Optimal three-ball inequalities and
quantitative uniqueness for the Stokes system
Ching-Lung Lin
^{∗}Gunther Uhlmann
^{†}Jenn-Nan Wang
^{‡}Abstract
In this paper we study the local behavior of a solution to the Stokes system with singular coefficients in R^{n}with n = 2, 3. One of the main results is the bound on the vanishing order of a nontrivial solution u satisfying the Stokes system, which is a quantitative version of the strong unique continuation property for u. Different from the previous known results, our strong unique continuation result only involves the velocity field u. Our proof relies on some delicate Carleman-type esti- mates. We first use these estimates to derive crucial optimal three-ball inequalities for u. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution u to the Stokes system from those three-ball inequalities. As an applica- tion, we derive a minimal decaying rate at infinity of any nontrivial u satisfying the Stokes equation under some a priori assumptions.
1 Introduction
Assume that Ω is a connected open set containing 0 in R^{n} with n = 2, 3. In this paper we are interested in the local behavior of u satisfying the following
∗Department of Mathematics, NCTS, National Cheng Kung University, Tainan 701, Taiwan. Email:cllin2@mail.ncku.edu.tw
†Department of Mathematics, University of Washington, Box 354350, Seattle 98195- 4350, USA. Email:gunther@math.washington.edu
‡Department of Mathematics, Taida Institute of Mathematical Sciences, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan. Email: jn- wang@math.ntu.edu.tw
Stokes system:
( ∆u + A(x) · ∇u + B(x)u + ∇p = 0 in Ω,
∇ · u = 0 in Ω, (1.1)
where A and B are measurable satisfying
|A(x)| ≤ λ_{1}|x|^{−1}| log |x||^{−3}, |B(x)| ≤ λ_{1}|x|^{−2}| log |x||^{−3} ∀ x ∈ Ω (1.2) and A · ∇u = (A · ∇u_{1}, · · · , A · ∇u_{n}).
For the Stokes system (1.1) with essentially bounded coefficients A(x), the weak unique continuation property has been shown by Fabre and Lebeau [6]. On the other hand, when A(x) satisfies |A(x)| = O(|x|^{−1+}) with >
0, the strong unique continuation property was proved by Regbaoui [19].
The results in [6] and [19] concern only the qualitative unique continuation theorem and both results require the vanishing property for u and p. In this work we aim to derive a quantitative estimate of the strong unique continuation for u satisfying (1.1) with an appropriate p.
For the second order elliptic operator, using Carleman or frequency func- tions methods, quantitative estimates of the strong unique continuation (in the form of doubling inequality) under different assumptions on coefficients were derived in [4], [7], [8], [14], [16]. For the power of Laplacian, a quanti- tative estimate was obtained in [17]. We refer to [16] and references therein for the development of this investigation.
Since there is no equation for p in the Stokes system (1.1), we apply the curl operator ∇× on the first equation and obtain
∆q + ∇ · F = 0, (1.3)
where q = ∇ × u and for n = 2, ∇ × u = ∂_{1}u_{2} − ∂_{2}u_{1}. For n = 3, ∇ · F is a vector function defined by (∇ · F )i = P3
j=1∂jFij, i = 1, 2, 3, where F_{ij} = P3
k,`=1A˜_{ijk`}(x)∂_{k}u_{`} +P3
k=1B˜_{ijk}(x)u_{k} with appropriate ˜A_{ijk`}(x) and B˜ijk(x) satisfying
| ˜A_{ijk`}(x)| ≤ C_{0}| log |x||^{−3}|x|^{−1}, | ˜B_{ijk}(x)| ≤ C_{0}| log |x||^{−3}|x|^{−2} ∀ x ∈ Ω.
(1.4) When n = 2, ∇ · F is a scalar and we simply drop i in the definition above.
Now we define ∇^{⊥}× G = ∇ × G for any three-dimensional vector function
G and ∇^{⊥}× g = (∂_{2}g, −∂_{1}g) for a scalar function g if n = 2. It is easy to check that ∆u = ∇(∇ · u) − ∇^{⊥}× (∇ × u) and thus we have
∆u + ∇^{⊥}× q = 0 (1.5)
if ∇·u = 0. However, the equations (1.3) and (1.5) do not give us a decoupled system. The frequency functions method does not seem to work in this case.
So we prove our results along the line of Carleman’s method. On the other hand, since the coefficient A(x) is more singular than the one considered in [19]. Carleman-type estimates derived in [19] can not be applied to the case here. Hence we need to derive new Carleman-type estimates for our purpose. The key is to use weights which are slightly less singular than the negative powers of |x| (see estimates (2.4) and (2.15)). The estimate (2.15) is to handle (1.3) and the idea is due to Fabre and Lebeau [6].
We can derive certain three-ball inequalities which are optimal in the sense explained in [5] using (2.4) and (2.15). We would like to remark that usually the three-ball inequality can be regarded as the quantitative estimate of the weak unique continuation property. However, when the three-ball in- equality is optimal, one is able to deduce the strong unique continuation from it. It seems reasonable to expect that one could derive a bound on the van- ishing order of a nontrivial solution from the optimal three-ball inequality.
A recent result by Bourgain and Kenig [3] (more precisely, Kenig’s lecture notes for 2006 CNA Summer School [13]) indicates that this is indeed possi- ble, at least for the Schr¨odinger operator. In this paper, we show that by the optimal three-ball inequality, we can obtain a bound on the vanishing order of a nontrivial solution to (1.1) containing ”nearly” optimal singular coeffi- cients. Finally, we would like to mention that quantitative estimates of the strong unique continuation are useful in studying the nodal sets of solutions for elliptic or parabolic equations [4], [9], [15], or the inverse problem [1].
We now state main results of this paper. Their proofs will be given in the subsequent sections. Assume that there exists 0 < R_{0} ≤ 1 such that BR0 ⊂ Ω. Hereafter Br denotes an open ball of radius r > 0 centered at the origin.
Theorem 1.1 There exists a positive number ˜R < 1, depending only on n, such that if 0 < R_{1} < R_{2} < R_{3} ≤ R_{0} and R_{1}/R_{3} < R_{2}/R_{3} < ˜R, then
Z
|x|<R_{2}
|u|^{2}dx ≤ C
Z
|x|<R_{1}
|u|^{2}dx
τZ
|x|<R_{3}
|u|^{2}dx
1−τ
(1.6)
for (u, p) ∈ (H^{1}(B_{R}_{0}))^{n+1} satisfying (1.1) in B_{R}_{0}, where the constant C de- pends on R_{2}/R_{3}, n, and 0 < τ < 1 depends on R_{1}/R_{3}, R_{2}/R_{3}, n. Moreover, for fixed R2 and R3, the exponent τ behaves like 1/(− log R1) when R1 is sufficiently small.
Remark 1.2 It is important to emphasize that C is independent of R_{1} and τ has the asymptotic (− log R_{1})^{−1}. These facts are crucial in deriving an vanishing order of a nontrivial (u, p) to (1.1). Due to the behavior of τ , the three-ball inequality is called optimal [5].
It should be emphasized that three-ball inequalities (1.6) involve only the velocity field u. This is important in the application to inverse problems for the Stokes system, for example, see [2]. Using (1.6), we can also derive an upper bound of the vanishing order for any nontrivial u satisfying (1.1), which is a quantitative form of the strong unique continuation property for u. Let us now pick any R2 < R3 such that R3 ≤ R_{0} and R2/R3 < ˜R.
Theorem 1.3 Let (u, p) ∈ (H^{1}(B_{R}_{0}))^{n+1} be a nontrivial solution to (1.1), then there exist positive constants K and m, depending on n and u, such that
Z
|x|<R
|u|^{2}dx ≥ KR^{m} (1.7)
for all R with R < R_{2}.
Remark 1.4 Based on Theorem 1.1, the constants K and m in (1.7) are given by
K = Z
|x|<R3
|u|^{2}dx and
m = ˜C log R
|x|<R3|u|^{2}dx R
|x|<R_{2}|u|^{2}dx
,
where ˜C is a positive constant depending on λ1, n and R2/R3.
From Theorem 1.3, we immediately conclude that if (u, p) ∈ (H_{loc}^{1} (Ω))^{n+1} satisfies (1.1) and for any N ∈ N, there exists CN > 0 such that
Z
|x|<r
|u|^{2}dx ≤ C_{N}r^{N},
then u vanishes identically in Ω. Consequently, p is a constant in Ω. This is a new strong unique continuation result for the Stokes system with singular coefficients.
By three-ball inequalities (1.6), one can also study the minimal decaying rate of any nontrivial velocity u to (1.1) with a suitable assumption on co- efficients A and B (see [3] for a related result for the Schr¨odinger equation).
Consider (u, p) satisfying (1.1) with Ω = R^{n}, n = 2, 3. Assume here that kuk_{L}^{∞}_{(R}^{n}_{)}+ kAk_{L}^{∞}_{(R}^{n}_{)}+ kBk_{L}^{∞}_{(R}^{n}_{)} ≤ λ_{2}. (1.8) Denote
M_{r}(t) = inf
|x|=t
Z
|y−x|<r
|u(y)|^{2}dy.
Then we can prove that
Theorem 1.5 Let (u, p) ∈ (H_{loc}^{1} (R^{n}))^{n+1} be a nontrivial solution to (1.1).
Assume that (1.8) holds. Then for any r < 1, there exists c > 0 such that M_{r}(t) ≥ r^{cζ}^{(1+ t}^{r )},
where c depends on λ_{2}, n, R
|x|<r|u|^{2}dx and ζ = 1 + 2 ˜C log(1/r) with ˜C given in Remark 1.4.
We can apply Theorem 1.5 to the stationary Navier-Stokes equation.
Corollary 1.6 Let (u, p) ∈ (H_{loc}^{1} (R^{n}))^{n+1} be a nontrivial solution of the stationary Navier-Stokes equation:
−∇u + u · ∇u + ρu + ∇p = 0, ∇ · u = 0, in R^{n} with n = 2, 3. Assume that
kuk_{L}^{∞}_{(R}^{n}_{)}+ kρk_{L}^{∞}_{(R}^{n}_{)} ≤ λ_{3}. Then for any r < 1, there exists ˜c > 0 such that
M_{r}(t) ≥ r^{˜}^{cζ}^{(1+ t}^{r )}, where ˜c depends on λ_{3}, n, andR
|x|<r|u|^{2}dx.
This paper is organized as follows. In Section 2, we derive suitable Carleman-type estimates. A technical interior estimate is proved in Sec- tion 3. Section 4 is devoted to the proofs of Theorem 1.1, 1.3. The proof of Theorem 1.5 is given in Section 5.
2 Carleman estimates
Similar to the arguments used in [10], we introduce polar coordinates in R^{n}\{0} by setting x = rω, with r = |x|, ω = (ω_{1}, · · · , ω_{n}) ∈ S^{n−1}. Further- more, using new coordinate t = log r, we can see that
∂
∂x_{j} = e^{−t}(ω_{j}∂_{t}+ Ω_{j}), 1 ≤ j ≤ n,
where Ω_{j} is a vector field in S^{n−1}. We could check that the vector fields Ω_{j} satisfy
n
X
j=1
ω_{j}Ω_{j} = 0 and
n
X
j=1
Ω_{j}ω_{j} = n − 1.
Since r → 0 iff t → −∞, we are mainly interested in values of t near −∞.
It is easy to see that
∂^{2}
∂x_{j}∂x_{`} = e^{−2t}(ω_{j}∂_{t}− ω_{j}+ Ω_{j})(ω_{`}∂_{t}+ Ω_{`}), 1 ≤ j, ` ≤ n.
and, therefore, the Laplacian becomes
e^{2t}∆ = ∂_{t}^{2}+ (n − 2)∂_{t}+ ∆_{ω}, (2.1) where ∆_{ω} = Σ^{n}_{j=1}Ω^{2}_{j} denotes the Laplace-Beltrami operator on S^{n−1}. We recall that the eigenvalues of −∆_{ω} are k(k + n − 2), k ∈ N, and the corre- sponding eigenspaces are Ek, where Ek is the space of spherical harmonics of degree k. It follows that
Z Z
|∆ωv|^{2}dtdω =X
k≥0
k^{2}(k + n − 2)^{2} Z Z
|vk|^{2}dtdω (2.2)
and
X
j
Z Z
|Ω_{j}v|^{2}dtdω =X
k≥0
k(k + n − 2) Z Z
|v_{k}|^{2}dtdω, (2.3) where v_{k} is the projection of v onto E_{k}. Let
Λ =
r(n − 2)^{2}
4 − ∆_{ω},
then Λ is an elliptic first-order positive pseudodifferential operator in L^{2}(S^{n−1}).
The eigenvalues of Λ are k + ^{n−2}_{2} and the corresponding eigenspaces are E_{k}. Denote
L^{±}= ∂_{t}+ n − 2 2 ± Λ.
Then it follows from (2.1) that
e^{2t}∆ = L^{+}L^{−}= L^{−}L^{+}.
Motivated by the ideas in [18], we will derive Carleman-type estimates with weights ϕ_{β} = ϕ_{β}(x) = exp(−β ˜ψ(x)), where β > 0 and ˜ψ(x) = log |x| + log((log |x|)^{2}). Note that ϕ_{β} is less singular than |x|^{−β}, For simplicity, we denote ψ(t) = t + log t^{2}, i.e., ˜ψ(x) = ψ(log |x|). From now on, the notation X . Y or X & Y means that X ≤ CY or X ≥ CY with some constant C depending only on n.
Lemma 2.1 There exist a sufficiently small r_{0} > 0 depending on n and a sufficiently large β_{0} > 1 depending on n such that for all u ∈ U_{r}_{0} and β ≥ β_{0}, we have that
β Z
ϕ^{2}_{β}(log |x|)^{−2}|x|^{−n}(|x|^{2}|∇u|^{2}+ |u|^{2})dx . Z
ϕ^{2}_{β}|x|^{−n}|x|^{4}|∆u|^{2}dx, (2.4) where U_{r}_{0} = {u ∈ C_{0}^{∞}(R^{n}\ {0}) : supp(u) ⊂ B_{r}_{0}}.
Proof. By the polar coordinate system described above, we have Z
ϕ^{2}_{β}|x|^{4−n}|∆u|^{2}dx
= Z Z
e^{−2βψ(t)}e^{4t}|∆u|^{2}dtdω
= Z Z
|e^{−βψ(t)}e^{2t}∆u|^{2}dtdω. (2.5)
If we set u = e^{βψ(t)}v and use (2.1), then
e^{−βψ(t)}e^{2t}∆u = ∂_{t}^{2}v + b∂_{t}v + av + ∆_{ω}v =: P_{β}v, (2.6) where a = (1 + 2t^{−1})^{2}β^{2}+ (n − 2)β + 2(n − 2)t^{−1}β − 2t^{−2}β and b = n − 2 + 2β + 4t^{−1}β. By (2.5) and (2.6), (2.4) holds if for t near −∞ we have
X
j+|α|≤1
β^{3−2(j+|α|)} Z Z
|t|^{−2}|∂_{t}^{j}Ω^{α}v|^{2}dtdω ≤ ˜C_{1} Z Z
|P_{β}v|^{2}dtdω, (2.7)
where ˜C_{1} is a positive constant depending on n.
From (2.6), using the integration by parts, for t < t_{0} and β > β_{0}, where t0 < −1 and β0 > 0 depend on n, we have that
Z Z
|P_{β}v|^{2}dtdω
= Z Z
|∂_{t}^{2}v|^{2}dtdω + Z Z
|b∂_{t}v|^{2}dtdω + Z Z
|av|^{2}dtdω + Z Z
|∆_{ω}v|^{2}dtdω
− Z Z
∂tb|∂tv|^{2}dtdω − 2 Z Z
a|∂tv|^{2}dtdω + Z Z
∂_{t}^{2}a|v|^{2}dtdω
− Z Z
∂_{t}(ab)|v|^{2}dtdω + 2X
j
Z Z
|∂_{t}Ω_{j}v|^{2}dtdω
+X
j
Z Z
∂tb|Ωjv|^{2}dtdω − 2X
j
Z Z
a|Ωjv|^{2}dtdω
≥ Z Z
|∆_{ω}v|^{2}dtdω + Z Z
{b^{2}− ∂_{t}b − 2a}|∂_{t}v|^{2}dtdω
+X
j
Z Z
{∂_{t}b − 2a}|Ω_{j}v|^{2}dtdω + Z Z
{a^{2}+ ∂_{t}^{2}a − ∂_{t}(ab)}|v|^{2}dtdω
≥ Z Z
|∆_{ω}v|^{2}dtdω +X
j
Z Z
{−4t^{−2}β − 2a}|Ω_{j}v|^{2}dtdω
+ Z Z
{a^{2}+ 11t^{−2}β^{3}}|v|^{2}dtdω + Z Z
β^{2}|∂_{t}v|^{2}dtdω. (2.8) In view of (2.8), using (2.2),(2.3), we see that
Z Z
|∆_{ω}v|^{2}dtdω − 2X
j
Z Z
a|Ω_{j}v|^{2}dtdω + Z Z
a^{2}|v|^{2}dtdω
= X
k≥0
Z Z
[a − k(k + n − 2)]^{2}|v_{k}|^{2}dtdω. (2.9)
Substituting (2.9) into (2.8) yields Z Z
|P_{β}v|^{2}dtdω
≥ X
k≥0
Z Z
{11t^{−2}β^{3} − 4t^{−2}βk(k + n − 2) + [a − k(k + n − 2)]^{2}}|v_{k}|^{2}dtdω
+ Z Z
β^{2}|∂_{t}v|^{2}dtdω
= X
k,k(k+n−2)≥2β^{2}
+ X
k,k(k+n−2)<2β^{2}
Z Z
{11t^{−2}β^{3}− 4t^{−2}βk(k + n − 2)
+[a − k(k + n − 2)]^{2}}|v_{k}|^{2}dtdω + Z Z
β^{2}|∂_{t}v|^{2}dtdω. (2.10) For k such that k(k + n − 2) < 2β^{2}, we have
11t^{−2}β^{3}− 4t^{−2}βk(k + n − 2) ≥ t^{−2}β^{3}+ t^{−2}βk(k + n − 2). (2.11) On the other hand, if 2β^{2} < k(k + n − 2), then, by taking t even smaller, if necessary, we get that
−4t^{−2}βk(k + n − 2) + [a − k(k + n − 2)]^{2} & t^{−2}βk(k + n − 2). (2.12) Finally, using formula (2.3) and estimates (2.11), (2.12) in (2.10), we imme- diately obtain (2.7) and the proof of the lemma is complete.
2
To handle the auxiliary equation corresponding to q, we need another Carleman estimate. The derivation here follows the line in [19].
Lemma 2.2 There exists a sufficiently small number t_{0} < 0 depending on n such that for all u ∈ V_{t}_{0}, β > 1, we have that
X
j+|α|≤1
β^{1−2(j+|α|)} Z Z
t^{−2}ϕ^{2}_{β}|∂_{t}^{j}Ω^{α}u|^{2}dtdω . Z Z
ϕ^{2}_{β}|L^{−}u|^{2}dtdω, (2.13)
where Vt0 = {u(t, ω) ∈ C_{0}^{∞}((−∞, t0) × S^{n−1})}.
Proof. If we set u = e^{βψ(t)}v, then simple integration by parts implies Z Z
ϕ^{2}_{β}|L^{−}u|^{2}dtdω
= Z Z
|∂_{t}v − Λv + βv + 2βt^{−1}v + (n − 2)v/2|^{2}dtdω
= Z Z
|∂_{t}v|^{2}dtdω + Z Z
| − Λv + βv + 2βt^{−1}v + (n − 2)v/2|^{2}dtdω +β
Z Z
t^{−2}|v|^{2}dtdω.
By the definition of Λ, we have Z Z
| − Λv + βv + 2βt^{−1}v + (n − 2)v/2|^{2}dtdω
= X
k≥0
Z Z
| − kv_{k}+ βv_{k}+ 2βt^{−1}v_{k}|^{2}dtdω
= X
k≥0
Z Z
(−k + β + 2βt^{−1})^{2}|v_{k}|^{2}dtdω,
where, as before, v_{k} is the projection of v on E_{k}. Note that (−k + β + 2βt^{−1})^{2}+ βt^{−2} ≥ 1
8β(2βt^{−1})^{2}+ 1
16β(β − k)^{2}. Considering β > (1/2)k and β ≤ (1/2)k, we can get that
Z Z
ϕ^{2}_{β}|L^{−}u|^{2}dtdω
= Z Z
|∂_{t}v|^{2}dtdω + Σ_{k≥0} Z Z
[(−k + β + 2βt^{−1})^{2}+ βt^{−2}]|v_{k}|^{2}dtdω
&
Z Z
|∂tv|^{2}dtdω + Σk≥0
Z Z
(β^{−1}t^{−2}k(k + n − 2) + βt^{−2})|vk|^{2}dtdω.
(2.14)
The estimate (2.13) then follows from (2.3).
2
Next we need a technical lemma. We then use this lemma to derive another Carleman estimate.
Lemma 2.3 There exists a sufficiently small number t_{1} < −2 depending on n such that for all u ∈ V_{t}_{1}, g = (g_{0}, g_{1}, · · · , g_{n}) ∈ (V_{t}_{1})^{n+1} and β > 0, we have that
Z Z
ϕ^{2}_{β}|u|^{2}dtdω . Z Z
ϕ^{2}_{β}(|L^{+}u + ∂_{t}g_{0}+
n
X
j=1
Ω_{j}g_{j}|^{2}+ kgk^{2})dtdω.
Proof. This lemma can be proved by exactly the same arguments used in
Lemma 2.2 of [19]. So we omit the proof here.
2
Lemma 2.4 There exist a sufficiently small number r_{1} > 0 depending on n and a sufficiently large number β_{1} > 2 depending on n such that for all w ∈ U_{r}_{1} and f = (f_{1}, · · · , f_{n}) ∈ (U_{r}_{1})^{n}, β ≥ β_{1}, we have that
Z
ϕ^{2}_{β}(log |x|)^{2}(|x|^{4−n}|∇w|^{2}+ |x|^{2−n}|w|^{2})dx . β
Z
ϕ^{2}_{β}(log |x|)^{4}|x|^{2−n}[(|x|^{2}∆w + |x|divf )^{2}+ kf k^{2}]dx, (2.15) where U_{r}_{1} is defined as in Lemma 2.1.
Proof. Replacing β by β + 1 in (2.15), we see that it suffices to prove Z
ϕ^{2}_{β}(log |x|)^{−2}(|x|^{2}|∇w|^{2}+ |w|^{2})|x|^{−n}dx . β
Z
ϕ^{2}_{β}[(|x|^{2}∆w + |x|divf )^{2}+ kf k^{2}]|x|^{−n}dx. (2.16) Working in polar coordinates and using the relation e^{2t}∆ = L^{+}L^{−}, (2.16) is equivalent to
X
j+|α|≤1
Z Z
β^{2−2(j+|α|)}t^{−2}ϕ^{2}_{β}|∂_{t}^{j}Ω^{α}u|^{2}dtdω
. β Z Z
ϕ^{2}_{β}(|L^{+}L^{−}w + ∂_{t}(
n
X
j=1
ω_{j}f_{j}) +
n
X
j=1
Ω_{j}f_{j}|^{2}+ kf k^{2})dtdω.(2.17) Applying Lemma 2.3 to u = L^{−}w and g = (Pn
j=1ω_{j}f_{j}, f_{1}, · · · , f_{n}) yields β
Z Z
ϕ^{2}_{β}|L^{−}w|^{2}dtdω . β
Z Z
ϕ^{2}_{β}(|L^{+}L^{−}w + ∂_{t}(
n
X
j=1
ω_{j}f_{j}) +
n
X
j=1
Ω_{j}f_{j}|^{2}+ kf k^{2})dtdω.(2.18)
Now (2.17) is an easy consequence of (2.13) and (2.18).
2
3 Interior estimates
To establish the three-ball inequality for (1.1), the following interior estimate is useful.
Lemma 3.1 Let (u, p) ∈ (H_{loc}^{1} (Ω))^{n+1} be a solution of (1.1). Then for any 0 < a_{3} < a_{1} < a_{2} < a_{4} such that B_{a}_{4}_{r} ⊂ Ω and |a_{4}r| < 1, we have
Z
a1r<|x|<a2r
|x|^{4}|∇q|^{2}+ |x|^{2}|q|^{2}+ |x|^{2}|∇u|^{2}dx ≤ C^{0} Z
a3r<|x|<a4r
|u|^{2}dx (3.1) where the constant C^{0} is independent of r and u. Here q = ∇ × u.
Proof. The proof of this lemma is motivated by ideas used in [11]. Let X = B_{a}_{4}_{r}\ ¯B_{a}_{3}_{r} and d(x) be the distant from x ∈ X to R^{n}\X. By the elliptic regularity, we obtain from (1.1) that u ∈ H_{loc}^{2} (Ω\{0}). It is trivial that
kvk_{H}^{1}_{(R}^{n}_{)} . k∆vkL^{2}(R^{n})+ kvk_{L}^{2}_{(R}^{n}_{)} (3.2) for all v ∈ H^{2}(R^{n}). By changing variables x → E^{−1}x in (3.2), we will have
P
|α|≤1E^{2−|α|}kD^{α}vk_{L}^{2}_{(R}^{n}_{)} . (k∆vkL^{2}(R^{n})+ E^{2}kvk_{L}^{2}_{(R}^{n}_{)}) (3.3) for all v ∈ H^{2}(R^{n}). To apply (3.3) on u, we need to cut-off u. So let ξ(x) ∈ C_{0}^{∞}(R^{n}) satisfy 0 ≤ ξ(x) ≤ 1 and
ξ(x) =
( 1, |x| < 1/4, 0, |x| ≥ 1/2.
Let us denote ξ_{y}(x) = ξ((x−y)/d(y)). For y ∈ X, we apply (3.3) to ξ_{y}(x)u(x) and use equation (1.5) to get that
E^{2} Z
|x−y|≤d(y)/4
|∇u|^{2}dx
≤ C_{1}^{0} Z
|x−y|≤d(y)/2
|∇q|^{2}dx + C_{1}^{0} Z
|x−y|≤d(y)/2
d(y)^{−2}|∇u|^{2}dx +C_{1}^{0}(E^{4}+ d(y)^{−4})
Z
|x−y|≤d(y)/2
|u|^{2}dx. (3.4)
Now taking E = M d(y)^{−1} for some positive constant M and multiplying d(y)^{4} on both sides of (3.4), we have
M^{2}d(y)^{2} Z
|x−y|≤d(y)/4
|∇u|^{2}dx
≤ C_{1}^{0} Z
|x−y|≤d(y)/2
d(y)^{4}|∇q|^{2}dx + C_{1}^{0} Z
|x−y|≤d(y)/2
d(y)^{2}|∇u|^{2}dx +C_{1}^{0}(M^{4}+ 1)
Z
|x−y|≤d(y)/2
|u|^{2}dx. (3.5)
Integrating d(y)^{−n}dy over X on both sides of (3.5) and using Fubini’s Theorem, we get that
M^{2} Z
X
Z
|x−y|≤d(y)/4
d(y)^{2−n}|∇u|^{2}dydx
≤ C_{1}^{0} Z
X
Z
|x−y|≤d(y)/2
d(y)^{4}|∇q(x)|^{2}d(y)^{−n}dydx +C_{1}^{0}
Z
X
Z
|x−y|≤d(y)/2
d(y)^{2−n}|∇u|^{2}dydx +2C_{1}^{0}M^{4}
Z
X
Z
|x−y|≤d(y)/2
|u|^{2}d(y)^{−n}dydx. (3.6) Note that |d(x) − d(y)| ≤ |x − y|. If |x − y| ≤ d(x)/3, then
2d(x)/3 ≤ d(y) ≤ 4d(x)/3. (3.7)
On the other hand, if |x − y| ≤ d(y)/2, then
d(x)/2 ≤ d(y) ≤ 3d(x)/2. (3.8)
By (3.7) and (3.8), we have ( R
|x−y|≤d(y)/4d(y)^{−n}dy ≥ (3/4)^{n}R
|x−y|≤d(x)/6d(x)^{−n}dy ≥ 8^{−n}R
|y|≤1dy, R
|x−y|≤d(y)/2d(y)^{−n}dy ≤ 2^{n}R
|x−y|≤3d(x)/4d(x)^{−n}dy ≤ (3/2)^{n}R
|y|≤1dy (3.9) Combining (3.6)–(3.9), we obtain
M^{2} Z
X
d(x)^{2}|∇u|^{2}dx
≤ C_{2}^{0} Z
X
d(x)^{2}|∇u(x)|^{2}dx + C_{2}^{0} Z
X
d(x)^{4}|∇q|^{2}dx + C_{2}^{0}M^{4} Z
X
|u|^{2}dx.
(3.10)
On the other hand, we have from (1.3) that
n
X
i=1
Z
|ξ_{y}(x)∇q_{i}|^{2}dx =
n
X
i=1
Z
∇q_{i}· ∇(ξ_{y}^{2}(x)¯q_{i})dx −
n
X
i=1
2 Z
ξ_{y}∇q_{i}· ¯q_{i}∇ξ_{y}dx
≤ C_{3}^{0}
n
X
i=1
| Z
(divF )_{i}ξ_{y}^{2}q_{i}dx| +
n
X
i=1
2 Z
|ξ_{y}∇q_{i}· ¯q_{i}∇ξ_{y}|dx
≤ C_{3}^{0}
n
X
i=1
| Z ^{n}
X
j=1
F_{ij} · ∂_{j}(ξ_{y}^{2}q_{i})dx| +1 4
n
X
i=1
Z
|ξ_{y}∇q_{i}|^{2}dx + 4 Z
|x−y|≤d(y)/2
d(y)^{−2}|q|^{2}dx
≤ C_{4}^{0} Z
|x−y|≤d(y)/2
|F |^{2}dx +1 4
n
X
i=1
Z
|ξ_{y}∇q_{i}|^{2}dx + C_{4}^{0} Z
|x−y|≤d(y)/2
d(y)^{−2}|q|^{2}dx
+1 4
n
X
i=1
Z
|ξ_{y}∇q_{i}|^{2}dx + C_{4}^{0} Z
|x−y|≤d(y)/2
d(y)^{−2}|q|^{2}dx.
(3.11) Therefore, we get that
Z
|x−y|≤d(y)/4
|∇q|^{2}dx
≤ Z
|ξ_{y}(x)∇q|^{2}dx
≤ C_{5}^{0} Z
|x−y|≤d(y)/2
|F |^{2}dx + C_{5}^{0} Z
|x−y|≤d(y)/2
d(y)^{−2}|q|^{2}dx.
(3.12) Multiply d(y)^{4} on both sides of (3.12), we obtain that
Z
|x−y|≤d(y)/4
d(y)^{4}|∇q|^{2}dx
≤ C_{6}^{0} Z
|x−y|≤d(y)/2
d(y)^{4}| ˜A|^{2}|∇u|^{2}dx + C_{6}^{0} Z
|x−y|≤d(y)/2
d(y)^{4}| ˜B|^{2}|u|^{2}dx +C_{6}^{0}
Z
|x−y|≤d(y)/2
d(y)^{2}|q|^{2}dx. (3.13)
Repeat (3.6)∼(3.10), we have that Z
X
d(x)^{4}|∇q|^{2}dx
≤ C_{7}^{0} Z
X
d(x)^{4}| ˜A|^{2}|∇u|^{2}dx + C_{7}^{0} Z
X
d(x)^{4}| ˜B|^{2}|u|^{2}dx +C_{7}^{0}
Z
X
d(x)^{2}|q|^{2}dx. (3.14)
Combining K×(3.14), (3.10) and R
Xd(x)^{2}|q|^{2}dx, we obtain that M^{2}
Z
X
d(x)^{2}|∇u|^{2}dx + K Z
X
d(x)^{4}|∇q|^{2}dx + Z
X
d(x)^{2}|q|^{2}dx
≤ Z
X
(C_{2}^{0}d(x)^{2} + C_{7}^{0}Kd(x)^{4}| ˜A|^{2})|∇u(x)|^{2}dx + C_{7}^{0}K Z
X
d(x)^{4}| ˜B|^{2}|u|^{2}dx +C_{2}^{0}M^{4}
Z
X
|u|^{2}dx + C_{2}^{0} Z
X
d(x)^{4}|∇q|^{2}dx + (C_{7}^{0}K + 1) Z
X
d(x)^{2}|q|^{2}dx.
(3.15) Taking K = 2C_{2}^{0}, one can eliminate R
Xd(x)^{4}|∇q|^{2}dx on the right hand side of (3.15). Observe that
Z
X
d(x)^{2}|q|^{2}dx ≤ C_{8}^{0} Z
X
d(x)^{2}|∇u(x)|^{2}dx.
So, by choosing M large enough, we can ignore R
Xd(x)^{2}|∇u(x)|^{2}dx on the right hand side of (3.15). Finally, we get that
M^{2} Z
X
d(x)^{2}|∇u|^{2}dx + K Z
X
d(x)^{4}|∇p|^{2}dx + Z
X
d(x)^{2}|q|^{2}dx
≤ C_{9}^{0} Z
X
|u|^{2}dx. (3.16)
We recall that X = B_{a}_{4}_{r}\ ¯B_{a}_{3}_{r} and note that d(x) ≥ ˜Cr if x ∈ B_{a}_{2}_{r}\ ¯B_{a}_{1}_{r}, where ˜C is independent of r. Hence, (3.1) is an easy consequence of (3.16).
2
4 Proof of Theorem 1.1 and Theorem 1.3
This section is devoted to the proofs of Theorem 1.1 and Theorem 1.3. To begin, we first consider the case where 0 < R_{1} < R_{2} < R < 1 and B_{R} ⊂ Ω.
The small constant R will be determined later. Since (u, p) ∈ (H^{1}(B_{R}_{0}))^{n+1}, the elliptic regularity theorem implies u ∈ H_{loc}^{2} (B_{R}_{0}\ {0}). Therefore, to use estimate (2.4), we simply cut-off u. So let χ(x) ∈ C_{0}^{∞}(R^{n}) satisfy 0 ≤ χ(x) ≤ 1 and
χ(x) =
0, |x| ≤ R_{1}/e,
1, R_{1}/2 < |x| < eR_{2}, 0, |x| ≥ 3R_{2},
where e = exp(1). We remark that we first choose a small R such that R ≤ min{r_{0}, r_{1}}/3 = ˜R_{0}, where r_{0} and r_{1} are constants appeared in (2.4) and (2.15). Hence ˜R_{0} depends on n. It is easy to see that for any multiindex
α (
|D^{α}χ| = O(R^{−|α|}_{1} ) for all R_{1}/e ≤ |x| ≤ R_{1}/2
|D^{α}χ| = O(R^{−|α|}_{2} ) for all eR_{2} ≤ |x| ≤ 3R_{2}. (4.1) Applying (2.4) to χu gives
C1β Z
(log |x|)^{−2}ϕ^{2}_{β}|x|^{−n}(|x|^{2}|∇(χu)|^{2}+|χu|^{2})dx ≤ Z
ϕ^{2}_{β}|x|^{−n}|x|^{4}|∆(χu)|^{2}dx.
(4.2) From now on, C_{1}, C_{2}, · · · denote general constants whose dependence will be specified whenever necessary. Next applying (2.15) to w = χq and f = |x|χF , we get that
C_{2} Z
ϕ^{2}_{β}(log |x|)^{2}(|x|^{4−n}|∇(χq)|^{2}+ |x|^{2−n}|χq|^{2})dx
≤ β Z
ϕ^{2}_{β}(log |x|)^{4}|x|^{2−n}[|x|^{2}∆(χq) + |x|div(|x|χF )]^{2}dx +β
Z
ϕ^{2}_{β}(log |x|)^{4}|x|^{2−n}k|x|χF k^{2}dx. (4.3)
Multiplying by M_{1} on (4.2) and combining (4.3), we obtain that M1β
Z
R1/2<|x|<eR2
(log |x|)^{−2}ϕ^{2}_{β}|x|^{−n}(|x|^{2}|∇u|^{2}+ |u|^{2})dx +
Z
R1/2<|x|<eR2
(log |x|)^{2}ϕ^{2}_{β}|x|^{−n}(|x|^{4}|∇q|^{2}+ |x|^{2}|q|^{2})dx
≤ M1β Z
ϕ^{2}_{β}(log |x|)^{−2}|x|^{−n}(|x|^{2}∇(χu)|^{2}+ |χu|^{2})dx +
Z
(log |x|)^{2}ϕ^{2}_{β}|x|^{−n}(|x|^{4}|∇(χq)|^{2}+ |x|^{2}|χq|^{2})dx
≤ M_{1}C_{3} Z
ϕ^{2}_{β}|x|^{−n}|x|^{4}|∆(χu)|^{2}dx +βC_{3}
Z
(log |x|)^{4}ϕ^{2}_{β}|x|^{−n}[|x|^{3}∆(χq) + |x|^{2}div(|x|χF )]^{2}dx +βC_{3}
Z
(log |x|)^{4}ϕ^{2}_{β}|x|^{−n}k|x|^{2}χF k^{2}dx. (4.4) By (1.2), (1.3), (1.4), and estimates (4.1), we deduce from (4.4) that
M1β Z
R1/2<|x|<eR2
(log |x|)^{−2}ϕ^{2}_{β}|x|^{−n}(|x|^{2}|∇u|^{2}+ |u|^{2})dx +
Z
R1/2<|x|<eR2
(log |x|)^{2}ϕ^{2}_{β}|x|^{−n}(|x|^{4}|∇q|^{2}+ |x|^{2}|q|^{2})dx
≤ C_{4}M_{1} Z
R1/2<|x|<eR2
ϕ^{2}_{β}|x|^{−n}|x|^{4}|∇q|^{2}dx +C_{4}β
Z
R1/2<|x|<eR2
(log |x|)^{−2}ϕ^{2}_{β}|x|^{−n}(|x|^{2}|∇u|^{2}+ |u|^{2})dx +C_{4}M_{1}
Z
{R1/e≤|x|≤R1/2}∪{eR2≤|x|≤3R2}
ϕ^{2}_{β}|x|^{−n}| ˜U |^{2}dx +C_{4}β
Z
{R1/e≤|x|≤R1/2}∪{eR2≤|x|≤3R2}
(log |x|)^{4}ϕ^{2}_{β}|x|^{−n}| ˜U |^{2}dx, (4.5)
where | ˜U (x)|^{2} = |x|^{4}|∇q|^{2}+|x|^{2}|q|^{2}+|x|^{2}|∇u|^{2}+|u|^{2} and the positive constant C_{4} only depends on n.
Now letting M_{1} = 2 + 2C_{4}, β ≥ 2 + 2C_{4}, and R small enough such that (log(eR))^{2} ≥ 2C_{4}M_{1}, then the first three terms on the right hand
side of (4.5) can be absorbed by the left hand side of (4.5). Also, it is easy to check that there exists ˜R_{1} > 0, depending on n, such that for all β > 0, both (log |x|)^{−2}|x|^{−n}ϕ^{2}_{β}(|x|) and (log |x|)^{4}|x|^{−n}ϕ^{2}_{β}(|x|) are decreas- ing functions in 0 < |x| < ˜R_{1}. So we choose a small R < ˜R_{2}, where R˜_{2} = min{exp(−2√
2C_{4}M_{1}− 1), ˜R_{1}/3, ˜R_{0}}. It is clear that ˜R_{2} depends on n.
With the choices described above, we obtain from (4.5) that
R^{−n}_{2} (log R_{2})^{−2}ϕ^{2}_{β}(R_{2}) Z
R1/2<|x|<R2
|u|^{2}dx
≤ Z
R1/2<|x|<eR2
(log |x|)^{−2}ϕ^{2}_{β}|x|^{−n}|u|^{2}dx
≤ C_{5}β Z
{R1/e≤|x|≤R1/2}∪{eR2≤|x|≤3R2}
(log |x|)^{4}ϕ^{2}_{β}|x|^{−n}| ˜U |^{2}dx
≤ C_{5}β(log(R_{1}/e))^{4}(R_{1}/e)^{−n}ϕ^{2}_{β}(R_{1}/e) Z
{R1/e≤|x|≤R1/2}
| ˜U |^{2}dx +C5β(log(eR2))^{4}(eR2)^{−n}ϕ^{2}_{β}(eR2)
Z
{eR_{2}≤|x|≤3R_{2}}
| ˜U |^{2}dx. (4.6)
Using (3.1), we can control | ˜U |^{2} terms on the right hand side of (4.6). In other words, it follows from (3.1) that
R^{−2β−n}_{2} (log R2)^{−4β−2} Z
R1/2<|x|<R2
|u|^{2}dx
≤ C_{6}2^{2β+n}(log(R_{1}/e))^{4}(R_{1}/e)^{−n}ϕ^{2}_{β}(R_{1}/e) Z
{R1/4≤|x|≤R1}
|u|^{2}dx +C_{6}2^{2β+n}(log(eR_{2}))^{4}(eR_{2})^{−n}ϕ^{2}_{β}(eR_{2})
Z
{2R_{2}≤|x|≤4R_{2}}
|u|^{2}dx
= C_{6}2^{2β+n}(log(R_{1}/e))^{−4β+4}(R_{1}/e)^{−2β−n} Z
{R1/4≤|x|≤R1}
|u|^{2}dx +C_{6}2^{2β+n}(log(eR_{2}))^{−4β+4}(eR_{2})^{−2β−n}
Z
{2R_{2}≤|x|≤4R_{2}}
|u|^{2}dx. (4.7)
Replacing 2β + n by β, (4.7) becomes R^{−β}_{2} (log R_{2})^{−2β+2n−2}
Z
R1/2<|x|<R2
|u|^{2}dx
≤ C_{7}2^{β}(log(R_{1}/e))^{−2β+2n+4}(R_{1}/e)^{−β} Z
{R1/4≤|x|≤R1}
|u|^{2}dx +C_{7}2^{β}(log(eR_{2}))^{−2β+2n+4}(eR_{2})^{−β}
Z
{2R2≤|x|≤4R2}
|u|^{2}dx. (4.8) Dividing R^{−β}_{2} (log R_{2})^{−2β+2n−2} on the both sides of (4.8) and providing β ≥ n + 2, we have that
Z
R1/2<|x|<R2
|u|^{2}dx
≤ C8(log R2)^{6}(2eR2/R1)^{β} Z
{R_{1}/4≤|x|≤R1}
|u|^{2}dx +C_{8}(log R_{2})^{6}(2/e)^{β}[(log R_{2}/ log(eR_{2}))^{2}]^{β−n−2}
Z
{2R2≤|x|≤4R2}
|u|^{2}dx
≤ C8(log R2)^{6}(2eR2/R1)^{β} Z
{R_{1}/4≤|x|≤R1}
|u|^{2}dx +C_{8}(log R_{2})^{6}(4/5)^{β}
Z
{2R2≤|x|≤4R2}
|u|^{2}dx. (4.9)
In deriving the second inequality above, we use the fact that log R_{2}
log(eR_{2}) → 1 as R_{2} → 0, and thus
2
e · log R_{2} log(eR_{2}) < 4
5
for all R_{2} < ˜R_{3}, where ˜R_{3}is sufficiently small. We now take ˜R = min{ ˜R_{2}, ˜R_{3}}, which depends on n.
Adding R
|x|<R_{1}/2|u|^{2}dx to both sides of (4.9) leads to Z
|x|<R2
|u|^{2}dx ≤ C_{9}(log R_{2})^{6}(2eR_{2}/R_{1})^{β} Z
|x|≤R1
|u|^{2}dx +C_{9}(log R_{2})^{6}(4/5)^{β}
Z
|x|≤1
|u|^{2}dx. (4.10)
It should be noted that (4.10) holds for all β ≥ ˜β with ˜β depending only on n. For simplicity, by denoting
E(R_{1}, R_{2}) = log(2eR_{2}/R_{1}), B = log(5/4), (4.10) becomes
Z
|x|<R2
|u|^{2}dx
≤ C_{9}(log R_{2})^{6}n
exp(Eβ) Z
|x|<R_{1}
|u|^{2}dx + exp(−Bβ) Z
|x|<1
|u|^{2}dxo . (4.11) To further simplify the terms on the right hand side of (4.11), we consider two cases. If R
|x|<R1|u|^{2}dx 6= 0 and exp (E ˜β)
Z
|x|<R1
|u|^{2}dx < exp (−B ˜β) Z
|x|<1
|u|^{2}dx,
then we can pick a β > ˜β such that exp (Eβ)
Z
|x|<R_{1}
|u|^{2}dx = exp (−Bβ) Z
|x|<1
|u|^{2}dx.
Using such β, we obtain from (4.11) that Z
|x|<R2
|u|^{2}dx
≤ 2C_{9}(log R_{2})^{6}exp (Eβ) Z
|x|<R1
|u|^{2}dx
= 2C_{9}(log R_{2})^{6}
Z
|x|<R1
|u|^{2}dx
_{E+B}^{B} Z
|x|<1
|u|^{2}dx
_{E+B}^{E}
. (4.12) If R
|x|<R1|u|^{2}dx = 0, then letting β → ∞ in (4.11) we haveR
|x|<R2|u|^{2}dx = 0 as well. The three-ball inequality obviously holds.
On the other hand, if exp (−B ˜β)
Z
|x|<1
|u|^{2}dx ≤ exp (E ˜β) Z
|x|<R_{1}
|u|^{2}dx,
then we have Z
|x|<R2
|u|^{2}dx
≤
Z
|x|<1
|u|^{2}dx
_{E+B}^{B} Z
|x|<1
|u|^{2}dx
_{E+B}^{E}
≤ exp (B ˜β)
Z
|x|<R1
|u|^{2}dx
_{E+B}^{B} Z
|x|<1
|u|^{2}dx
_{E+B}^{E}
. (4.13)
Putting together (4.12), (4.13), and setting C10= max{2C9(log R2)^{6}, exp ( ˜β log(5/4))}, we arrive at
Z
|x|<R2
|u|^{2}dx ≤ C_{10}
Z
|x|<R1
|u|^{2}dx
_{E+B}^{B} Z
|x|<1
|u|^{2}dx
_{E+B}^{E}
. (4.14) It is readily seen that _{E+B}^{B} ≈ (log(1/R_{1}))^{−1} when R_{1} tends to 0.
Now for the general case, we consider 0 < R_{1} < R_{2} < R_{3} < 1 with R_{1}/R_{3} < R_{2}/R_{3} ≤ ˜R, where ˜R is given as above. By scaling, i.e. defining u(y) := u(Rb _{3}y), p(y) := Rb _{3}p(R_{3}y) and bA(y) = A(R_{3}y), (4.14) becomes
Z
|y|<R_{2}/R3
|u(y)|b ^{2}dy ≤ C_{11}( Z
|y|<R_{1}/R3
|u(y)|b ^{2}dy)^{τ}( Z
|y|<1
|bu(y)|^{2}dy)^{1−τ}, (4.15) where
τ = B/[E(R_{1}/R_{3}, R_{2}/R_{3}) + B],
C_{11} = max{2C_{9}(log R_{2}/R_{3})^{6}, exp ( ˜β log(5/4))}.
Note that C_{11}is independent of R_{1}. Restoring the variable x = R_{3}y in (4.15) gives
Z
|x|<R2
|u|^{2}dx ≤ C_{11}( Z
|x|<R1
|u|^{2}dx)^{τ}( Z
|x|<R3
|u|^{2}dx)^{1−τ}. The proof of Theorem 1.1 is complete.
We now turn to the proof of Theorem 1.3. We fix R_{2}, R_{3} in Theorem 1.1.
By dividing R
|x|<R_{2}|u|^{2}dx on the three-ball inequality (1.5), we have that 1 ≤ C(
Z
|x|<R_{1}
|u|^{2}dx/
Z
|x|<R_{2}
|u|^{2}dx)^{τ}( Z
|x|<R_{3}
|u|^{2}dx/
Z
|x|<R_{2}
|u|^{2}dx)^{1−τ}. (4.16)
Raising both sides by 1/τ yields that Z
|x|<R3
|u|^{2}dx ≤ ( Z
|x|<R1
|u|^{2}dx)(C Z
|x|<R3
|u|^{2}dx/
Z
|x|<R2
|u|^{2}dx)^{1/τ}. (4.17) In view of the formula for τ , we can deduce from (4.17) that
Z
|x|<R3
|u|^{2}dx ≤ ( Z
|x|<R1
|u|^{2}dx)(1/R_{1})^{C log(}^{˜}
R
|x|<R3|u|^{2}dx/R
|x|<R2|u|^{2}dx)
, (4.18) where ˜C is a positive constant depending on n and R_{2}/R_{3}. Consequently, (4.18) is equivalent to
( Z
|x|<R_{3}
|u|^{2}dx)R^{m}_{1} ≤ Z
|x|<R_{1}
|u|^{2}dx for all R_{1} sufficiently small, where
m = ˜C log R
|x|<R_{3}|u|^{2}dx R
|x|<R2|u|^{2}dx
. We now end the proof of Theorem 1.3.
5 Proof of Theorem 1.5
We prove Theorem 1.5 in this section. Let us first choose a > max{2, ˜R^{−1}}, where ˜R is given in Theorem 1.1. By doing so, we can see that if we set R_{2} = ar and R_{3} = a^{2}r, then R_{2}/R_{3} < ˜R for r > 0. Now let 0 < r < 1 and define R_{2}, R_{3} accordingly. Let |˜x| = t. We pick a sequence of points 0 = x_{0}, x_{1}, · · · , x_{N} = ˜x such that |x_{j+1} − x_{j}| ≤ r. We shall prove the desired estimate iteratively. To see how the iteration goes, let us assume that R
|x−x_{l}|<r|u|^{2}dx ≥ r^{m}^{l} for some m_{l} > 0 since u is nontrivial. By Theorem 1.3 and Remark 1.4, we have that
Z
|x−x_{l+1}|<r
|u|^{2}dx ≥ Z
|x−x_{l+1}|<R_{3}
|u|^{2}dx · r^{m}, (5.1) where
m = ˜C log R
|x−x_{l+1}|<R3|u|^{2}dx R
|x−x_{l+1}|<R2|u|^{2}dx
.