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ON the DIMENSION of P-HARMONIC MEASURE

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ON the DIMENSION of P-HARMONIC MEASURE Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 459{505 ON THE DIMENSION OF p-HARMONIC MEASURE Bj¨orn Bennewitz and John L. Lewis University of Kentucky, Department of Mathematics Lexington, Kentucky, 40506-0027, U.S.A. [email protected]; [email protected] 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 2 Denote points in Euclidean 2-space R by x = (x1; x2): Let ; be the standard inner product on R2 and let x = x; x 1=2 be the Euclideanh ·norm· i of x. Set B(x; r) = y R2 : x y < r whenevj j her xi R2 and r > 0. Let dx denote Lebesgue measuref 2 on Rj2 −andj defineg k dimensional2 Hausdorff measure, in R2 , 2 0 < k 2, as follows: For fixed δ > 0 and E R , let L(δ) = B(xi; ri) be such that≤ E B(x ; r ) and 0 < r < δ , i = 1⊆; 2; : : : . Set f g ⊆ i i i S k k φδ (E) = inf α(k) ri ; L(δ)X where α(k) denotes the volume of the unit ball in Rk . Then k k H (E) = lim φδ (E); 0 < k 2: δ!0 ≤ If O is open and 1 q , let W 1;q(O) be the space of equivalence classes of ≤ ≤ 1 functions u with distributional gradient u = (ux1 ; ux2 ), both of which are q th power integrable on O . Let r u = u + u k k1;q k kq kr kq 1;q be the norm in W (O) where q denotes the usual Lebesgue q -norm in O . 1 k · k Let C0 (O) be infinitely differentiable functions with compact support in O and 1;q 1 1;q let W0 (O) be the closure of C0 (O) in the norm of W (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 < , 1 2000 Mathematics Subject Classification: Primary 35J65; Secondary 31A15. Both authors were partially supported by NSF grant 0139748. 460 Bj¨orn Bennewitz and John L. Lewis 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 \ 2 \ (1:1) u p−2 u; θ dx = 0 Z jr j hr r i 1;p whenever θ W0 (Ω N). Observe that if u is smooth and u = 0 in Ω N , then ( 2u p−2 u)\ 0; in the classical sense, where denotesr 6 divergence.\ We assumer · jrthatj urhas ≡zero boundary values on @Ω in the∇·Sobolev sense. More 1 1;p specifically if ζ C0 (N), then u ζ W0 (Ω N). Extend u to N Ω by putting u 0 on N Ω2. Then u W 1;p(N2) and it follo\ ws from (1.1), as inn [HKM], that there≡ exists na positive finite2 Borel measure µ on R2 with support contained in @Ω and the property that (1:2) u p−2 u; φ dx = φ dµ Z jr j hr r i − Z 1 whenever φ C0 (N). For the reader's convenience we outline another proof of existence for2µ under the assumption that u is continuous in N . We claim that it suffices to show u p−2 u; φ dx 0 Z jr j hr r i ≤ 1 whenever φ 0 and φ C0 (N). Indeed once this claim is established, we get existence of ≥µ as in (1.2)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 "; 0])" η" φ can be shown to be an admissible test − − function in (1.1) for small η > 0. From (1.1) we see that (η + max[u "; 0])" η" u p−2 u; φ dx 0: Z − − jr j hr r i ≤ fu≥"g\N Using dominated convergence, letting first η and then " 0 we get our claim. We note that if @Ω is smooth enough, then ! (1:3) dµ = u p−1 dH1 : jr j j@Ω In this paper we study for given p, 1 < p < , the Hausdorff dimension of µ (denoted H-dim µ) defined as follows: 1 H-dim µ = inf k : there exists E Borel @Ω with H k(E) = 0 and µ(E) = µ(@Ω) : f ⊂ g To outline previous work, let p = 2 and u be the Green's function for Ω with pole at some point x Ω. Then µ is called harmonic measure for Ω relative 0 2 On the dimension of p-harmonic measure 461 to x0 . 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 ≤ 1 the importance of gn log gn dH where gn is the Green's function for @Ωn jr j jr j Ωn with pole at x0R Ω0 and (Ωn) is an increasing sequence of smooth domains whose union is Ω. Later2 Makarov [M] proved for any simply connected domain Ω R2 , that H-dim µ = 1. Jones and Wolff [JW] proved that H-dim µ 1 when Ω ⊂ R2 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. f g 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: 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. 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. ≤ ≤ 1 ≥ ≤ 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 R3 . The first step in his construction was to determine the sign of (1:5) g~ ( ; ") ln g~( ; ") dH2 Z@D (") jr j · jr · j where D (") is a certain domaine with smooth boundary and g~( ; ") is Green's · function for the Laplacian in D (") with pole at . In analogy with Wolff and e for p fixed, 1 < p < , put D(") = x R2 : x 1> " (x ) where C1(R) e 2 1 0 with 0 in R 1( 1; 1). Let g(f; "2) be a positive weakg solution 2to the p- Laplace ≡equation inn D−(") with g( ; "·) = 0 continuously on @D(") and g(x; ") = x + !(x; ") where !( ; ) is infinitely· differentiable in D (") ( " ; " ) and 2 · · × − 0 0 ! k(1 + x )−2; x D("); jr j ≤ j j 2 for some constant k independent of " ( " ; " ). Existence of g( ; ") and !( ; ") 2 − 0 0 · · can be deduced by Picard iteration and Schauder type estimates for "0 > 0, 462 Bj¨orn Bennewitz and John L. Lewis sufficiently small. In fact proceeding operationally, assume g(x; ") = x2 + !(x; ") exists. Assuming that g = 0 in D(") and writing out the p-Laplace equation for g , one deduces thatrif w6 (x ; x ) = ! x ; pp 1 x , then w satisfies 1 2 1 − 2 w w w w ∆w(x) = F w ; x2 ; w ; x1x2 ; x2x1 ; x2x2 = H(w; x); x1 pp 1 x1x1 pp 1 pp 1 p 1 − − − − x D("), where 2 2 F (q ; q ; q ; q ; q ; q ) = (p 2) q q q 1 2 11 12 21 22 − − i j ij i;jX=1 2 2 2 2q + q2 q 2(p 2) q q : − 2 i ii − − i ij i;jX=1 Xi=1 i;jX=1 Also, w = pp 1 x2 on @D("). Let w0 be the bounded solution to Laplace's equation in−D(")−with w = pp 1 x on @D("). Proceeding by induction, let 0 − − 2 wn+1 for n = 1; 2; : : :, be the bounded solution to ∆w (x) = H(w ; x) in D(") with w = p 1 x continuously on @D("): n+1 n n+1 − − 2 p Using Schauder type estimates one can show for "0 > 0, sufficiently small that limn!1 wn = w exists with the desired smoothness properties. Thus g( ; ") ex- ists. · From (1.3) we get that the analogue of (1.5) for p fixed, 1 < p < , is 1 (1:6) I(") = g( ; ") p−1 ln g( ; ") dH1: Z@D(") jr · j jr · j Following Wolff one calculates that I(0) = I 0(0) = 0 and 2 00 p 2 d I (0) = − dx1: p 1 ZRdx − 1 Now if "0 is small enough, then I has three continuous derivatives on ( "0; "0) and so by Taylor's theorem, − (1:7) I(") > 0 when p > 2 and I(") < 0 for 1 < p < 2, when " is sufficiently small.
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