Korn inequality on irregular domains

J. Math. Anal. Appl. 423 (2015) 41–59

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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Korn inequality on irregular domains Renjin Jiang a,∗ , Aapo Kauranen b a

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Beijing 100875, People’s Republic of China b Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Finland

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 1 July 2013 Available online 5 October 2014 Submitted by T. Witelski

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality. © 2014 Elsevier Inc. All rights reserved.

Keywords: Korn inequality Divergence equation Poincaré inequality s-John domain Quasihyperbolic metric

1. Introduction Let p > 1 and Ω be a bounded domain of Rn , n ≥ 2. For each vector v = (v1 , · · · , vn ) ∈ W 1,p (Ω)n , let Dv denote its gradient matrix, and (v) denote the symmetric part of Dv, i.e., (v) = (i,j (v))1≤i,j≤n with   1 ∂vi ∂vj . i,j (v) = + 2 ∂xj ∂xi Korn’s (second) inequality states that, if Ω is sufficient regular (e.g., Lipschitz), then there exists C > 0 such that ˆ  ˆ ˆ   (v)p dx + |v|p dx . |Dv|p dx ≤ C (Kp ) Ω

Ω

Ω

The Korn inequality (Kp ) is a fundamental tool in the theory of linear elasticity equations; see [3,1,7,9,11, 13,22,30] and the references therein. Notice that Korn’s inequality (Kp ) fails for p = 1 even on a cube; see the example from [6]. * Corresponding author. E-mail addresses: [email protected] (R. Jiang), aapo.p.kauranen@jyu.fi (A. Kauranen). http://dx.doi.org/10.1016/j.jmaa.2014.09.076 0022-247X/© 2014 Elsevier Inc. All rights reserved.

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On R2 and p = 2, several different inequalities (including the Friedrichs inequality) are actually equivalent to Korn’s inequality (Kp ) on simply connected Lipschitz domains; see [14,30] for example. Friedrichs [11] proved the Korn inequality (Kp ) for p = 2 on domains with a finite number of corners or edges on ∂Ω, Nitsche [27] proved the Korn inequality (Kp ) for p = 2 on Lipschitz domains, while Mjasnikov and Mosolov [26] and Ting [31] proved (Kp ) for all p ∈ (1, ∞); Kondratiev and Oleinik [22] studied the Korn inequality (K2 ) on star-shaped domains. Recently, Acosta, Durán and Muschietti [3] proved the Korn inequality (Kp ) holds for all p ∈ (1, ∞) on John domains. Weighted Korn inequality on irregular domains (in particular, Hölder domains) has received considerable interest recently; see [3,1,2,7,22] and references therein. Motivated by this, in this paper, we study weighted Korn inequality on some irregular domains including s-John domains (s ≥ 1) and domains satisfying quasihyperbolic boundary conditions. It is well known that one can deduce the (weighted) Korn inequality from the divergence equation, see [7,8] for instance. In particular, it was shown in [8, Proposition 3.2] that the validity of Poincaré inequality ˆ

  u(x) − uΩ p dx ≤ C

Ω

ˆ

  ∇u(x)p dist(x, ∂Ω)b dx,

Ω

implies certain regularity of solutions to the divergence equation div u = f . Then by using duality, one gets the (weighted) Korn inequality. The proofs and results in [8] can be easily generalized to our setting, which enables us to deduce a (weighted) Korn inequality; see Section 2 below. For more on the recent progress on the divergence equation, see [3,4,8,18]. In Section 3 we discuss s-John domains and domains satisfying quasihyperbolic boundary conditions. By using Poincaré inequalities on these domains, we deduce the (weighted) Korn inequalities on them. Moreover, we will show the obtained (weighted) Korn inequalities are essentially sharp by presenting some counter-examples in Section 4. The weighted Poincaré inequality on s-John domains is well known (see [12,20]), however, we could not find the results we needed for domains with quasihyperbolic boundary condition; for some related results see [10]. To this end, we will in Section 3 establish the weighted Poincaré inequality on such domains, which may have independent interest. Another interesting question is what is the geometric counterpart of the Korn inequality. In general the Korn inequality (Kp ) does not imply any Poincaré inequality. Indeed, if Ω1 , Ω2 ⊂ Rn , Ω1 ∩ Ω2 = ∅, are two domains that support the Korn inequality (Kp ), then Ω := Ω1 ∪ Ω2 admits the Korn inequality (Kp ) as well. However, Poincaré inequality does not have this property. In Section 2.2, we will show that, if the following Korn inequality ˆ

ˆ |Dv| dx ≤ C p

Ω

Ω

  (v)p dx +



ˆ |v| dx p

p) (K

Q

holds for some cube Q ⊂⊂ Ω, then there is a Poincaré inequality on Ω. The paper is organized as follows. In Section 2, we will show that, abstractly, weighted Poincaré inequality  p ) also implies a Poincaré inequality. In implies a weighted Korn inequality; conversely, Korn inequality (K Section 3, we establish the Korn inequality on s-John domains and domains satisfying quasihyperbolic boundary conditions, and present examples for the sharpness of the Korn inequality in Section 4. Throughout the paper, we denote by C positive constants which are independent of the main parameters, but which may vary from line to line. Corresponding to a function space X, we denote its n-vector valued spaces by X n . We will usually omit the superscript n or n × n for simplicity. Let Q(x, r) ⊂ Rn denote the cube with the center x and side-length r, and for a constant C > 0, let CQ denote the cube Q(x, Cr).

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

43

2. Korn inequality and Poincaré inequality In this section, we show that, abstractly, the weighted Poincaré inequality implies Korn inequality; and conversely, certain Korn inequality implies a weaker Poincaré inequality; see the discussions after Theorem 2.2. Throughout the paper, let ρ(x) be the distance from x to the boundary ∂Ω, i.e., ρ(x) := dist(x, ∂Ω). Let a, b ∈ R and p ∈ [1, ∞), the weighted Lebesgue space Lp (Ω, ρa ) is defined as set of all measurable functions f in Ω such that ˆ f Lp (Ω,ρa ) :=

  f (x)p ρ(x)a dx

1/p < ∞.

Ω

´ We denote by Lp0 (Ω, ρa ) the set of functions f ∈ Lp (Ω, ρa ) with Ω f (x)ρ(x)a dx = 0. Let D (Ω) denote the set of smooth functions compactly supported in Ω. Let u ∈ L1loc (Ω) and 1 ≤ i ≤ n, fi ∈ L1loc (Ω) is a weak partial derivative of u, if ˆ u(x)

∂ φ(x) dx = − ∂xi

Ω

ˆ fi (x)φ(x) dx Ω

∂ holds for each φ ∈ D (Ω). In what follows, we will denote such fi by ∂x u. The weighted Sobolev space i 1,p a b p a p W (Ω, ρ , ρ ) is then defined as the set of all u ∈ L (Ω, ρ ) with ∇u ∈ L (Ω, ρb )n . For u ∈ W 1,p (Ω, ρa , ρb ), define its norm by

uW 1,p (Ω,ρa ,ρb ) := uLp (Ω,ρa ) + ∇uLp (Ω,ρb ) . The Sobolev space W01,p (Ω, ρa , ρb ) is then defined as the completion of D (Ω) with respect to the norm of W 1,p (Ω, ρa , ρb ). We denote W 1,p (Ω, ρa , ρa ) (resp. W01,p (Ω, ρa , ρa )) by W 1,p (Ω, ρa ) (resp. W01,p (Ω, ρa )), and denote W 1,p (Ω, ρa ) (resp. W01,p (Ω, ρa )) by W 1,p (Ω) (resp. W01,p (Ω)) if a = 0. Notice that as ρa and ρb are continuous positive functions in Ω, the subspace C ∞ (Ω) ∩ W 1,p (Ω, ρa , ρb ) is dense in W 1,p (Ω, ρa , ρb ); see [12, Theorem 3]. Let p ≥ 1 and a ≥ 0. We say that the (Pp,a,b )-Poincaré inequality holds, if there exists C > 0 such that for every u ∈ W 1,p (Ω, ρa , ρb ), it holds ˆ

  u(x) − uΩ,a p ρ(x)a dx ≤ C

Ω

where we denote uΩ,a :=

´ Ω

1 ρa dx

ˆ

  ∇u(x)p ρ(x)b dx,

(Pp,a,b )

Ω

´ Ω

uρa dx and uΩ := uΩ,a for a = 0.

2.1. Korn inequality from weighted Poincaré inequality In this subsection, we will show that the weighted Poincaré inequality implies the Korn inequality, and in the following Section 4 we will provide examples which show sharpness of our results. We prove Korn inequality by first establishing suitable solutions to divergence equations with essentially same arguments as in [8, Theorem 4.1]. From this weighted Korn inequality follows with arguments similar to those in [7].

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

44

Theorem 2.1. Let Ω be a bounded domain of Rn , n ≥ 2. Let 1 < p < ∞, 0 ≤ a < ∞ and b ∈ R. Suppose the (Pp,a,b )-Poincaré inequality holds on Ω. Then for an arbitrarily fixed cube Q ⊂⊂ Ω, there exists C = C(p, a, b, Ω, Q) such that for every v ∈ W 1,p (Ω, ρa )n , the following inequality holds ˆ

  Dv(x)p ρ(x)a dx ≤ C

ˆ

Ω

  (v)(x)p ρ(x)b−p dx +

Ω

ˆ

   Dv(x)p ρ(x)a dx .

 p,a,b−p ) (K

Q

 p,a,b−p ) implies (Kp ); see Kondratiev and Oleinik [22]. Indeed, as Remark 2.1. If a = 0 and b = p, then (K Q ⊂⊂ Ω, it holds dist(Q, ∂Ω) ≤ ρ(x) ≤ diam(Ω) for each x ∈ Q. Since the Korn inequality (Kp ) holds on cubes, we always have ˆ

ˆ

ˆ |Dv| ρ dx ≤ C(a, Ω, Q)

|Dv| dx ≤ C(a, Ω, Q)

p a

Q

p

Q

ˆ

≤ C(p, a, b, Ω, Q)

  (v)p ρb−p dx +

Q

ˆ

  (v)p dx +

Q

 |v|p ρa dx .



ˆ |v| dx p

Q

Q

 p,a,b−p ) above implies that Thus (K 

DvLp (Ω,ρa ) ≤ C(p, a, b, Ω, Q) (v) Lp (Ω,ρb−p ) + vLp (Q,ρa ) and hence 

DvLp (Ω,ρa ) ≤ C(p, a, b, Ω, Q) (v) Lp (Ω,ρb−p ) + vLp (Ω,ρa ) ,

(Kp,a,b−p )

which is the usual Korn inequality (Kp ) if a = 0 and b = p. We employ the divergence equation to prove the previous theorem. Let p, q ∈ (1, ∞) satisfying 1/q + 1/p = 1, and Ω be a bounded domain in Rn . A vector function u is called a weak solution of the divergence equation div u = f ρa for some f ∈ Lp0 (Ω, ρa ), if for every φ ∈ W 1,q (Ω, ρa , ρb ) it holds that ˆ

ˆ u(x) · ∇φ(x) dx = −

Ω

f (x)φ(x)ρ(x)a dx.

(divp,a,b )

Ω

Further, if u is in some Sobolev space with zero boundary values such that, for every φ ∈ W 1,q (Ω, ρa , ρb ), it holds that ˆ ˆ div u(x)φ(x) dx = f (x)φ(x)ρ(x)a dx, Ω

Ω

then we call u a solution to div u = f ρa . Since C ∞ (Ω) ∩W 1,q (Ω, ρa , ρb ) is dense in W 1,q (Ω, ρa , ρb ), it is not a restriction to require the test function φ to be in C ∞ (Ω) ∩ W 1,q (Ω, ρa , ρb ). Proposition 2.1. Let Ω be a bounded domain in Rn , p, q ∈ (1, ∞) with 1/p + 1/q = 1, 0 ≤ a < ∞ and b ∈ R. Suppose that Ω supports a (Pp,a,b )-Poincaré inequality, then for each f ∈ Lq0 (Ω, ρa ), there exists a solution u ∈ W01,q (Ω, ρ−qb/p , ρq−qb/p )n of the equation

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

45

div u = f ρa satisfying uLq (Ω,ρ−qb/p ) + DuLq (Ω,ρq−qb/p ) ≤ Cf Lq (Ω,ρa ) ,

(2.1)

where C = C(n, p, a, b) > 0. To prove this, let us introduce some auxiliary results. Lemma 2.1. Let Ω be a bounded domain of Rn , n ≥ 2. Let 1 ≤ p < ∞, 0 ≤ a < ∞ and b ∈ R. Suppose the (Pp,a,b )-Poincaré inequality holds on Ω. Then a + p − b ≥ 0. Proof. Let {Qj }j be a Whitney decomposition of Ω. We choose {χj }j ⊂ D (Ω) such that χj ∈ D (2Qj ), 0 ≤ χj ≤ 1, χj = 1 on 12 Qj , |∇χj | ≤ C(Qj )−1 , where C is a constant independent of j. Notice that (Qj ) → 0 as j → ∞. Moreover, since a ≥ 0, we have ˆ

n  ρa dx ≤ C diam(Ω)a (Qj ) → 0

2Qj

as j → ∞. Hence, there exists N0 ∈ N, such that for each j > N0 , it holds ˆ ρa dx <

1 2

2Qj

ˆ ρa dx, Ω

which further implies that for each j > N0 and each x ∈ Qj ,   χj (x) − (χj )Ω,a  = χj (x) − ´

1 a dx ρ Ω

´

ˆ

ρa dx 1 2Qj ´ ≥ . χj ρ dx ≥ 1 − a dx 2 ρ Ω a

2Qj

Hence by the (Pp,a,b )-Poincaré inequality, we see that for j > N0 , ˆ (Qj )a+n ≤ C

ˆ

  χj (x) − (χj )Ω,a p ρ(x)a dx

ρ(x)a dx ≤ C Qj

ˆ

≤C

Ω

  ∇χj (x)p ρ(x)b dx ≤ C(Qj )n+b−p ,

Ω

where the constants C depends only on the dimension and the Poincaré inequality, and in particular does not depend on j. By the fact (Qj ) → 0 as j → ∞, we see that b − p ≤ a, as desired. 2 The following technique of decomposition of functions is essentially established in [8, Proposition 4.2]. Let {Qj }j be a Whitney decomposition of Ω. Lemma 2.2. Let Ω be a bounded domain in Rn , p ∈ (1, ∞) with 1/p + 1/q = 1, 0 ≤ a < ∞ and b ∈ R. Suppose that Ω supports a (Pp,a,b )-Poincaré inequality, then for each f ∈ Lq0 (Ω, ρa ), f can be decomposed as f (x)ρ(x)a =

 j

fj (x),

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

46

where {fj } satisfies: (i) supp fj ⊂ 2Qj ; ´ (ii) 2Qj fj (x) dx = 0; ´ ´ q q−qb/p dx ≤ C Ω |f (x)|q ρ(x)a dx, for some C = C(Ω, p, a, b). (iii) j 2Qj |fj (x)| ρ(x) Proof. The proof is very similar to the proof of [8, Proposition 4.2], so we sketch a proof for completeness. For f ∈ Lq0 (Ω, ρa ), by using the (Pp,a,b )-Poincaré inequality, similarly as in [8, Proposition 3.2], we conclude ˜ = f ρa such that ˜ of the equation div u that there exists a weak solution u ˜ uLq (Ω,ρ−qb/p ) ≤ Cf Lq (Ω,ρa ) .

(2.2)

Let {χj }j be a partition of unity associated with the {Qj }j , where χj ∈ D (2Qj ), χj = 1 on |∇χj | ≤ C(Qj )−1 and j χj = 1 on Ω. For each j, we set for all x ∈ Ω ˜ · ∇χj (x). fj (x) := div(˜ uχj )(x) = f (x)ρ(x)a χj (x) + u

1 2 Qj ,

(2.3)

It follows that j fj (x) = f (x)ρ(x)a and (i), (ii) hold. Finally, notice that Lemma 2.1 implies that aq + q − qb/p =

p ap − a p ap + p − b ≥ ≥ a. p−1 p p−1 p

Thus, by the properties of the Whitney decomposition and the fact ˜ uLq (Ω,ρ−qb/p ) ≤ Cf Lq (Ω,ρa ) , we obtain ˆ   fj (x)q ρ(x)q−qb/p dx j 2Q j

≤C

 ˆ     

q χj (x)f (x)ρ(x)a + u ˜ (x)(Qj )−1 ρ(x)q−qb/p dx j 2Q j

ˆ

≤ C(diam Ω)

aq+q−qb/p−a

ˆ ≤ C(Ω, p, a, b)

  f (x)q ρ(x)a dx + C

Ω

ˆ

q  u ˜ (x) ρ(x)−qb/p dx

Ω

  f (x)q ρ(x)a dx,

Ω

which proves (iii), and hence the lemma. 2 Proof of Proposition 2.1. The case a = 0 is obtained in [8, Theorem 4.1]; the proof of the case a > 0 is essentially the same as the case a = 0 in [8], we outline the proof here. For f ∈ Lq0 (Ω, ρa ), by Lemma 2.2, we see that f can be decomposed as f (x)ρ(x)a = j fj (x), where {fj }j satisfy (i)–(iii) of Lemma 2.2. For each j, by [4, Theorem 2], there exists uj ∈ W01,q (2Qj )n such that div uj = fj and Duj Lq (2Qj ) ≤ C(q)fj Lq (2Qj ) , which together with the Sobolev inequality implies that uj Lq (2Qj ) ≤ C(q)(Qj )Duj Lq (2Qj ) ≤ C(q)(Qj )fj Lq (2Qj ) .

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

Denote u :=

47



j uj , which is well-defined since the dilations of Whitney cubes have bounded overlaps. k Let us show that u ∈ W01,q (Ω, ρ−qb/p , ρq−qb/p )n . To this end, for each k ∈ N, set vk := i=1 uj . Since, by the definition of Whitney decomposition, ρ(x) ∼ (Qj ) for each x ∈ 2Qj and each j, we further deduce that

vk qLq (Ω,ρ−qb/p ) + Dvk qLq (Ω,ρq−qb/p ) ≤C

ˆ k    

  uj (x)q ρ(x)−qb/p dx + Duj (x)q ρ(x)q−qb/p dx j=1 2Q

≤C

k 

j

(Qj )−qb/p uj qLq (2Qj ) + (Qj )q−qb/p Duj qLq (2Qj )



j=1

≤C

k 

ˆ (Qj )

j=1

≤C

  fj (x)q dx

q−qb/p 2Qj

ˆ ˆ k      fj (x)q ρ(x)q−qb/p dx ≤ C f (x)q ρ(x)a dx. j=1 2Q

(2.4)

Ω

j

Obviously, the same estimate holds also for u. Hence, u, vk ∈ W 1,q (Ω, ρ−qb/p , ρq−qb/p )n for each k ∈ N. k On the other hand, the fact supp vk ⊂ j=1 2Qj ⊂⊂ Ω implies vk ∈ W01,q (Ω, ρ−qb/p , ρq−qb/p )n . Applying the same estimate as in (2.4) to u − vk yields u −

vk qLq (Ω,ρ−qb/p )

q + D(v − vk ) q

L (Ω,ρq−qb/p )

∞ 

≤C

ˆ q−qb/p

(Qj )

j=k+1

  fj (x)q dx,

2Qj

which together with Lemma 2.2 implies that vk → u in W 1,q (Ω, ρ−qb/p , ρq−qb/p )n as k → ∞. Hence, u ∈ W01,q (Ω, ρ−qb/p , ρq−qb/p )n . Finally, let us show that u is a solution of div u = f ρa . We conclude from the facts u ∈ 1,q W0 (Ω, ρ−qb/p , ρq−qb/p )n , u = j uj in Lq (Ω, ρ−qb/p ) and ∇φ ∈ Lp (Ω, ρb ) that ˆ

ˆ div u(x)φ(x) dx = − Ω

u(x) · ∇φ(x) dx = −

ˆ

Ω

=

ˆ

uj (x) · ∇φ(x) dx

j Ω

fj (x) · φ(x) dx.

j Ω

Above, in the second equality we can change the order of summation and integral since, by the facts supp uj ⊂ 2Qj , (2.4) and finite overlap of expanded Whitney cubes, we have  ˆ   uj (x) · ∇φ(x) dx ≤ uj Lq (2Q

−qb/p ) j ,ρ

j Ω

∇φLp (2Qj ,ρb )

j

≤



uj qLq (2Qj ,ρ−qb/p )

1/q  

j

ˆ ≤C Ω

1/p ∇φpLp (2Qj ,ρb )

j

  f (x)q ρ(x)a dx

1/q ∇φLp (Ω,ρb ) < ∞.

(2.5)

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

48

˜ = f ρa given in the proof of Lemma 2.2, and {χj }j be the partition ˜ be the weak solution of div u Let u of unity there. From the definition of fj in (2.3), it follows that ˆ div u(x)φ(x) dx =

ˆ

 ˜ (x) · ∇χj (x) φ(x) dx f (x)ρ(x)a χj (x) + u

j Ω

Ω

=

ˆ

 ˜ (x) · ∇(χj φ)(x) − χj (x)˜ f (x)ρ(x)a χj (x)φ(x) + u u(x) · ∇φ(x) dx

j Ω

=

ˆ

 f (x)ρ(x)a χj (x)φ(x) − f (x)ρ(x)a χj (x)φ(x) − χj (x)˜ u(x) · ∇φ(x) dx

j Ω

=−

ˆ

χj (x)˜ u(x) · ∇φ(x) dx

j Ω

ˆ =− ˆ

˜ (x) · ∇φ(x) dx u Ω

f (x)φ(x)ρ(x)a dx.

= Ω

In the second last equality above, we can change the order of summation and integral since, similarly to (2.5), using (2.2) and properties of a partition of unity, we can deduce that ˆ 1/q ˆ     χj (x)˜ f (x)q ρ(x)a dx u(x) · ∇φ(x) dx ≤ C ∇φLp (Ω,ρb ) < ∞. j Ω

Ω

The proof is therefore completed. 2 Finally, let us prove Theorem 2.1. ∂vi Proof of Theorem 2.1. Recall that Dv = ( ∂x )1≤i,j≤n , 1 ≤ i, j ≤ n, and (v) = (i,j (v))1≤i,j≤n with j

i,j =

  1 ∂vi ∂vj + 2 ∂xj ∂xi

and the identity ∂i,k (v) ∂i,j (v) ∂j,k (v) ∂ 2 vi = + − . ∂xj ∂xk ∂xj ∂xk ∂xi

(2.6)

For each f ∈ Lq0 (Ω, ρa ), by using Proposition 2.1, we see that there exists a solution u ∈ W01,q (Ω, ρ−qb/p , ρq−qb/p )n of the equation div u = f ρa . For each k ∈ {1, 2, · · · , n}, let uk be the k-th component of u. Using the identity (2.6) yields ˆ      f (x)ρ(x)a ∂vj (x) − ∂vj  ∂xi ∂xi Ω

 Ω,a

 ˆ      ∂vj ∂vj dx =  div u(x) (x) − ∂xi ∂xi Ω

ˆ    ∂vj =  u(x) · ∇ (x) dx ∂xi Ω

 Ω,a

  dx

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

≤

49

 n ˆ 2     uk (x) ∂ vj (x) dx   ∂xk ∂xi

k=1 Ω

   n ˆ    ∂j,k (v) ∂j,i (v) ∂k,i (v) k  = (x) dx + −  u (x) ∂xi ∂xk ∂xj k=1 Ω

 ˆ  k n    ∂u (x)   ≤C  ∂xi j,k (v)(x) dx i,j,k=1 Ω

≤ CDuLq (Ω,ρq−qb/p ) (v) Lp (Ω,ρb−p ) , which together with Proposition 2.1 implies that ˆ      f (x)ρ(x)a ∂vj (x) − ∂vj  ∂xi ∂xi Ω

 Ω,a

  dx ≤ Cf Lq (Ω,ρa ) (v) Lp (Ω,ρb−p ) .

This implies that, for each f˜ ∈ Lq (Ω, ρa ), ˆ      f˜(x)ρ(x)a ∂vj (x) − ∂vj  ∂xi ∂xi Ω

 Ω,a

 ˆ       ∂vj a ∂vj ˜ ˜   dx =  f (x) − fΩ,a ρ(x) (x) − ∂xi ∂xi Ω

 Ω,a

  dx

≤ Cf˜ − f˜Ω,a Lq (Ω,ρa ) (v) Lp (Ω,ρb−p ) ≤ Cf˜Lq (Ω,ρa ) (v) Lp (Ω,ρb−p ) .

The fact (Lq (Ω, ρa ))∗ = Lp (Ω, ρa ) further implies that   ∂vj ∂vj ≤ C (v) Lp (Ω,ρb−p ) . ∂xi − ∂xi p Ω,a L (Ω,ρa ) Now for an arbitrarily fixed cube Q ⊂⊂ Ω, we choose a ψ ∈ C0∞ (Q) such that supp ψ ⊂ Q, and |∇ψ| ≤ C/(Q)n+1 . Write ∂vj ∂vj = − ∂xi ∂xi



∂vj ∂xi



ˆ  + Ω,a

Q

∂vj ∂xi

 − Ω,a

 ˆ ∂vj ∂vj ψ dx + ψ dx. ∂xi ∂xi Q

Then by the Hölder inequality, we obtain   ˆ      ∂vj ∂vj  ψ dx ≤ C(a, p, Q, Ω) (v) Lp (Ω,ρb−p ) , −  ∂xi Ω,a ∂xi Q

and  ˆ ∂vj   ∂vj   .  ∂xi (x)ψ(x) dx ≤ C(a, p, Q, Ω) ∂xi p L (Q,ρa ) Q

Combining (2.7), (2.8) and the above estimates, we obtain that   ∂vj ∂vj (v) p , ≤ C(p, a, b, Ω, Q) + ∂xi p ∂xi p L (Ω,ρb−p ) L (Ω,ρa ) L (Q,ρa )

(2.7) ´ Q

ψ dx = 1

(2.8)

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50

which is 

DvLp (Ω,ρa ) ≤ C(p, a, b, Ω, Q) (v) Lp (Ω,ρb−p ) + DvLp (Q,ρa ) .

 p,a,b−p ) (K

The proof is completed. 2 2.2. Korn inequality implies Poincaré inequality From the previous subsection, we know that the weighted Poincaré inequality implies Korn inequality, and in this section we will prove a partial converse result. Theorem 2.2. Let Ω be a bounded domain of Rn , n ≥ 2. Let 1 < p < ∞ and Q ⊂ Ω be a closed cube. Suppose that for all v ∈ W 1,p (Ω)n it holds that 

DvLp (Ω) ≤ C (v) Lp (Ω) + DvLp (Q) ,

p) (K

then there exists C > 0 such that for all u ∈ W 1,p (Ω), it holds ˆ

  u(x) − uΩ p dx ≤ C

Ω

ˆ

  ∇u(x)p dx.

(Pp,0,0 )

Ω

Combining Theorems 2.1 and 2.2, we see that on a bounded domain Ω, it holds that  p ) =⇒ (Pp,0,0 ). (Pp,0,p ) =⇒ (K Notice that the Poincaré inequality (Pp,0,0 ) is weaker than (Pp,0,p ). We do not know whether the Korn  p ) implies the Poincaré inequality (Pp,0,p ). inequality (K We will need the following characterization of weighted Poincaré inequality from Hajłasz and Koskela [12, Theorem 1] (for non-weighted cases see Maz’ya [25]). A subset A ⊂ Ω is admissible if A is open and ∂A ∩ Ω is a smooth submanifold. Theorem 2.3. (See [12].) Let Ω be a bounded domain in Rn , n ≥ 2. Let 1 ≤ p ≤ q < ∞ and 0 ≤ a < ∞, b ∈ R. Then the following conditions are equivalent. (i) There exists a constant C > 0 such that, for every u ∈ C ∞ (Ω) it holds that ˆ

  u(x) − uΩ,a q ρ(x)a dx

p/q

ˆ ≤C

Ω

  ∇u(x)p ρ(x)b dx.

Ω

(ii) For an arbitrary cube Q ⊂⊂ Ω, there exists a constant C = C(Q) > 0 such that ˆ

p/q ρ(x)a dx

A

ˆ ≤ C inf u

  ∇u(x)p ρ(x)b dx

(2.9)

Ω

for every admissible set A ⊂ Ω with A ∩Q = ∅. Here the infimum is taken over the set of all u ∈ C ∞ (Ω) that satisfy u|A = 1 and u|Q = 0.

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We next prove Theorem 2.2. Proof of Theorem 2.2. We only need to verify that the second condition of Theorem 2.3 holds. Assume that  p ) holds. Fix a y = (y1 , y2 , · · · , yn ) ∈ Ω. (K Let A ⊂ Ω with A ∩ Q = ∅ be an admissible set, and u ∈ C ∞ (Ω) that satisfies u|A = 1 and u|Q = 0. For each x = (x1 , · · · , xn ) ∈ Ω, let v = (v1 , v2 , 0, · · · , 0) with 

v1 (x1 , · · · , xn ) = (x2 − y2 )u(x1 , · · · , xn ), v2 (x1 , · · · , xn ) = (y1 − x1 )u(x1 , · · · , xn ).

Then for each x = (x1 , · · · , xn ) ∈ A, ⎛

0 ⎜ −1 ⎜ Dv(x) = ⎜ ⎜ 0 ⎝ ··· 0

⎞ 1 0 ··· 0 0 0 ··· 0⎟ ⎟ 0 0 ··· 0⎟ ⎟, ⎠ 0 0

··· 0

and for x ∈ Q, Dv(x) = 0. These imply that ˆ DvpLp (Ω) ≥

dx A

and D(v)Lp (Q) = 0. On the other hand, for every x = (x1 , x2 , · · · , xn ) ∈ Ω, it holds ⎛

∂u (x2 − y2 ) ∂x 1 ∂u ⎜ −u + (y1 − x1 ) ∂x ⎜ 1 ⎜ Dv(x) = ⎜ 0 ⎝ ··· 0

∂u u + (x2 − y2 ) ∂x 2 ∂u (y1 − x1 ) ∂x 2 0

∂u (x2 − y2 ) ∂x 3 ∂u (y1 − x1 ) ∂x 3 0

0

0

∂u · · · (x2 − y2 ) ∂x n ∂u · · · (y1 − x1 ) ∂x n ··· 0

···

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

0

which implies that (v)

Lp (Ω)

≤ C |∇u|(· − y) Lp (Ω) ≤ C diam(Ω)∇uLp (Ω) .

 p ) implies that The Korn inequality (K ˆ



p dx ≤ DvpLp (Ω) ≤ C (v) Lp (Ω) + D(v) Lp (Q)

A

ˆ ≤C

  ∇u(x)p dx,

Ω

for every u ∈ C ∞ (Ω) that satisfies u|A = 1 and u|Q = 0. Then the Poincaré inequality (Pp,0,0 ) holds by using Theorem 2.3 with p = q and a = b = 0. 2 Remark 2.2. Similarly, if the following Korn inequality 

DvLp (Ω) ≤ C(Ω, Q) (v) Lp (Ω) + vLp (Q) , holds for some cube Q ⊂⊂ Ω, then the (Pp,0,0 )-Poincaré inequality also holds.

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52

Remark 2.3. Theorem 2.2 can be generalized to the weighted cases by similar proofs as: if the Korn inequality ˆ

  Dv(x)p ρ(x)a dx ≤ C

Ω

ˆ

  (v)(x)p ρ(x)b−p dx +

Ω

ˆ

  Dv(x)p ρ(x)a dx



 p,a,b−p ) (K

Q

holds for some Q ⊂⊂ Ω, then the weighted Poincaré inequality ˆ

  u(x) − uΩ,a p ρ(x)a dx ≤ C

Ω

ˆ

  ∇u(x)p ρ(x)b−p dx

Ω

holds. 3. Korn inequality on irregular domains In this section, we are going to study the Korn inequality on some irregular domains. If Ω is an α-Hölder domain for some α ∈ (0, 1], it is then proved in [1] that there is a constant C = C(n, p, Ω, α) > 0 such that for every v ∈ W 1,p (Ω, ρa )n , it holds ˆ

ˆ |Dv|p ρa dx ≤ C Ω

  (v)p ρb−p dx +

Ω

ˆ

 |v|p ρa dx ,

Ω

where 0 ≤ a = b − αp. See [1,2] for more on this aspect and the counterexample for sharpness. We next focus on two kinds of irregular domains: s-John domains and quasihyperbolic domains. 3.1. s-John domains Let us first recall the definition of s-John domain. Definition 3.1 (s-John domain). A bounded domain Ω ⊂ Rn with a distinguished point x0 ∈ Ω is called an s-John domain, 1 ≤ s < ∞, if there exists a constant C > 0 such that for all x ∈ Ω, there is a curve γ : [0, l] → Ω parameterized by arclength such that γ(0) = x, γ(l) = x0 , and d(γ(t), Rn \ Ω) ≥ Cts . If s = 1 then we say that Ω is a John domain for simplicity. John domains were introduced by Martio and Sarvas [24], F. John [19] had earlier considered a similar class of domains. In the following theorem, case (i) was established in [15] for the case a = 0, b = p and in [5] for general cases, and case (ii) was established in [12, Theorem 9]. Theorem 3.1. Suppose that Ω is an s-John domain, 1 ≤ s < ∞ and 0 ≤ a < ∞. Then there is a constant C = C(n, p, Ω, a, b) > 0 such that ˆ

1/p |u − uΩ,a | ρ dx p a

Ω

for each u ∈ C ∞ (Ω), if (i) s = 1 and a ≥ b − p; (ii) s > 1 and n + a > s(n + b − 1) − p + 1.

ˆ ≤C

1/p |∇u| ρ dx p b

Ω

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53

Notice that for the case s > 1, there is no result for the endpoint case, i.e., n + a = s(n + b − 1) − p + 1. Kilpeläinen and Malý [20,21] have established the critical Poincaré inequalities on s-John domains for p < q, but the case p = q is still open. We have the corresponding Korn inequality on s-John domains. Theorem 3.2. Suppose that Ω is an s-John domain with 1 ≤ s < ∞, and 0 ≤ a < ∞. Then there is a constant C = C(n, p, Ω, a, b) > 0 such that for every v ∈ W 1,p (Ω, ρa )n , it holds ˆ  ˆ ˆ   (v)p ρb−p dx + |v|p ρa dx , |Dv|p ρa dx ≤ C (Kp,a,b−p ) Ω

Ω

Ω

where a ≥ b − p if s = 1, and n + a > s(n + b − 1) − p + 1 if s > 1. Moreover, for 0 ≤ a < ∞ and n + a < s(n + b − 1) − p + 1, there exists a domain Ω which does not support the Korn inequality (Kp,a,b−p ). Proof. By using Theorem 2.1 and the Poincaré inequality in Theorem 3.1, we see that the Korn inequality (Kp,a,b−p ) holds for a ≥ b − p if s = 1, and n + a > s(n + b − 1) − p + 1 if s > 1. The converse part follows from Example 4.1(1) in Section 4. 2 Remark 3.1. For the case s = 1, i.e., on the John domain, we can then take a = 0 and b = p, and obtain the usual Korn inequality. This gives another proof of [3, Theorem 4.2]. 3.2. Quasihyperbolic domains Let Ω be a proper domain in Rn , n ≥ 2. By quasihyperbolic metric we mean that for all x, y ∈ Ω, ˆ 1 ds(z), k(x, y) := inf γ dist(z, ∂Ω) γ

where the infimum is taken over all curves γ joining x to y in Ω. The quasihyperbolic metric arises naturally in the theory of conformal geometry and plays an important role for example in the study of the boundary behavior of quasiconformal maps. Let β ∈ (0, 1]. Our domain Ω is said to satisfy a β-quasihyperbolic boundary condition (for short, β-QHBC), if for some fixed base point x0 there exists C0 < ∞ such that for every x ∈ Ω k(x, x0 ) ≤

dist(x0 , ∂Ω) 1 log + C0 . β dist(x, ∂Ω)

Changing the base point x0 changes the constant C0 . We first establish the following weighted Poincaré inequality on these domains; for non-weighted cases see [28,23,17], and recent paper [16] for (q, p)-Poincaré inequality with q < p. Let W be a Whitney decomposition of Ω. We may and do assume that the base-point x0 is the center of some Q ∈ W. For each Q ∈ W, we choose a quasihyperbolic geodesic γ joining x0 to the center of Q and let P (Q) denote the collection of all of Whitney cubes that intersect γ. The shadow of the cube Q ∈ W is the set S(Q) :=



Q1 .

Q1 ∈W Q∈P (Q1 )

We have the following estimate for the shadow of a cube from [17].

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Lemma 3.1. (See [17].) Let Ω satisfy the β-quasihyperbolic boundary condition, for some 0 < β ≤ 1. There exists a constant C = C(n, C0 ) such that for all Q ∈ W

 1−β 2β diam S(Q) ≤ C dist(x0 , ∂Ω) 1+β diam(Q) 1+β . Theorem 3.3. Let Ω ⊂ Rn be a proper subdomain satisfying a β-quasihyperbolic boundary condition, for some 0 < β ≤ 1. Then there is a constant C = C(n, p, q, β, Ω) > 0 such that ˆ

1/q |u − uΩ,a | ρ dx q a

ˆ ≤C

Ω

1/p |∇u| ρ dx p b

Ω

for each u ∈ C ∞ (Ω), where 1 ≤ p ≤ q < ∞, 0 ≤ a < ∞, p−n−b a + n 2β + > 0; q 1+β p additionally, q ≤

np n−p

if p < n.

Proof. For p = 1, the same proof as [12, Proof of Theorem 7] applies with Lemma 3.1 replacing the s-John condition there. For p > 1, the proof is similar to the proof of Theorem 3.2 in [23] with small modifications from [17]. We will verify condition (ii) of Theorem 2.3. Let W be a Whitney decomposition of Ω. Let A be an admissible set and Q0 some fixed cube. Let u be a smooth test function which equals 1 on A and 0 on Q0 . We split our set A to two parts 

 Ag =

x ∈ A : uQ ≤

1 for some Whitney cube Q  x 2

and Ab = A \ Ag . For all points x ∈ Ag with x ∈ Q ∈ W, from the properties of the Whitney decomposition, we have ρ(x) ∼ (Q), and hence  p  ˆ ˆ  pq  pq   1 u(x) − uQ q dx ρ(x)a dx ≤ C(Q)ap/q 2 Q∩A

Q

≤ C(Q)

ap pn q +p−n+ q −b

ˆ

  ∇u(x)p ρ(x)b dx

Q

≤ C diam(Ω)

ap pn q +p−n+ q −b

ˆ

  ∇u(x)p ρ(x)b dx,

Q pn p−n−b a+n 2β where ap ) ≥ 0. q + p − n + q − b ≥ p( q 1+β + p Summing over all such cubes Q, as q ≥ p, we obtain

ˆ Ω

|∇u|p ρ(x)b dx ≥ C −1

ˆ ρ(x)a dx

 pq .

(3.1)

Ag

Next we estimate the integral over the bad set. For each x ∈ Ab , let P (Q(x)) consist of the collection of all of the Whitney cubes which intersect the quasihyperbolic geodesic joining x0 to the center of Q(x), then a straightforward chaining argument shows that

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59



C

  ∇u(y)dy ≥ 1;

diam Q

Q∈P (Q(x))

55

Q

see [29, Lemma 8] for instance. Hence, by using the Hölder inequality, we have ˆ

ˆ ≤

a

ρ(x) dx

C

Ab

ρ(x)

C

C

≤

C

C

1− n p

ˆ



ˆ

ρ(x)a dx(diam Q)1− p − p n

b

  ∇u(y)p ρ(y)b dy

p

ˆ a

ρ(x) dx

 b (1− n p − p )p

1/p

(diam Q)

 1   ˆ  p1 p   ∇u(y)p ρ(y)b dy Q∈W Q

S(Q)∩Ab

  Q∈W

1/p

Q

S(Q)∩Ab

 

dx

Q

S(Q)∩Ab

ˆ

1/p

  ∇u(y)p dy

ρ(x) dx(diam Q)

Q∈W

≤

  ∇u(y)p dy

Q

a

Q∈W

Hölder

ˆ

(diam Q)

ˆ

 Q∈W

≤

1− n p

Q∈P (Q(x))

Ab

=



a

p

ˆ ρ(x)a dx

|Q|( n − p − pn )p 1

1

b



 p1  ˆ

  ∇u(y)p ρ(y)b dy

 p1 .

Ω

S(Q)∩Ab

This together with the following Lemma 3.2 gives that ˆ

1/q

ˆ

  ∇u(y)p ρ(y)b dy

≤C

ρ(x)a dx Ab

 p1 .

Ω

The proof is completed by combining the above estimate together with (3.1). 2 Lemma 3.2. With the assumptions of Theorem 3.3 and 1 < p < ∞, we have  Q∈W

p

ˆ ρ(x) dx

Q∈W

ˆ ≤C

a

p

ˆ a

ρ(x) dx

 pq

ρ(x) dx

.

Ab

S(Q)∩Ab

Proof. Since q ≥ p > 1, we have p − 1 − 

1 1 b (n −p − pn )p

|Q|

a

p q

≥ 0 and hence

|Q|( n − p − pn )p 1

1

b



S(Q)∩Ab

ˆ ≤

a

ρ(x) dx

Q∈W

Ab

ˆ ρ(x)a dx

= ˆ

ρ(x)a dx Ab

a

 pq

ˆ

ρ(x)a dx|Q|( n − p − pn )p 1

ρ(x) dx

S(Q)

|Q| p + pn − n 1

p −1− pq 

b

ˆ ρ(x)a dx

1

S(Q)∩Ab

´ 1    ( S(Q) ρ(x)a dx) q p ˆ |Q| p + pn − n 1

Q∈W Q ∈S(Q)

1

S(Q)∩Ab

1  p −1− pq   (´ ρ(x)a dx) q p S(Q)

Q∈W

Ab

=

p −1− pq   ˆ

b

1

Q ∩Ab

ρ(x)a dx

b



R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

56

ˆ a

=

ρ(x) dx

p −1− pq 

´ 1    ( S(Q) ρ(x)a dx) q p ˆ |Q|

Q ∈W Q∈P (Q )

Ab

ˆ ≤C

p



ρ(x)a dx

 −1− pq

ˆ

+1

1 b 1 p + pn − n

 ρ(x)a dx

=C

Ab

ρ(x)a dx

Q ∩Ab

p q

.

Ab

Above in estimating the last inequality, we use Lemma 3.1 to see that ˆ  ˆ 2β a a ρ(y) dy = ρ(y)a dy ≤ C|Q|( n +1) 1+β , Q ∈S(Q)Q

S(Q)

and [23, Lemma 2.6] to obtain ´ 1    ( S(Q) ρ(x)a dx) q p Q∈P (Q )

|Q|

1 b 1 p + pn − n

≤C



2β



|Q|(( n +1) (1+β)q + n − p − pn )p ≤ C(a, b, p, q, β, Ω, n), a

1

1

b

(3.2)

Q∈P (Q )

as 

 a 2β 1 1 b +1 + − − > 0. n (1 + β)q n p np

The proof is completed. 2 Remark 3.2. If a = b = 0, then the Poincaré inequality obtained above coincides with [17, Theorem 1]. One can modify [23, Example 5.5] to show that the Poincaré inequality from Theorem 3.3 is sharp, in the sense that the inequality ˆ

1/p |u − uΩ,a |q ρa dx

ˆ ≤C

Ω

does not hold if

a+n 2β q 1+β

+

p−n−b p

1/p |∇u|p ρb dx

Ω

< 0.

We have the following Korn inequality for domain satisfying a β-QHBC. Theorem 3.4. Let Ω ⊂ Rn be a proper subdomain satisfying a β-QHBC, for some 0 < β ≤ 1. Let 1 < p < ∞. Then there is a constant C = C(n, p, q, β, Ω) > 0 such that for every v ∈ W 1,p (Ω, ρa )n , it holds ˆ  ˆ ˆ  p b−p p a p a   (v) ρ |Dv| ρ dx ≤ C dx + |v| ρ dx , (Kp,a,b ) Ω

Ω

Ω

2β where 0 ≤ a < ∞, b ∈ R satisfying (a + n) 1+β > n + b − p. 2β Moreover, for 0 ≤ a < ∞ and (a + n) 1+β < n + b − p, the Korn inequality (Kp,a,b−p ) fails on Ω.

Proof. By using Theorem 2.1 and the Poincaré inequality (Theorem 3.3) with p = q, we see that the Korn 2β inequality (Kp,a,b−p ) holds if (a + n) 1+β + p − n − b > 0. The converse part follows from Example 4.1(2) in Section 4. 2 Remark 3.3. Notice that in the Poincaré inequality (Theorem 3.3) and the Korn inequality (Theorem 3.4), 2β there are no results for the borderline case (a + n) 1+β + p − n − b = 0. However, we believe the Poincaré inequality and the Korn inequality are true at the borderline.

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57

Fig. 1. A rooms-and-corridors domain.

4. Examples We next give examples to indicate the sharpness of Theorems 3.2 and 3.4 for n = 2. It is easy to check that the example works also for higher dimension. Example 4.1. Let Ω be a domain of the union of sequences of rectangles Ω = Q0 ∪ C1 ∪ Q1 ∪ C2 ∪ Q2 ∪ C3 ∪ . . . . The rectangles are arranged as in Fig. 1. This is possible if the sidelengths converge to 0 fast enough. The sidelength of Q0 is one and that of square Qi is ri . The height of the rectangle Ci is riτ and width is riσ for all i ≥ 1, where σ, τ ≥ 1 are fixed real numbers. The domain is called “A rooms-and-corridors domain”. We can control the boundary accessibility by choosing the constants σ and τ . Here are two relevant choices. (i) Ω is an s-John domain if s = σ and Ω is not an s-John domain if s < σ (independent of τ ); see [23, Example 5.5]. 1 (ii) Ω is a β-QHBC domain if σ ≤ τ , β = 2σ−1 and Ω is not a β-QHBC domain for any β > 0 if 1 ≤ τ < σ; see [23, Example 5.5] and [17]. For each i ∈ N, define the vector function ui (x, y) on Ω as follows: ⎧ riτ , −2(x − xi )), ∀(x, y) ∈ Qi ; ⎨ (2y + y2 2 ui (x, y) = (− rτ , rτ (x − xi )y), ∀(x, y) ∈ Ci ; i i ⎩ (0, 0), ∀(x, y) ∈ Ω \ (Ci ∪ Qi ). Above (xi , −ri /2 − riτ ) is the center of the cube Qi . It is immediate that ui is Lipschitz continuous in Ω. Direct computation gives that when (x, y) ∈ Qi ,  Dui (x, y) = and hence (ui )(x, y) = 0; when (x, y) ∈ Ci ,

0 −2

2 0

 ,

R. Jiang, A. Kauranen / J. Math. Anal. Appl. 423 (2015) 41–59

58

 Dui (x, y) =

0 2y/riτ

−2y/riτ 2(x − xi )/riτ

 ,

and  (ui )(x, y) =

0 0 0 2(x − xi )/riτ

 .

Meanwhile, for (x, y) ∈ Ω \ (Ci ∪ Qi ), (ui )(x, y) = Dui (x, y) = 0. From the above calculations, we deduce that ˆ ˆ |Dui |p ρa dx dy  ρa dx dy ∼ ria+2 , Ω

Qi

and since ρ(x) = riσ − |x − xi | in the corridor Ci , we see that ˆ

  (ui )p ρ(x)b−p dx dy ∼

Ω

ˆ Ci

  b−p |x − xi | p

σ σ(b+1)+τ (1−p) dx dy ∼ ri . ri − |x − xi | riτ

Moreover, ˆ

ˆ

rip ρa dx dy ∼ ria+p+2 .

|ui |p ρa dx dy ∼ Ω

Qi

The above estimates imply that if the Korn inequality ˆ  ˆ ˆ   (ui )p ρb−p dx dy + |ui |p ρa dx dy |Dui |p ρa dx dy ≤ C Ω

Ω

(Kp,a,b−p )

Ω

holds, then for each i, it holds σ(b+1)+τ (1−p)

ria+2  ri

+ ria+p+2 .

(4.1)

By choosing different parameters σ and τ , we obtain: (1) Sharpness of Theorem 3.2. Let 1 = τ ≤ σ, from Example 4.1(i) we know Ω is an s-John domain and s = σ. In this case, if the Korn inequality (Kp,a,b−p ) holds on Ω, then (4.1) becomes σ(b+1)+1−p

ria+2  ri

+ ria+p+2 .

This is true for all i. Thus a + 2 ≥ σ(b + 1) + 1 − p and we see that Korn inequality (Kp,a,b−p ) fails if a + 2 < s(b + 1) + 1 − p, therefore our Theorem 3.2 is essentially sharp. 1 (2) Sharpness of Theorem 3.4. Let 1 ≤ τ = σ, then Ω is a β-QHBC domain with β = 2σ−1 according to Example 4.1(ii). Suppose the Korn inequality (Kp,a,b−p ) holds on Ω, then (4.1) becomes σ(b+2−p)

ria+2  ri

+ ria+p+2

for each i. This implies that a + 2 ≥ σ(b + 2 − p). We see that Korn inequality (Kp,a,b−p ) fails if 2β 1 1+β (a + 2) = σ (a + 2) < b + 2 − p, which implies that Theorem 3.4 is essentially sharp.

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Acknowledgments The authors wish to thank their advisor Professor Pekka Koskela for posing the problem and helpful discussions. The authors also wish to thank both referees for their very carefully reading of the paper, and also their many valuable remarks which essentially improved the presentation of this article. Jiang was partially supported by NSFC (No. 11301029), the Fundamental Research Funds for Central Universities of China (No. 2013YB60) and the Project sponsored by SRF for ROCS, SEM, and Kauranen was supported by The Finnish National Graduate School in Mathematics and its Applications. References [1] G. Acosta, R.G. Durán, A.L. Lombardi, Weighted Poincaré and Korn inequalities for Hölder α domains, Math. Methods Appl. Sci. 29 (2006) 387–400. [2] G. Acosta, R.G. Durán, F. López García, Korn inequality and divergence operator: counter-examples and optimality of weighted estimates, Proc. Amer. Math. Soc. 141 (2013) 217–232. [3] G. Acosta, R.G. Durán, M.A. Muschietti, Solutions of the divergence operator on John domains, Adv. Math. 206 (2006) 373–401. [4] J. Bourgain, H. Brezis, On the equation div Y = f and application to control of phases, J. Amer. Math. Soc. 16 (2003) 393–426. [5] S.K. Chua, R.L. Wheeden, Self-improving properties of inequalities of Poincaré type on s-John domains, Pacific J. Math. 250 (2011) 67–108. [6] S. Conti, D. Faraco, F. Maggi, A new approach to counterexamples to L1 estimates: Korn’s inequality, geometric rigidity, and regularity for gradients of separately convex functions, Arch. Ration. Mech. Anal. 175 (2005) 287–300. [7] R.G. Durán, F. López García, Solutions of the divergence and Korn inequalities on domains with an external cusp, Ann. Acad. Sci. Fenn. Math. 35 (2010) 421–438. [8] R.G. Durán, M.A. Muschietti, E. Russ, P. Tchamitchian, Divergence operator and Poincaré inequalities on arbitrary bounded domains, Complex Var. Elliptic Equ. 55 (2010) 795–816. [9] G. Duvaut, J.-L. Lions, Inequalities in Mechanics and Physics, Springer, 1976. [10] D.E. Edmunds, R. Hurri-Syrjänen, Weighted Hardy inequalities, J. Math. Anal. Appl. 310 (2005) 424–435. [11] K.O. Friedrichs, On the boundary-value problems of the theory of elasticity and Korn’s inequality, Ann. of Math. 48 (2) (1947) 441–471. [12] P. Hajłasz, P. Koskela, Isoperimetric inequalities and imbedding theorems in irregular domains, J. Lond. Math. Soc. (2) 58 (1998) 425–450. [13] C.O. Horgan, Korn’s inequalities and their applications in continuum mechanics, SIAM Rev. 37 (1995) 491–511. [14] C.O. Horgan, L.E. Payne, On inequalities of Korn, Friedrichs and Babuška–Aziz, Arch. Ration. Mech. Anal. 82 (1983) 165–179. [15] R. Hurri-Syrjänen, An improved Poincaré inequality, Proc. Amer. Math. Soc. 120 (1994) 213–222. [16] R. Hurri-Syrjänen, N. Marola, A.V. Vähäkangas, Poincaré inequalities in quasihyperbolic boundary condition domains, arXiv:1201.5789. [17] R. Jiang, A. Kauranen, A note on “Quasihyperbolic boundary conditions and Poincaré domains”, Math. Ann. 357 (2013) 1199–1204. [18] R. Jiang, A. Kauranen, P. Koskela, Solvability of the divergence equation implies John via Poincaré inequality, Nonlinear Anal. 101 (2014) 80–88. [19] F. John, Rotation and strain, Comm. Pure Appl. Math. 4 (1961) 391–414. [20] T. Kilpeläinen, J. Malý, Sobolev inequalities on sets with irregular boundaries, Z. Anal. Anwend. 19 (2000) 369–380. [21] T. Kilpeläinen, J. Malý, A correction to: Sobolev inequalities on sets with irregular boundaries, http://users.jyu.fi/~terok/ preprints/correction.pdf. [22] V.A. Kondratiev, O.A. Oleinik, On Korn’s inequalities, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989) 483–487. [23] P. Koskela, J. Onninen, J.T. Tyson, Quasihyperbolic boundary conditions and Poincaré domains, Math. Ann. 323 (2002) 811–830. [24] O. Martio, J. Sarvas, Injectivity theorems in plane and space, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1978/1979) 383–401. [25] V.G. Maz’ya, Sobolev Spaces, Springer, 1985. [26] P.P. Mosolov, V.P. Mjasnikov, A proof of Korn’s inequality, Dokl. Akad. Nauk SSSR 201 (1971) 36–39. [27] J.A. Nitsche, On Korn’s second inequality, RAIRO J. Numer. Anal. 15 (1981) 237–248. [28] W. Smith, D.A. Stegenga, Hölder domains and Poincaré domains, Trans. Amer. Math. Soc. 319 (1990) 67–100. [29] W. Smith, D.A. Stegenga, Exponential integrability of the quasihyperbolic metric on Hölder domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991) 345–360. [30] A. Tiero, On inequalities of Korn, Friedrichs, Magenes–Stampacchia–Nečas and Babuška–Aziz, Z. Anal. Anwend. 20 (2001) 215–222. [31] T.W. Ting, Generalized Korn’s inequalities, Tensor 25 (1972) 295–302.