Optimal three-ball inequalities and

Optimal three-ball inequalities and

quantitative uniqueness for the Stokes system

Ching-Lung Lin

Jenn-Nan Wang

Abstract

In this paper we study the local behavior of a solution to the Stokes system with singular coefficients. One of the main results is the bound on the vanishing order of a nontrivial solution to the Stokes system, which is a quantitative version of the strong unique continuation prop- erty. Our proof relies on some delicate Carleman-type estimates. We first use these estimates to derive crucial optimal three-ball inequal- ities. Taking advantage of the optimality, we then derive an upper bound on the vanishing order of any nontrivial solution to the Stokes system from those three-ball inequalities.

1 Introduction

Assume that Ω is a connected open set containing 0 in Rⁿwith n ≥ 2. In this paper we are interested in the local behavior of (u, p) satisfying the following Stokes system:

( ∆u + A(x) · ∇u + ∇p = 0

divu = 0, (1.1)

where A is measurable satisfying

|A(x)| ≤ C₀| log |x||⁻³|x|⁻¹ ∀ x ∈ Ω (1.2)

∗Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan.

Email:cllin2@mail.ncku.edu.tw

†Department of Mathematics, Taida Institute of Mathematical Sciences, NCTS (Taipei), National Taiwan University, Taipei 106, Taiwan. Email: jn- wang@math.ntu.edu.tw

and A · ∇u = (A · ∇u₁, · · · , 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 [5]. On the other hand, when A(x) satisfies |A(x)| = O(|x|^−1+) with  >

0, the strong unique continuation property was proved by Regbaoui [18].

The results in [5] and [18] concern only the qualitative unique continuation theorem. In this work we aim to derive a quantitative estimate of the strong unique continuation for (1.1).

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 [3], [6], [7], [13], [15]. For the power of Laplacian, a quanti- tative estimate was obtained in [16]. We refer to [15] and references therein for the development of this investigation.

Since there is no equation for p in the Stokes system (1.1), to prove the unique continuation theorem for (1.1), one usually apply the divergence on the first equation and obtain

∆p + div(A(x) · ∇u) = 0. (1.3)

However, the first equation of (1.1) and (1.3) 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 [18]. Carleman-type estimates derived in [18] 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 [5]. It is tempting to derive doubling inequalities for (1.1) by (2.4) and (2.15) using the ideas in [15] or [16]. But this seems hard to reach with estimates (2.4), (2.15). One of the difficulties is the appearance of the parameter β on the right hand side of (2.15).

Even though we are not able to prove doubling inequalities for (1.1), we can derive certain three-ball inequalities which are optimal in the sense explained in [4] 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 inequality

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 vanishing order of a nontrivial solution from the optimal three-ball inequality. A recent result by Bourgain and Kenig [2] (more precisely, Kenig’s lecture notes for 2006 CNA Summer School [12]) indicates that this is indeed possible, 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 coefficients.

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 [8], [14], 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 < R₀ ≤ 1 such that B_R0 ⊂ Ω. Hereafter B_r denotes an open ball of radius r > 0 centered at the origin. Also, we let U (x) = [|x|⁴|∇p|²+ |x|²|p|²+ |u|²]^1/2.

Theorem 1.1 There exists a positive number ˜R < 1, depending only on n, such that if 0 < R₁ < R₂ < R₃ ≤ R₀ and R₁/R₃ < R₂/R₃ < ˜R, then

Z

|x|<R₂

|U |²dx ≤ C

Z

|x|<R₁

|U |²dx

τZ

|x|<R₃

|U |²dx

1−τ

(1.4)

for (u, p) ∈ (H¹(BR0))ⁿ⁺¹ satisfying (1.1) in BR0, where the constant C de- pends on R₂/R₃, n, and 0 < τ < 1 depends on R₁/R₃, R₂/R₃, 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 R1 and τ has the asymptotic (− log R₁)⁻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 [4].

Remark 1.3 We want to say a few words about the appearance of ∇p term in U . In the derivation of the three-ball inequality (1.4), it is crucial to control ∇u in a smaller region by quantities of u and p in a bigger region (see (3.1)). Roughly speaking, this is an interior estimate for ∇u. In view of the first equation of (1.1), ∇p needs be included in this estimate.

Theorem 1.4 Let (u, p) ∈ (H¹(B_R0))ⁿ⁺¹ be a nontrivial solution to (1.1), i.e, (u, p) 6= (0, 0), then there exist positive constants K and m, depending on n and (u, p), such that

Z

|x|<R

|U |²dx ≥ KR^m (1.5)

for all R sufficiently small.

Remark 1.5 Based on Theorem 1.1, the constants K and m in (1.5) are given by

K = Z

|x|<R3

|U |²dx and

m = ˜C log R

|x|<R3|U |²dx R

|x|<R₂|U |²dx

 , where ˜C is a positive constant depending on n and R2/R3.

Corollary 1.6 Let (u, p) ∈ (H_loc¹ (Ω))ⁿ× L²_loc(Ω) be a solution of (1.1) with A satisfying (1.2). Assume that (u, p) vanishes of infinite order at the origin, i.e., for all N > 0,

Z

|x|<R

(|u|²+ |p|²)dx = O(R^N) as R → 0. (1.6) Then (u, p) ≡ 0 in Ω.

This corollary is a small improvement of the strong unique continuation prop- erty for the Stokes system proved in [18] where |A(x)| = O(|x|^−1+) with

 > 0.

This paper is organized as follows. In Section 2, we derive suitable Carleman-type estimates. A technical interior estimate is proved in Section 3.

Section 4 is devoted to the proofs of Theorem 1.1, 1.4, and Corollary 1.6.

2 Carleman estimates

Similar to the arguments used in [9], we introduce polar coordinates in Rⁿ\{0} by setting x = rω, with r = |x|, ω = (ω₁, · · · , ω_n) ∈ Sⁿ⁻¹. Fur- thermore, using new coordinate t = log r, we can see that

∂

∂x_j = e^−t(ωj∂t+ Ωj), 1 ≤ j ≤ n,

where Ω_j is a vector field in Sⁿ⁻¹. We could check that the vector fields Ω_j satisfy

n

X

j=1

ω_jΩ_j = 0 and

n

X

j=1

Ω_jω_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<sub>`</sub> = e<sup>−2t</sup>(ωj∂t− ωj+ Ωj)(ω`∂t+ Ω`), 1 ≤ j, ` ≤ n.

and, therefore, the Laplacian becomes

e^2t∆ = ∂_t²+ (n − 2)∂_t+ ∆_ω, (2.1) where ∆_ω = Σⁿ_j=1Ω²_j denotes the Laplace-Beltrami operator on Sⁿ⁻¹. We recall that the eigenvalues of −∆_ω are k(k + n − 2), k ∈ N, and the corre- sponding eigenspaces are E_k, where E_k is the space of spherical harmonics of degree k. It follows that

Z Z

|∆_ωv|²dtdω =X

k≥0

k²(k + n − 2)² Z Z

|v_k|²dtdω (2.2)

and

X

j

Z Z

|Ω_jv|²dtdω =X

k≥0

k(k + n − 2) Z Z

|v_k|²dtdω, (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 L²(Sⁿ⁻¹).

The eigenvalues of Λ are k + ⁿ⁻²₂ 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 [17], 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 ϕβ is less singular than |x|^−β, For simplicity, we denote ψ(t) = t + log t², 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 r0 > 0 depending on n and a sufficiently large β₀ > 1 depending on n such that for all u ∈ U_r0 and β ≥ β₀, we have that

β Z

ϕ²_β(log |x|)⁻²|x|⁻ⁿ(|x|²|∇u|²+ |u|²)dx . Z

ϕ²_β|x|⁻ⁿ|x|⁴|∆u|²dx, (2.4) where U_r0 = {u ∈ C₀^∞(Rⁿ\ {0}) : supp(u) ⊂ B_r0}.

Proof. By the polar coordinate system described above, we have Z

ϕ²_β|x|⁴⁻ⁿ|∆u|²dx

= Z Z

e^−2βψ(t)e^4t|∆u|²dtdω

= Z Z

|e^−βψ(t)e^2t∆u|²dtdω. (2.5)

If we set u = e^βψ(t)v and use (2.1), then

e^−βψ(t)e^2t∆u = ∂_t²v + b∂_tv + av + ∆_ωv =: P_βv, (2.6) where a = (1 + 2t⁻¹)²β²+ (n − 2)β + 2(n − 2)t⁻¹β − 2t⁻²β and b = n − 2 + 2β + 4t⁻¹β. By (2.5) and (2.6), (2.4) holds if for t near −∞ we have

X

j+|α|≤1

β^3−2|α|

Z Z

|t|⁻²|∂_t^jΩ^αv|²dtdω ≤ ˜C₁ Z Z

|P_βv|²dtdω, (2.7)

where ˜C₁ is a positive constant depending on n.

From (2.6), using the integration by parts, for t < t₀ and β > β₀, where t₀ < −1 and β₀ > 0 depend on n, we have that

Z Z

|P_βv|²dtdω

= Z Z

|∂_t²v|²dtdω + Z Z

|b∂_tv|²dtdω + Z Z

|av|²dtdω + Z Z

|∆_ωv|²dtdω

− Z Z

∂_tb|∂_tv|²dtdω − 2 Z Z

a|∂_tv|²dtdω + Z Z

∂_t²a|v|²dtdω

− Z Z

∂_t(ab)|v|²dtdω + 2X

j

Z Z

|∂_tΩ_jv|²dtdω

+X

j

Z Z

∂_tb|Ω_jv|²dtdω − 2X

j

Z Z

a|Ω_jv|²dtdω

≥ Z Z

|∆_ωv|²dtdω + Z Z

{b²− ∂_tb − 2a}|∂_tv|²dtdω

+X

j

Z Z

{∂_tb − 2a}|Ω_jv|²dtdω + Z Z

{a²+ ∂_t²a − ∂_t(ab)}|v|²dtdω

≥ Z Z

|∆_ωv|²dtdω +X

j

Z Z

{−4t⁻²β − 2a}|Ω_jv|²dtdω

+ Z Z

{a²+ 11t⁻²β³}|v|²dtdω + Z Z

β²|∂tv|²dtdω. (2.8) In view of (2.8), using (2.2),(2.3), we see that

Z Z

|∆_ωv|²dtdω − 2X

j

Z Z

a|Ω_jv|²dtdω + Z Z

a²|v|²dtdω

= X

k≥0

Z Z

[a − k(k + n − 2)]²|v_k|²dtdω. (2.9)

Substituting (2.9) into (2.8) yields Z Z

|P_βv|²dtdω

≥ X

k≥0

Z Z

{11t⁻²β³ − 4t⁻²βk(k + n − 2) + [a − k(k + n − 2)]²}|v_k|²dtdω

+ Z Z

β²|∂_tv|²dtdω

= X

k,k(k+n−2)≥2β²

+ X

k,k(k+n−2)<2β²

Z Z

{11t⁻²β³− 4t⁻²βk(k + n − 2)

+[a − k(k + n − 2)]²}|v_k|²dtdω + Z Z

β²|∂_tv|²dtdω. (2.10) For k such that k(k + n − 2) < 2β², we have

11t⁻²β³− 4t⁻²βk(k + n − 2) ≥ t⁻²β³+ t⁻²βk(k + n − 2). (2.11) On the other hand, if 2β² < k(k + n − 2), then, by taking t even smaller, if necessary, we get that

−4t⁻²βk(k + n − 2) + [a − k(k + n − 2)]² & t⁻²βk(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 the pressure p, we need another Carleman estimate. The derivation here follows the line in [18].

Lemma 2.2 There exists a sufficiently small number t₀ < 0 depending on n such that for all u ∈ V_t0, β > 1, we have that

X

j+|α|≤1

β^{1−2(j+|α|)} Z Z

t⁻²ϕ²_β|∂_t^jΩ^αu|²dtdω . Z Z

ϕ²_β|L⁻u|²dtdω, (2.13)

where Vt0 = {u(t, ω) ∈ C₀^∞((−∞, t0) × Sⁿ⁻¹)}.

Proof. If we set u = e^βψ(t)v, then simple integration by parts implies Z Z

ϕ²_β|L⁻u|²dtdω

= Z Z

|∂_tv − Λv + βv + 2βt⁻¹v + (n − 2)v/2|²dtdω

= Z Z

|∂_tv|²dtdω + Z Z

| − Λv + βv + 2βt⁻¹v + (n − 2)v/2|²dtdω +β

Z Z

t⁻²|v|²dtdω.

By the definition of Λ, we have Z Z

| − Λv + βv + 2βt⁻¹v + (n − 2)v/2|²dtdω

= X

k≥0

Z Z

| − kv_k+ βv_k+ 2βt⁻¹v_k|²dtdω

= X

k≥0

Z Z

(−k + β + 2βt⁻¹)²|v_k|²dtdω,

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

ϕ²_β|L⁻u|²dtdω

= Z Z

|∂_tv|²dtdω + Σ_k≥0 Z Z

[(−k + β + 2βt⁻¹)²+ βt⁻²]|v_k|²dtdω

&

Z Z

|∂tv|²dtdω + Σk≥0

Z Z

(β⁻¹t⁻²k(k + n − 2) + βt⁻²)|vk|²dtdω.

(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 t₁ < −2 depending on n such that for all u ∈ V_t1, g = (g₀, g₁, · · · , g_n) ∈ (V_t1)ⁿ⁺¹ and β > 0, we have that

Z Z

ϕ²_β|u|²dtdω . Z Z

ϕ²_β(|L⁺u + ∂_tg₀+

n

X

j=1

Ω_jg_j|²+ kgk²)dtdω.

Proof. This lemma can be proved by exactly the same arguments used in

Lemma 2.2 of [18]. So we omit the proof here.

2

Lemma 2.4 There exist a sufficiently small number r₁ > 0 depending on n and a sufficiently large number β₁ > 2 depending on n such that for all w ∈ U_r1 and f = (f₁, · · · , f_n) ∈ (U_r1)ⁿ, β ≥ β₁, we have that

Z

ϕ²_β(log |x|)²(|x|⁴⁻ⁿ|∇w|²+ |x|²⁻ⁿ|w|²)dx . β

Z

ϕ²_β(log |x|)⁴|x|²⁻ⁿ[(|x|²∆w + |x|divf )²+ kf k²]dx, (2.15) where U_r1 is defined as in Lemma 2.1.

Proof. Replacing β by β + 2 in (2.15), we see that it suffices to prove Z

ϕ²_β(log |x|)⁻²(|x|²|∇w|²+ |w|²)|x|⁻ⁿdx . β

Z

ϕ²_β[(|x|²∆w + |x|divf )²+ kf k²]|x|⁻ⁿdx. (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

t⁻²ϕ²_β|∂_t^jΩ^αu|²dtdω

. β Z Z

ϕ²_β(|L⁺L⁻w + ∂_t(

n

X

j=1

ω_jf_j) +

n

X

j=1

Ω_jf_j|²+ kf k²)dtdω.(2.17) Applying Lemma 2.3 to u = L⁻w and g = (Pn

j=1ω_jf_j, f₁, · · · , f_n) yields β

Z Z

ϕ²_β|L⁻w|²dtdω . β

Z Z

ϕ²_β(|L⁺L⁻w + ∂_t(

n

X

j=1

ω_jf_j) +

n

X

j=1

Ω_jf_j|²+ kf k²)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_loc¹ (Ω))ⁿ⁺¹ be a solution of (1.1). Then for any 0 < a₃ < a₁ < a₂ < a₄ such that B_a4r ⊂ Ω and |a₄r| < 1, we have

Z

a1r<|x|<a2r

|x|²|∇u|²dx ≤ C Z

a3r<|x|<a4r

(|x|⁴|∇p|²+ |u|²)dx, (3.1) where the constant C is independent of r and (u, p).

Proof. Let X = B_a4r\B_a3r and d(x) be the distant from x ∈ X to Rⁿ\X.

By the elliptic regularity, we obtain from (1.1) that u ∈ H_loc² (Ω\{0}). It is trivial that

kvk_H¹_(Rⁿ₎ . k∆vkL²(Rⁿ)+ kvk_L²_(Rⁿ₎ (3.2) for all v ∈ H²(Rⁿ). By changing variables x → B⁻¹x in (3.2), we will have

P

|α|≤1B^2−|α|kD^αvk_L²_(Rⁿ₎ . (k∆vkL²(Rⁿ)+ B²kvk_L²_(Rⁿ₎) (3.3) for all v ∈ H²(Rⁿ). To apply (3.3) on u, we need to cut-off u. So let ξ(x) ∈ C₀^∞(Rⁿ) 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 (1.1) to get that

B² Z

|x−y|≤d(y)/4

|∇u|²dx .

Z

|x−y|≤d(y)/2

(|A|²+ d(y)⁻²)|∇u|²dx + Z

|x−y|≤d(y)/2

|∇p|²dx +(B⁴+ d(y)⁻⁴)

Z

|x−y|≤d(y)/2

|u|²dx. (3.4)

Now taking B = M d(y)⁻¹ for some positive constant M and multiplying d(y)⁴ on both sides of (3.4), we have

M²d(y)² Z

|x−y|≤d(y)/4

|∇u|²dx .

Z

|x−y|≤d(y)/2

(d(y)⁴|A|²+ d(y)²)|∇u|²dx +

Z

|x−y|≤d(y)/2

d(y)⁴|∇p|²dx + (M⁴+ 1) Z

|x−y|≤d(y)/2

|u|²dx. (3.5)

Integrating d(y)⁻ⁿdy over X on both sides of (3.5) and using Fubini’s Theorem, we get that

M² Z

X

Z

|x−y|≤d(y)/4

d(y)²⁻ⁿ|∇u|²dydx .

Z

X

Z

|x−y|≤d(y)/2

(d(y)²+ d(y)⁴|A|²)|∇u(x)|²d(y)⁻ⁿdydx +

Z

X

Z

|x−y|≤d(y)/2

d(y)⁴⁻ⁿ|∇p|²dydx +M⁴

Z

X

Z

|x−y|≤d(y)/2

|u|²d(y)⁻ⁿdydx. (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)⁻ⁿdy ≥ (3/4)ⁿR

|x−y|≤d(x)/6d(x)⁻ⁿdy ≥ 8⁻ⁿR

|y|≤1dy, R

|x−y|≤d(y)/2d(y)⁻ⁿdy ≤ 2ⁿR

|x−y|≤3d(x)/4d(x)⁻ⁿdy ≤ (3/2)ⁿR

|y|≤1dy (3.9)

Combining (3.6)–(3.9), we obtain M²

Z

X

d(x)²|∇u|²dx .

Z

X

(d(x)²+ d(x)⁴|A|²)|∇u(x)|²dx + Z

X

d(x)⁴|∇p|²dx +M⁴

Z

X

|u|²dx. (3.10)

In view of (1.2), we can take M large enough to absorb the first term on the right hand side of (3.10). Thus we conclude that

Z

X

d(x)²|∇u|²dx . Z

X

(d(x)⁴|∇p|²+ |u|²)dx. (3.11) We recall that X = B_a4r\B_a3r and note that d(x) ≥ ˜Cr if x ∈ B_a2r\B_a1r, where ˜C is independent of r. Hence, (3.1) is an easy consequence of (3.11).

2

4 Proof of Theorem 1.1 and Theorem 1.4

This section is devoted to the proofs of Theorem 1.1 and Theorem 1.4. To begin, we first consider the case where 0 < R₁ < R₂ < R < 1 and B_R ⊂ Ω.

The small constant R will be determined later. Since (u, p) ∈ (H¹(B_R0))ⁿ⁺¹, the elliptic regularity theorem implies u ∈ H_loc² (B_R0\ {0}). Therefore, to use estimate (2.4), we simply cut-off u. So let χ(x) ∈ C₀^∞(Rⁿ) satisfy 0 ≤ χ(x) ≤ 1 and

χ(x) =







0, |x| ≤ R₁/e,

1, R₁/2 < |x| < eR₂, 0, |x| ≥ 3R₂,

where e = exp(1). We remark that we first choose a small R such that R ≤ min{r₀, r₁}/3 = ˜R₀, where r₀ and r₁ are constants appeared in (2.4) and (2.15). Hence ˜R0 depends on n. It is easy to see that for any multiindex

α (

|D^αχ| = O(R^−|α|₁ ) for all R₁/e ≤ |x| ≤ R₁/2

|D^αχ| = O(R^−|α|₂ ) for all eR₂ ≤ |x| ≤ 3R₂. (4.1)

Applying (2.4) to χu gives C1β

Z

(log |x|)⁻²ϕ²_β|x|⁻ⁿ(|x|²|∇(χu)|²+|χu|²)dx ≤ Z

ϕ²_β|x|⁻ⁿ|x|⁴|∆(χu)|²dx.

(4.2) From now on, C1, C2, · · · denote general constants whose dependence will be specified whenever necessary. Next applying (2.15) to w = χp and f =

|x|χA · ∇u, we get that C₂

Z

ϕ²_β(log |x|)²(|x|⁴⁻ⁿ|∇(χp)|²+ |x|²⁻ⁿ|χp|²)dx

≤ β Z

ϕ²_β(log |x|)⁴|x|²⁻ⁿ[|x|²∆(χp) + |x|div(|x|χA · ∇u)]²dx +β

Z

ϕ²_β(log |x|)⁴|x|²⁻ⁿk|x|χA · ∇uk²dx. (4.3) Multiplying by M₁ on (4.2) and combining (4.3), we obtain that

M1β Z

R1/2<|x|<eR2

(log |x|)⁻²ϕ²_β|x|⁻ⁿ(|x|²|∇u|²+ |u|²)dx +

Z

R1/2<|x|<eR2

(log |x|)²ϕ²_β|x|⁻ⁿ(|x|⁴|∇p|²+ |x|²|p|²)dx

≤ M1β Z

ϕ²_β(log |x|)⁻²|x|⁻ⁿ(|x|²∇(χu)|²+ |χu|²)dx +

Z

(log |x|)²ϕ²_β|x|⁻ⁿ(|x|⁴|∇(χp)|²+ |x|²|χp|²)dx

≤ M₁C₃ Z

ϕ²_β|x|⁻ⁿ|x|⁴|∆(χu)|²dx +βC₃

Z

(log |x|)⁴ϕ²_β|x|⁻ⁿ[|x|³∆(χp) + |x|²div(|x|χA · ∇u)]²dx +βC₃

Z

(log |x|)⁴ϕ²_β|x|⁻ⁿk|x|²χA · ∇uk²dx. (4.4)

By (1.1), (1.2), (1.3), and estimates (4.1), we deduce from (4.4) that M1β

Z

R1/2<|x|<eR2

(log |x|)⁻²ϕ²_β|x|⁻ⁿ(|x|²|∇u|²+ |u|²)dx +

Z

R1/2<|x|<eR2

(log |x|)²ϕ²_β|x|⁻ⁿ(|x|⁴|∇p|²+ |x|²|p|²)dx

≤ C4M1

Z

R1/2<|x|<eR2

(log |x|)⁻²ϕ²_β|x|⁻ⁿ|x|²|∇u|²dx +C₄M₁

Z

R1/2<|x|<eR2

ϕ²_β|x|⁻ⁿ|x|⁴|∇p|²dx +C₄β

Z

R1/2<|x|<eR2

(log |x|)⁻²ϕ²_β|x|⁻ⁿ|x|²|∇u|²dx +C₄M₁

Z

{R1/e≤|x|≤R1/2}∪{eR2≤|x|≤3R2}

ϕ²_β|x|⁻ⁿ| ˜U |²dx +C₄β

Z

{R1/e≤|x|≤R1/2}∪{eR2≤|x|≤3R2}

(log |x|)⁴ϕ²_β|x|⁻ⁿ| ˜U |²dx, (4.5)

where | ˜U (x)|² = |x|⁴|∇p|²+|x|²|p|²+|x|²|∇u|²+|u|² and the positive constant C4 only depends on n.

Now letting M₁ = 2 + 2C₄, β ≥ 2 + 2C₄, and R small enough such that (log(eR))² ≥ 2C₄M₁, 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 ˜R₁ > 0, depending on n, such that for all β > 0, both (log |x|)⁻²|x|⁻ⁿϕ²_β(|x|) and (log |x|)⁴|x|⁻ⁿϕ²_β(|x|) are decreas- ing functions in 0 < |x| < ˜R₁. So we choose a small R < ˜R₂, where R˜2 = min{exp(−2√

2C4M1− 1), ˜R1/3, ˜R0}. It is clear that ˜R2 depends on n.

With the choices described above, we obtain from (4.5) that

R⁻ⁿ₂ (log R₂)⁻²ϕ²_β(R₂) Z

R1/2<|x|<R2

| ˜U |²dx

≤ Z

R1/2<|x|<eR2

(log |x|)⁻²ϕ²_β|x|⁻ⁿ| ˜U |²dx

≤ C₅β Z

{R1/e≤|x|≤R1/2}∪{eR2≤|x|≤3R2}

(log |x|)⁴ϕ²_β|x|⁻ⁿ| ˜U |²dx

≤ C₅β(log(R₁/e))⁴(R₁/e)⁻ⁿϕ²_β(R₁/e) Z

{R₁/e≤|x|≤R1/2}

| ˜U |²dx +C₅β(log(eR₂))⁴(eR₂)⁻ⁿϕ²_β(eR₂)

Z

{eR2≤|x|≤3R2}

| ˜U |²dx. (4.6)

Using (3.1), we can control |∇u| terms on the right hand side of (4.6). In other words, it follows from (3.1) that

R^−2β−n₂ (log R₂)^−4β−2 Z

R1/2<|x|<R2

|U |²dx

≤ C₆2^2β+n(log(R₁/e))⁴(R₁/e)⁻ⁿϕ²_β(R₁/e) Z

{R1/4≤|x|≤R1}

|U |²dx +C₆2^2β+n(log(eR₂))⁴(eR₂)⁻ⁿϕ²_β(eR₂)

Z

{2R₂≤|x|≤4R₂}

|U |²dx

= C₆2^2β+n(log(R₁/e))^−4β+4(R₁/e)^−2β−n Z

{R1/4≤|x|≤R1}

|U |²dx +C62^2β+n(log(eR2))^−4β+4(eR2)^−2β−n

Z

{2R₂≤|x|≤4R₂}

|U |²dx. (4.7)

Recall that |U (x)|² = |x|⁴|∇p|²+ |x|²|p|²+ |u|². Replacing 2β + n by β, (4.7) becomes

R₂^−β(log R₂)^{−2β+2n−2} Z

R1/2<|x|<R2

|U |²dx

≤ C₇2^β(log(R₁/e))^−2β+2n+4(R₁/e)^−β Z

{R₁/4≤|x|≤R1}

|U |²dx +C₇2^β(log(eR₂))^−2β+2n+4(eR₂)^−β

Z

{2R2≤|x|≤4R2}

|U |²dx. (4.8)

Dividing R^−β₂ (log R₂)^{−2β+2n−2} on the both sides of (4.8) and providing β ≥ n + 2, we have that

Z

R1/2<|x|<R2

|U |²dx

≤ C₈(log R₂)⁶(2eR₂/R₁)^β Z

{R₁/4≤|x|≤R1}

|U |²dx +C₈(log R₂)⁶(2/e)^β[(log R₂/ log(eR₂))²]^β−n−2

Z

{2R2≤|x|≤4R2}

|U |²dx

≤ C₈(log R₂)⁶(2eR₂/R₁)^β Z

{R1/4≤|x|≤R1}

|U |²dx +C₈(log R₂)⁶(4/5)^β

Z

{2R₂≤|x|≤4R₂}

|U |²dx. (4.9)

In deriving the second inequality above, we use the fact that log R₂

log(eR₂) → 1 as R₂ → 0, and thus

2

e · log R2

log(eR₂) < 4 5

for all R₂ < ˜R₃, where ˜R₃is sufficiently small. We now take ˜R = min{ ˜R₂, ˜R₃}, which depends on n.

Adding R

|x|<R₁/2|U |²dx to both sides of (4.9) leads to Z

|x|<R₂

|U |²dx ≤ C9(log R2)⁶(2eR2/R1)^β Z

|x|≤R₁

|U |²dx +C₉(log R₂)⁶(4/5)^β

Z

|x|≤1

|U |²dx. (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(R1, R2) = log(2eR2/R1), B = log(5/4),

(4.10) becomes Z

|x|<R₂

|U |²dx

≤ C₉(log R₂)⁶n

exp(Eβ) Z

|x|<R1

|U |²dx + exp(−Bβ) Z

|x|<1

|U |²dxo . (4.11) To further simplify the terms on the right hand side of (4.11), we consider two cases. If R

|x|<R1|U |²dx 6= 0 and exp (E ˜β)

Z

|x|<R1

|U |²dx < exp (−B ˜β) Z

|x|<1

|U |²dx,

then we can pick a β > ˜β such that exp (Eβ)

Z

|x|<R₁

|U |²dx = exp (−Bβ) Z

|x|<1

|U |²dx.

Using such β, we obtain from (4.11) that Z

|x|<R2

|U |²dx

≤ 2C₉(log R₂)⁶exp (Eβ) Z

|x|<R₁

|U |²dx

= 2C₉(log R₂)⁶

Z

|x|<R1

|U |²dx

_E+B^B Z

|x|<1

|U |²dx

_E+B^E

. (4.12)

IfR

|x|<R1|U |²dx = 0, then letting β → ∞ in (4.11) we haveR

|x|<R2|U |²dx = 0 as well. The three-ball inequality obviously holds.

On the other hand, if exp (−B ˜β)

Z

|x|<1

|U |²dx ≤ exp (E ˜β) Z

|x|<R1

|U |²dx,

then we have Z

|x|<R2

|U |²dx

≤

Z

|x|<1

|U |²dx

_E+B^B Z

|x|<1

|U |²dx

_E+B^E

≤ exp (B ˜β)

Z

|x|<R₁

|U |²dx

_E+B^B Z

|x|<1

|U |²dx

_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 |²dx ≤ C₁₀

Z

|x|<R1

|U |²dx

_E+B^B Z

|x|<1

|U |²dx

_E+B^E

. (4.14) It is readily seen that _E+B^B ≈ (log(1/R₁))⁻¹ when R₁ tends to 0.

Now for the general case, we consider 0 < R₁ < R₂ < R₃ < 1 with R₁/R₃ < R₂/R₃ ≤ ˜R, where ˜R is given as above. By scaling, i.e. defining u(y) := u(Rb 3y), p(y) := Rb 3p(R3y) and bA(y) = A(R3y), (4.14) becomes

Z

|y|<R₂/R3

| bU (y)|²dy ≤ C₁₁( Z

|y|<R₁/R3

| bU (y)|²dy)^τ( Z

|y|<1

| bU (y)|²dy)^1−τ, (4.15) where

τ = B/[E(R₁/R₃, R₂/R₃) + B],

C₁₁ = max{2C₉(log R₂/R₃)⁶, exp ( ˜β log(5/4))},

and bU (y) = |y|⁴|∇p(y)|b ²+ |y|²|p(y)|b ²+ |bu(y)|². Note that C11 is independent of R₁. Restoring the variable x = R₃y in (4.15) gives

Z

|x|<R2

|U |²dx ≤ C₁₁( Z

|x|<R1

|U |²dx)^τ( Z

|x|<R3

|U |²dx)^1−τ. The proof of Theorem 1.1 is complete.

We now turn to the proof of Theorem 1.4. We fix R₂, R₃ in Theorem 1.1 and define











eu(x) := u(x)/qR

|x|<R2|U |²dx, p(x) := p(x)/e qR

|x|<R2|U |²dx,

V (x) = |x|⁴|∇p(x)|e ²+ |x|²|p(x)|e ²+ |eu(x)|².

Note that R

|x|<R2|V |²dx = 1. From the three-ball inequality (1.4), we have that

1 ≤ C(

Z

|x|<R1

|V |²dx)^τ( Z

|x|<R3

|V |²dx)^1−τ. (4.16) Raising both sides by 1/τ yields that

Z

|x|<R3

|V |²dx ≤ ( Z

|x|<R1

|V |²dx)(C Z

|x|<R3

|V |²dx)^1/τ. (4.17)

In view of the formula for τ , we can deduce from (4.17) that Z

|x|<R₃

|V |²dx ≤ ( Z

|x|<R₁

|V |²dx)(1/R₁)^{C log(˜}

R

|x|<R3|V |²dx)

, (4.18)

where ˜C is a positive constant depending on n and R₂/R₃. Consequently, (4.18) is equivalent to

( Z

|x|<R₃

|U |²dx)R^m₁ ≤ Z

|x|<R₁

|U |²dx for all R₁ sufficiently small, where

m = ˜C log R

|x|<R3|U |²dx R

|x|<R2|U |²dx

 .

We now end the proof of Theorem 1.4.

Finally, we come to the proof of Corollary 1.6. In view of Theorem 1.4, it is enough to show that

p ∈ H_loc¹ (Ω) (4.19)

and for all N > 0 Z

|x|<R

|∇p|²dx = O(R^N) as R → 0. (4.20)

It is only a matter of checking that the arguments used in [18, page 1898-1899]

can be applied to prove (4.19) and (4.20). To avoid unnecessary repetition, we only sketch the main steps here. By virtue of (1.3), u ∈ H_loc¹ (Ω), and (1.2), we get that p ∈ H_loc¹ (Ω \ {0}) by elliptic regularity. Using elliptic

regularity and the first equation of (1.1), we have u ∈ H_loc² (Ω \ {0}). By the vanishing assumption (1.6), we can derive that

Z

R<|x|<2R

|∇p|²dx = O(R^N) as R → 0

for all N > 0. It follows that p is the sum of a function in H_loc¹ (Ω) and a distribution supported at 0. But no distribution supported at 0 is in L²_loc(Ω).

Thus, p ∈ H_loc¹ (Ω) and (4.20) holds.

Acknowledgements

The authors were supported in part by the National Science Council of Tai- wan.

References

[1] G. Alessandrini, E. Beretta, E. Rosset, and S. Vessella, Optimal stabil- ity for elliptic boundary value problems with unknow boundaries, Ann.

Scuola Norm. Sup. Pisa Cl. Sci, 29 (2000), 755-786.

[2] J. Bourgain and C. Kenig, On localization in the continuous Anderson- Bernoulli model in higher dimension, Invent. Math. 161 (2005), 1389- 426.

[3] H. Donnelly and C. Fefferman, Nodal sets of eigenfunctions on Rieman- nian manifolds, Invent. Math. 93 (1988), 161-183.

[4] L. Escauriaza, F.J. Fern´andez, and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223.

[5] C. Fabre and G. Lebeau, Prolongement unique des solutions de l’´equation de Stokes, Comm. in PDE, 21 (1996), 573-596.

[6] N. Garofalo and F.H. Lin, Monotonicity properties of variational inte- grals, A_p weights and unique continuation, Indiana Univ. Math. J. 35 (1986), 245-267.

[7] N. Garofalo and F.H. Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math., 40, 347-366, 1987.

[8] R. Hardt and L. Simon, Nodal sets for solutions of elliptic equations, J.

Diff. Geom., 30 (1989), 505-522.

[9] L. H¨ormander, Uniqueness theorems for second order elliptic differential equations, Comm. in P.D.E. 8, No. 1, (1983), 21-64.

[10] L. H¨ormander, ”The analysis of linear partial differential operators”, Vol. 3, Springer-Verlag, Berlin/New York, 1985.

[11] D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrodinger operators. With an appendix by E. M. Stein, Ann. of Math. (2), 121 (1985), 463-494.

[12] C. Kenig, Lecture Notes for 2006 CNA Summer School: Probabilistic and Analytical Perspectives on Contemporary PDEs, Center for Nonlin- ear Analysis, Carnegie Mellon University.

[13] F.H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136.

[14] F.H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308.

[15] C.L. Lin, G. Nakamura and J.N. Wang Quantitative uniqueness for sec- ond order elliptic operators with strongly singular coefficients, Preprint.

[16] C.L. Lin, S. Nagayasu and J.N. Wang Quantitative uniqueness for the power of Laplacian with singular coefficients, Preprint.

[17] R. Regbaoui, Strong uniqueness for second order differential operators, J. Diff. Eq. 141 (1997), 201–217.

[18] R. Regbaoui, Strong unique continuation for stokes equations, Comm.

in PDE 24 (1999), 1891-1902.