John domain and the weak boundary Harnack principle

Transcription

1 John domain and the weak boundary Harnack principle Hiroaki Aikawa Department of Mathematics, Hokkaido University Summer School in Conformal Geometry, Potential Theory, and Applications NUI Maynooth, Ireland June

2 Contents 1. Introduction 3 2. Weak boundary Harnack principle for a John domain 9 3. Domar s argument Union of convex sets 24 References 28 2

3 1. Introduction Exposition of [AHL06]: H. Aikawa, K. Hirata, and T. Lundh, Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan 58 (2006), no. 1, Let E {x = (x 1,..., x n ) : x n = 0} be closed. D = R n \ E is called a Denjoy domain. D Let P be the family of positive harmonic functions in D vanishing on D. Benedicks [Ben80] proved the following: E Contents 3

4 dim P = 1 or 2. i.e., 1 or 2 minimal Martin boundary points at. Criterion in terms of harmonic measure β E (x) = ω(x, K x, K x \ E). K x : cube center at x, side α x. 0 x 1 K x dim P = 1 dim P = 2 x 1 x 1 β E (x) x n 1 dx 1 dx n 1 =. β E (x) x n 1 dx 1 dx n 1 <. Monotonicity: If E E, dim P E = 2, then dim P E = 2. Contents 4

5 Location Topics Authors C 2 surface dim P 2 Ancona [Anc84] Hyperplane Harmonic Measure Benedicks [Ben80] Lipschitz surface dim P 2 WBHP Ancona [Anc84] Real line Lebesgue Measure Segawa [Seg88] Hyperplane Lebesgue Measure Gardiner [Gar89] C 1,1 surface Harmonic Measure Chevallier [Che89] C 1,α surface Harmonic Measure Ancona [Anc90] Lipschitz surface Non Monotonicity Ancona [Anc90] Real line Quasi-conformal Segawa [Seg90] Sectorial Harmonic Measure Cranston- Salisbury [CS93] Quasi-Sectorial Schrödinger Equation Lömker [Löm00] Contents 5

6 Weak boundary Harnack principle. Ancona [Anc84]. B(x, r), S (x, r) the open ball and the sphere with center at x and radius r. P ξ : kernel functions h at ξ, i.e., h > 0 harmonic on D, h = 0 q.e on D, bounded outside ξ. E S : Lipschitz surface. h 0, h 1, h 2 P ξ. Then ( h0 (y + R h 0 (x) A ) h 1 (y + R )h 1(x) + h 0(y R ) ) h 2 (y R )h 2(x) for x D B(ξ, R) \ B(ξ, R/2). y + R y R Contents 6

7 If h 0, h 1, h 2 P ξ, then i s.t. Hence dim P ξ 2. h i A j i h j ; Contents 7

8 Sectorial domain. Cranston-Salisbury [CS93]. y j R If h 0,..., h N P ξ. Then N h 0 (y j R h 0 (x) A ) h j (y j R )h j(x) j=1 for x D B(ξ, R) \ B(ξ, R/2); i s.t. h i A h j ; Hence dim P ξ N. Quasi-sectorial domain (higher dimension) Lömker [Löm00]. Contents 8 j i

9 2. Weak boundary Harnack principle for a John domain John domain. twisted cone condition: x D, γ : x x0 s.t. δ D (y) c J l(γ(x, y)) for all y γ, x 0 y twisted cone D x Contents 9

10 Denjoy domain Sectorial domain Quasi-Sectorial = John domain Theorem 1 Let D be a John domain with John constant c J. Let ξ D. Then (i) dim P ξ N(c J ) <. (ii) If c J > 3/2, then dim P ξ 2. Remark 1 c J > 3/2 is sharp for all n 2. Contents 10

11 x 0 ξ Quasihyperbolic metric: k D (x, y) = inf γ γ ds(z) δ D (z). where inf is taken over all curves γ connecting x to y in D. k D (x, y) length of Harnack chain. Contents 11

12 If h > 0 is harmonic on D, then exp( Ak D (x, y)) h(x) h(y) exp(ak D(x, y)) Local reference points: y 1 R,..., y N R S (ξ, R) D s.t. δ D(y i R ) R and min {k D R (x, y i R )} A log R i=1,...,n for x B(ξ, ηr) D, where D R = D B(ξ, AR). δ D (x) + A ξ y j R If h P ξ, then 0-extension to D c is subharmonic in R n \ {ξ}. Contents 12

13 Lemma 1 (Domar [Dom57]) Let u 0 be subharmonic in Ω s.t. I = (log + u) n 1+ε dx < Ω for ε > 0. Then u(x) exp(ai 1/ε dist(x, Ω) n/ε ). Lemma 2 τ > 0 s.t. D B(ξ,R) ( R δ D (x) ) τ dx AR n. Contents 13

14 Lemma 3 Let h P ξ for ξ D. Then h(x) A x ξ λ. By the tract argument [FH76], dim P ξ N(c J ) <. Proof. By local reference points ( ) λ R N h(x) A δ D (x) i=1 h(y i R ). Apply Domar s argument (Lemma 1) to Ω = B(ξ, AR) \ B(ξ, A 1 R) with the help of Lemma 2. Then (1) h(x) A N h(y i R ) on S (ξ, R), i=1 Contents 14

15 and hence on D \ B(ξ, R) by the maximum principle. Since δ D (y i R ) R, we have h(y i R ) AR λ. Hence i.e. h(x) A x ξ λ on D. h(x) AR λ on D \ B(ξ, R), Contents 15

16 By the box argument introduced by Bass-Burdzy [BB91] (see [Aik01, Lemma 2]) we have ω(x, D S (ξ, AR), D B(ξ, AR)) AR 2 n N G R (x, y i R ) i=1 for x D B(ξ, R), where G R is the Green function for D B(ξ, A R). Combine with (1). Then h(x) AR 2 n N N G R (x, y i R ) i=1 j=1 h(y j R ). Apply this inequality to h(x) = G R (x, y). Then G R (x, y) AR 2 n N N G R (x, y i R ) G R (y j R, y). i=1 j=1 Contents 16

17 Ancona s ingenious tricks [Anc84] and [Anc07] erase cross terms: N G R (x, y) AR 2 n G R (x, y i R )G R(y i R, y). Weak boundary Harnack principle Let h 0, h 1,..., h N P ξ. Then N h 0 (y i R h 0 (x) A ) h i (y i R ) h i(x) for x D. In particular, dim P ξ N. i=1 i=1 Contents 17

18 3. Domar s argument Lemma (Domar [Dom57]) Let u 0 be subharmonic in Ω s.t. I = (log + u) n 1+ε dx < for Ω ε > 0. Then u(x) exp(ai 1/ε dist(x, Ω) n/ε ). Lemma 4 Let L n = (e 2 / B(0, 1) ) 1/n. Let u 0 be subharmonic in B(x, R). If u(x) t > 0 and (2) R L n {y B(x, R) : e 1 t < u(y) et} 1/n, then x B(x, R) s.t. u(x ) > et. Contents 18

19 Proof. Observe that (2) is equivalent to {y B(x, R) : e 1 t < u(y) et} B(x, R) 1 e 2. If u et on B(x, R), then the mean value property yields 1 t u(x) u(y)dy B(x, R) B(x,R) ( ) 1 = udy + udy B(x, R) B(x,R) {u e 1 t} B(x,R) {u>e 1 t} e 1 t + 1 et < t. e2 This is a contradiction. Contents 19

20 Proof of Domar s Lemma. It is sufficient to show that (3) δ Ω (x) AI 1/n (log u(x)) ε/n, whenever u(x) > e 2. Fix x 1 Ω with u(x 1 ) > e 2 and let us prove (3) with x = x 1. Let R j = L n {y Ω : e j 2 u(x 1 ) < u(y) e j u(x 1 )} 1/n for j 1. Choose {x j } as follows: If δ Ω (x 1 ) < R 1, then we stop. If δ Ω (x 1 ) R 1, then B(x 1, R 1 ) Ω, so that there exists x 2 B(x 1, R 1 ) such that u(x 2 ) > eu(x 1 ) by Lemma 4. Next we consider δ Ω (x 2 ). If δ Ω (x 2 ) < R 2, then we stop. If δ Ω (x 2 ) R 2, then B(x 2, R 2 ) Ω, so that there exists x 3 B(x 2, R 2 ) such that u(x 3 ) > e 2 u(x 1 ) by Lemma 4. Repeat this procedure to obtain a finite or infinite sequence {x j }. Contents 20

21 We claim (4) δ Ω (x 1 ) 2 R j. Suppose first {x j } is finite. If δ Ω (x 1 ) < R 1, then (4) trivially holds. If δ Ω (x 1 ) R 1, then we have an integer J 2 such that δ Ω (x 1 ) R 1,..., δ Ω (x J 1 ) R J 1, δ Ω (x J ) < R J, x 2 B(x 1, R 1 ), x 3 B(x 2, R 2 ),..., x J B(x J 1, R J 1 ). Hence we have (4) as δ Ω (x 1 ) x 1 x x J 1 x J + δ Ω (x J ) < R R J 1 + R J. Suppose next {x j } is infinite. Since u(x j ) > e j u(x 1 ), the local boundedness of a subharmonic function shows that x j Ω. Hence, J 2 s.t. δω (x J ) 1 2 δ Ω(x 1 ). j=1 Contents 21

22 Then δ Ω (x 1 ) x 1 x x J 1 x J + δ Ω (x J ) R R J δ Ω(x 1 ), so that (4) follows. In view of (4) we observe that (3) follows from (5) j=1 R j AI 1/n (log u(x 1 )) ε/n. To show (5), let j 1 be the integer such that e j 1 j 1 2 and < u(x 1 ) e j 1+1. Then R j L n {y Ω : e j 1+ j 2 < u(y) e j 1+ j+1 } 1/n. Contents 22

23 Since the family of intervals {(e j 1+ j 2, e j 1+ j+1 ]} j overlaps at most 3 times, it follows from Hölder s inequality that R j 3L n j=1 3L n A j ε/n 1 j= j 1 {y Ω : e j 1 < u(y) e j } 1/n 1 (n 1)/n j (n 1+ε)/(n 1) j= j 1 ( (log + u) n 1+ε dy Ω A(log u(x 1 )) ε/n I 1/n. ) 1/n Thus (5) follows. The lemma is proved. j n 1+ε {y Ω : e j 1 < u(y) e j } j= j 1 1/n Contents 23

24 4. Union of convex sets John const c J is close to 1 = D is better. Yet two minimal Marin boundary points. Condition for 1 minimal Marin boundary point? Ancona [Anc79, Théorème]: D is admissible: (A1) D = λ B(x λ, ρ 0 ). (A2) Let ξ D. If D B 1, B 2 with radius ρ 0 tangential at ξ, then D Γ θ (ξ, y) B(ξ, r), a truncated circular cone with aperture θ > 0, radius r > 0 and axis on the tangent hyperplane. B 1 ξ B Γ θ (ξ,y) B(ξ,r) Contents 24

25 Theorem A (Ancona) If D is a bounded admissible domain, then D = D. Generalize both (A1) and (A2). (I) D = λ C λ ; C λ are open convex sets s.t. B(z λ, ρ 0 ) C λ B(z λ, A 1 ρ 0 ) C λ ρ z λ A 1 ρ 0 Contents 25

26 (II) For ξ D θ 1 sin 1 (1/A 1 ), ρ 1 ρ 0 cos θ 1 s.t. C (ξ) = Γ θ1 (ξ, y) B(ξ, 2ρ 1 ) is connected. y D, Γ θ1 (ξ,y) B(ξ,2ρ 1 ) D Ω C (ξ) θ ξ Theorem 2 Let D satisfy (I) and (II). Then D = D. Contents 26

27 Remark 2 Denjoy domain = D = λ B(x λ, ρ 0 ). Lipschitz Denjoy domains sectorial domain = D = λ C λ with (I). Remark 3 The bounds θ 1 sin 1 (1/A 1 ) and ρ 1 ρ 0 cos θ 1 are sharp. Contents 27

28 References [AHL06] [Aik01] [Anc79] [Anc84] [Anc90] H. Aikawa, K. Hirata, and T. Lundh, Martin boundary points of a John domain and unions of convex sets, J. Math. Soc. Japan 58 (2006), no. 1, [3] H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), no. 1, [16] A. Ancona, Une propriété de la compactification de Martin d un domaine euclidien, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, [24], Régularité d accès des bouts et frontière de Martin d un domaine euclidien, J. Math. Pures Appl. (9) 63 (1984), no. 2, [5, 6, 16], Sur la frontière de Martin des domaines de Denjoy, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), no. 2, [5] [Anc07], Sur la théorie du potentiel dans les domaines de John, Publ. Mat. 51 (2007), no. 2, [16] [BB91] R. F. Bass and K. Burdzy, A boundary Harnack principle in twisted Hölder domains, Ann. of Math. (2) 134 (1991), no. 2, [16] Contents 28

29 [Ben80] [Che89] [CS93] [Dom57] [FH76] [Gar89] M. Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in R n, Ark. Mat. 18 (1980), no. 1, [3, 5] N. Chevallier, Frontière de Martin d un domaine de R n dont le bord est inclus dans une hypersurface lipschitzienne, Ark. Mat. 27 (1989), no. 1, [5] M. C. Cranston and T. S. Salisbury, Martin boundaries of sectorial domains, Ark. Mat. 31 (1993), no. 1, [5, 8] Y. Domar, On the existence of a largest subharmonic minorant of a given function, Ark. Mat. 3 (1957), [12, 18] S. Friedland and W. K. Hayman, Eigenvalue inequalities for the Dirichlet problem on spheres and the growth of subharmonic functions, Comment. Math. Helv. 51 (1976), no. 2, [13] S. J. Gardiner, Minimal harmonic functions on Denjoy domains, Proc. Amer. Math. Soc. 107 (1989), no. 4, [5] [Löm00] A. Lömker, Martin boundaries of quasi-sectorial domains, Potential Anal. 13 (2000), no. 1, [5, 8] Contents 29

30 [Seg88] S. Segawa, Martin boundaries of Denjoy domains, Proc. Amer. Math. Soc. 103 (1988), no. 1, [5] [Seg90], Martin boundaries of Denjoy domains and quasiconformal mappings, J. Math. Kyoto Univ. 30 (1990), no. 2, [5] Contents 30