfors, L. V.: Lectures on Quasiconformal Mappings. - D. Van Nostrand, 1966.

[ALR] Alessandrini, G., D. Lupo, and E. Rosset: Local behaviour and geometric properties of solutions to degenerate quasilinear elliptic equations in the plane. - Appl. Anal. 50, 1993, 191–215.

[Ba] Batakis, A.: Harmonic measure of some Cantor type sets. - Ann. Acad. Sci. Fenn. Math. 21, 1996, 255–270.

[Bi] Billingsley, P.: Ergodic Theory and Information. - John Wiley & Sons, Inc., 1965.

[BCGJ] Bishop, C., L. Carleson, J. Garnett, and P. Jones: Harmonic measures supported on curves. - Pacific J. Math. 138, 1989, 233–236.

[B] Bourgain, J.: On the Hausdorff dimension of harmonic measure in higher dimensions. - Invent. Math. 87, 1987, 477–483.

[CFMS] Caffarelli, L., E. Fabes, S. Mortola, and S. Salsa: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. - Indiana J. Math. 30, 1981, 621–640.

[C] Carleson, L.: On the support of harmonic measure for sets of Cantor type. - Ann. Acad. Sci. Fenn. Math. 10, 1985, 113–123.

[CM] Carleson, L., and N. Makarov: Some results connected with Brennan’s conjecture. - Ark. Mat. 32, 1994, 33–62.

[D] DiBenedetto, E.: C^{1+α} local regularity of weak solutions of degenerate elliptic equations. - Nonlinear Anal. 7, 1983, 827–850.

[EL] Eremenko, A., and J. Lewis: Uniform limits of certain A-harmonic functions with applications to quasiregular mappings. - Ann. Acad. Sci. Fenn. Math. 16, 1991, 361– 375.

[E] Evans, L. C.: Partial Differential Equations. - Grad. Stud. Math. 19, American Mathematical Society, 1998.

[Fe] Federer, H.: Geometric Measure Theory. - Springer-Verlag, 1969.

[F] Friedman, A.: Partial Differential Equations. - Robert Krieger Publishing Company, 1983.

[F1] Friedman, A.: On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations. - J. Math. Mechanics 7, 1958, 43–59.

[GZ] Gariepy, R., and W. Ziemer: A regularity condition at the boundary for solutions of quasilinear elliptic equations. - Arch. Ration. Mech. Anal. 6, 1977, 25–39.

[G] Gehring, F.: Uniform domains and the ubiquitous quasidisk. - Jahresber. Deutsch. Math.- Verein. 89, 1987, 88–103.

[GT] Gilbarg, D., and N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. - 2nd edition, Springer-Verlag, 1983.

[HKM] Heinonen, J., T. Kilpeläinen, and O. Martio: Nonlinear Potential Theory of Degenerate Elliptic Equations. - Oxford University Press, 1993.

[H] Hopf, E.: Über den Funktionalen insbesondere den analytischen Character der Lösungen elliptischer Differentialgleichungen zweiter Ordung. - Math. Z. 34, 1932, 194–233.

[IM] Iwaniec, T., and J. Manfredi: Regularity of p harmonic functions on the plane. - Rev. Mat. Iberoamericana 5, 1989, 1–19.

[JK] Jerison, D., and C. Kenig: Boundary behavior of harmonic functions in nontangentially accessible domains. - Adv. Math. 46, 1982, 80–147.

[JW] Jones, P., and T. Wolff: Hausdorff dimension of harmonic measure in the plane. - Acta Math. 161, 1988, 131–144.

[KW] Kaufmann, R., and J. M. Wu: On the snowflake domain. - Ark. Mat. 23, 1985, 177–183.

[KZ] Kilpeläinen, T., and X. Zhong: Growth of entire A-subharmonic functions. - Ann. Acad. Sci. Fenn. Math. 28, 2003, 181–192.

[K] Krol′, I. N.: On the behavior of the solutions of a quasilinear equation near null salient points of the boundary. - Proc. Steklov Inst. Math. 125, 1973, 130–136.

[L] Lewis, J.: Approximation of Sobolev functions in Jordan domains. - Ark. Mat. 25, 1987, 255–264.

[L1] Lewis, J. L.: Regularity of the derivatives of solutions to certain degenerate elliptic equations. - Indiana Univ. Math. J. 2, 1983, 849–858.

[LVV] Lewis, J., G. Verchota, and A. Vogel: On Wolff snowflakes. - Pacific J. Math. (to appear).

[Li] Lieberman, G. M.: Boundary regularity for solutions of degenerate elliptic equations. - Nonlinear Anal. 12, 1988, 1203–1219.

[LMW] Llorente, J., J. Manfredi, and J. M. Wu: p harmonic measure is not subadditive. - Submitted.

[M] Makarov, N.: Distortion of boundary sets under conformal mapping. - Proc. London Math. Soc. 51, 1985, 369–384.

[Ma] Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. - Cambridge Univ. Press, 1995.

[Mar] Martio, O.: Potential theoretic aspects of nonlinear elliptic partial differential equations. - Ber. [Report] 44, Universität Jyväskylä Mathematisches Institut, Jyväskylä, 1989, 1–23.

[P] Pommerenke, Ch.: Boundary Behaviour of Conformal Maps. - Grundlehren Math. Wiss. 299, Springer-Verlag, 1992.

[Re] Reshetnyak, Y.: Space Mappings with Bounded Distortion. - Transl. Math. Monogr. 73, American Mathematical Society, 1989.

[R] Rudin, W.: Real and Complex Analysis. - McGraw-Hill, 1987.

[S] Serrin, J.: Local behavior of solutions of quasilinear elliptic equations. - Acta Math. 111, 1964, 247–302.

[Sh] Shields, P.: The Ergodic Theory of Discrete Sample Paths. - Grad. Stud. Math. 13, American Mathematical Society, 1996.

[T] Tolksdorff, P.: Everywhere regularity for some quasi-linear systems with lack of ellipticity. - Ann. Math. Pura Appl. 134, 1984, 241–266.

[V] Volberg, A.: On the dimension of harmonic measure of Cantor repellers. - Michigan Math. J, 40, 1993, 239–258.

[W] Wolff, T.: Counterexamples with harmonic gradients in R^{3}. - In: Essays in Honor of Elias M. Stein, Princeton Math. Ser. 42, 1995, 321–384.

[W1] Wolff, T.: Plane harmonic measures live on sets of σ finite linear length. - Ark. Mat. 31, 1993, 137–172.

Received 20 October 2004