Increasing stability in an inverse problem for the acoustic equation

In this paper, we study the issue of stability for determining the refractive index in the acoustic equation by boundary measurements. It is well known that this inverse problem is ill posed. However, one anticipates that the stability will increase if one increases the frequency. This phenomenon was observed numerically in the inverse obstacle scattering problem [6]. Several rigorous justifications of the increasing stability phenomena in different settings were obtained by Isakov et al [8–10, 13, 14]. In particular, in [10], Isakov considered the Helmholtz equation with a potential

$$\begin{equation} {-}\Delta u-k^2u+c(x)u=0\quad \mbox{in}\quad \Omega . \end{equation} \tag{ 1.1 }$$

He obtained stability estimates of determining c(x) by the Dirichlet-to-Neumann map for different ranges of ks. All of these results demonstrate the increasing stability phenomena in k. For the case of the inverse source problem for Helmholtz equation and a homogeneous background it was shown in [3] that the ill-posedness of the inverse problem decreases as the frequency increases.

In this paper, we study the acoustic wave equation. Let $\Omega \subset \mathbb {R}^{n}$ be a bounded domain, where n ⩾ 3. Let ∂Ω be smooth. We assume q ∈ H^s(Ω) for some s > (n/2) + 1 and consider the equation

$$\begin{equation} ( \Delta + k^{2} q(x)) u(x) = 0\quad \mbox{in}\quad \Omega , \end{equation} \tag{ 1.2 }$$

where the function q(x) is the refractive index. Since the kernel of the operator Δ + k²q(x) on H¹₀(Ω) is not necessarily trivial, we define the boundary measurements to be the Cauchy data corresponding to (1.2)

$$\begin{equation*} \mathcal {C}_{q} = \left\lbrace \left( u|_{\partial \Omega }, \frac{\partial u}{\partial \nu } \biggr |_{\partial \Omega } \right) , \ \mbox{where} \ u \ \mbox{is a solution to (1.2)} \right\rbrace . \end{equation*}$$

Hereafter, ν is the unit outer normal vector of ∂Ω. Assume that $\mathcal {C}_{q_1}$ and $\mathcal {C}_{q_2}$ are two Cauchy data associated with refraction indices q₁ and q₂, respectively. To measure the distance between two Cauchy data, we define

$$\begin{eqnarray*} &&\fl \mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2} ) = \max \left\lbrace \max _{(f,g) \in \mathcal {C}_{q_{1}}} \min _{( \widetilde{f}, \widetilde{g} ) \in \mathcal {C}_{q_{2}}} \frac{\Vert (f,g) - ( \widetilde{f}, \widetilde{g} ) \Vert _{ H^{1/2} \oplus H^{-1/2} }}{\Vert (f,g) \Vert _{H^{1/2} \oplus H^{-1/2}}},\right.\\ &&\left. \max _{(f,g) \in \mathcal {C}_{q_{2}}} \min _{( \widetilde{f}, \widetilde{g} ) \in \mathcal {C}_{q_{1}}} \frac{\Vert (f,g) - ( \widetilde{f} , \widetilde{g} ) \Vert _{H^{1/2} \oplus H^{-1/2} }}{\Vert (f,g) \Vert _{H^{1/2} \oplus H^{-1/2}}} \right\rbrace , \end{eqnarray*}$$

where

$$\begin{equation*} \Vert (f,g) \Vert _{H^{1/2} \oplus H^{-1/2}} =\bigl ( \Vert f \Vert _{H^{1/2} ( \partial \Omega )}^2 +\Vert g \Vert _{H^{-1/2} ( \partial \Omega ) }^2 \bigr )^{1/2}. \end{equation*}$$

The uniqueness of this inverse problem is well known [16]. This inverse problem is notoriously ill posed. For this aspect, Alessandrini proved that the stability estimate for this problem is of log type [1] and Mandache showed that the log-type stability is optimal [11]. In this paper, we would like to focus on how the stability behaves when the frequency k increases. Now we state the main result.

Theorem 1.1. Assume that q₁(x) and q₂(x) are two refraction indices with associated Cauchy data $\mathcal {C}_{q_1}$ and $\mathcal {C}_{q_2}$, respectively. Let s > (n/2) + 1, M > 0. Suppose $\Vert q_{l} \Vert _{H^{s} ( \Omega )} \le M$ (l = 1, 2) and $\mathop {\mathrm{supp}}( q_{1} - q_{2} ) \subset \Omega$. Denote by $\widetilde{q}$ the zero extension of q₁ − q₂. Then there exists a constant C₁, depending only on n, s and Ω, such that if k² ⩾ 1/(C₁M) and $\mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2}) \le 1 / e$ then

$$\begin{equation} \fl \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} \le {C}{k^{2}} \exp ( C k^{2} ) \mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2}) + C \left( k^{2} + \log \frac{1}{\mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2})} \right)^{- ( 2 s - n )} \end{equation} \tag{ 1.3 }$$

holds, where C > 0 depends only on n, s, Ω, M and $\mathop {\mathrm{supp}}( q_{1} - q_{2} )$.

Remark 1.2. 

• (1)  
  The estimate (1.3) consists of two parts—Lipschitz and logarithmic estimates. As k increases, the logarithmic part decreases and the Lipschitz part becomes dominant. In other words, the ill posedness is alleviated when k is large.
• (2)  
  We would like to remark on the constant Ck²exp (Ck²) appearing in the Lipschitz part of (1.3). k² comes from k²q in the equation, which appears naturally (see also [10, equation (8)]), while exp (Ck²) is due to the fact that we use the complex geometrical optics solutions in the proof. Even so, we expect that there is an exponential growth of the constant with frequency since we do not assume any geometrical restriction on q(x) other than regularity. We do not know whether the constant Ck²exp (Ck²) is optimal. For the wave equation it has been shown by Burq for the obstacle problem [5] that the local energy decay is log-slow and this is due to the presence of trapped rays. Note that in our case we can have trapped rays. For the case of simple sound speeds, we expect that there is no exponential increase in the constant. In [15], a Hölder stability estimate was obtained for the hyperbolic DN map for generic simple metrics. For very general metrics, there is no known modulus of continuity for the hyperbolic DN map; see [2] for convergence results.However, in practice, k is fixed and so is the constant. Therefore, one should expect to obtain a better resolution of q from boundary measurements when the chosen k is large.
• (3)  
  Unlike the result in [10, theorem 2.1] (for equation (1.1)) where the stability estimates were derived in different ranges of k, estimate (1.3) is valid for all the range of k provided k² ⩾ 1/(C₁M). Similar to our arguments here, the analysis in [10] relies on complex geometrical optics solutions and Alessandrini's identity. We also want to point out that in [10] the constant appearing in the Lipschitz part of the stability estimate grows polynomially in k (see (8) there).

We consider the problem in dimension n ⩾ 3 here. For n = 2, a global uniqueness for the Schrödinger equation with a general potential was proved by Bukhgeim [4] who used complex geometrical optics solutions with quadratic phases in the proof. Based on Bukhgeim's result, a logarithmic-type stability estimate was derived by Novikov and Santacesaria [12]. However, the phenomenon of increasing stability in the wave number k for (1.1) and (1.2) is yet to be verified. This is of course an interesting open problem.

We would also like to say a few words on performing our measurements in practice. For simplicity, we assume that n = 3 and q(x) = 1 outside of a ball B. Let v = v(x, ω, k) be the total field satisfying the following scattering problem:

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}}\Delta v+k^2q(x)v=0\qquad \mbox{in}\quad \mathbb {R}^3,\\\ms v={\rm e}^{{\rm i}kx\cdot \omega }+v^s,\\\ms \lim\limits_{r\rightarrow \infty }r \left( \dsty\frac{\partial v^s}{\partial r}-{\rm i}kv^s \right)=0,\end{array} \right. \end{equation} \tag{ 1.4 }$$

where |ω| = 1 and r = |x|. We now choose Ω such that $\bar{B}\subset \Omega$ and measure the Cauchy data of v on ∂Ω for all v solving (1.4). Using the denseness property of span{v(x, ω, ·)} on the solution set of (1.2) in Ω, we can determine, at least in theory, the Cauchy data $\mathcal {C}_q$ for (1.2).

The proof of theorem 1.1 makes use of Alessandrini's arguments [1] and the CGO solutions constructed in [16]. The main task is to keep track of how k appears in the proof of the stability estimates.

In this section, we construct CGO solutions to the equation (1.2) by using the idea in [16]. The main point is to express the dependence of constants on k explicitly. We first state two easy consequences from the results in [16].

Lemma 2.1 (see [16, proposition 2.1 and corollary 2.2]). Let s ⩾ 0 be an integer. Let ε₀ > 0. Let $\xi \in \mathbb {C}^{n}$ satisfy ξ · ξ = 0 and |ξ| ⩾ ε₀. Then for any f ∈ H^s(Ω) there exists w ∈ H^s(Ω) such that w is a solution to

$$\begin{equation*} \Delta w + \xi \cdot \nabla w = f \quad \mbox{in}\quad \Omega \end{equation*}$$

and satisfies the estimate

$$\begin{equation*} \Vert w \Vert _{H^{s} ( \Omega )} \le \frac{C_{0}}{| \xi |} \Vert f \Vert _{H^{s} ( \Omega )} , \end{equation*}$$

where a positive constant C₀ depends only on n, s, ε₀ and Ω.

By using this lemma, we can obtain a solution to the equation

$$\begin{equation} \Delta \psi + \xi \cdot \nabla \psi + g \psi = f \end{equation} \tag{ 2.1 }$$

satisfying some decaying property as in the following lemma.

Lemma 2.2 ([16, theorem 2.3 and corollary 2.4]). Let s > n/2 be an integer. Let ε₀ > 0. Let $\xi \in \mathbb {C}^{n}$ satisfy ξ · ξ = 0 and |ξ| ⩾ ε₀. Let f, g ∈ H^s(Ω). Then there exists C₁ > 0 depending only on n, s, ε₀ and Ω such that if

$$\begin{equation*} | \xi | \ge C_{1} \Vert g \Vert _{H^{s} ( \Omega )} \end{equation*}$$

then there exists a solution ψ ∈ H^s(Ω) to the equation (2.1) satisfying the estimate

$$\begin{equation*} \Vert \psi \Vert _{H^{s} ( \Omega )} \le \frac{2 C_{0}}{| \xi |} \Vert f \Vert _{H^{s} ( \Omega )} , \end{equation*}$$

where C₀ is the positive constant in lemma 2.1.

The CGO solutions needed are constructed as follows.

Proposition 2.3. Let s > n/2 be an integer. Let ε₀ > 0. Let $\xi \in \mathbb {C}^{n}$ satisfy ξ · ξ = 0 and |ξ| ⩾ ε₀. Define the constants C₀ and C₁ as in lemma 2.2. Then if

$$\begin{equation*} | \xi | \ge C_{1} k^{2} \Vert q \Vert _{H^{s} ( \Omega )} \end{equation*}$$

then there exists a solution u to the equation (1.2) with the form of

$$\begin{equation} u(x) = \exp \left( \frac{\xi }{2} \cdot x \right) ( 1 + \psi (x)) , \end{equation} \tag{ 2.2 }$$

where ψ has the estimate

$$\begin{equation*} \Vert \psi \Vert _{H^{s} ( \Omega )} \le \frac{2 C_{0} k^{2}}{| \xi |} \Vert q \Vert _{H^{s} ( \Omega )} . \end{equation*}$$

Proof. Substituting (2.2) into (1.2), we have

$$\begin{equation*} \Delta \psi + \xi \cdot \nabla \psi + k^{2} q \psi = - k^{2} q. \end{equation*}$$

Then by lemma 2.2, we obtain this proposition. □

This section is devoted to the proof of theorem 1.1. Let u₁ and u₂ be the solutions of (1.2) corresponding to q₁ and q₂, respectively. The following inequality was proved in [7] (see (4.3.12) there).

Proposition 3.1. 

$$\begin{eqnarray*} &&\fl k^{2} \left| \int _{\Omega } ( q_{2} - q_{1} ) u_{1} u_{2}\, \mathrm{d}x \right| \le \left\Vert \left( u_{1} , \frac{\partial u_1}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \cdot \mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2} )\cdot \left\Vert \left( u_{2} , \frac{\partial u_2}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}}. \end{eqnarray*}$$

Now we would like to estimate the Fourier transform of the difference of two qs. We denote $\mathcal {F}(f)$ the Fourier transformation of a function f.

Lemma 3.2. Let s > (n/2) + 1 be an integer and M > 0. Assume $\Vert q_{l} \Vert _{H^{s} ( \Omega )} \le M$, $\mathop {\mathrm{supp}}( q_{1} - q_{2} ) \subset \Omega$ and k² ⩾ 1/C₁M, where C₁ is the constant defined in lemma 2.2 corresponding to ε₀ = 1. Let $\widetilde{q}$ be the zero extension of q₁ − q₂ and a₀ ⩾ C₁. Suppose that χ ∈ C^∞₀(Ω) satisfies χ ≡ 1 near $\mathop {\mathrm{supp}}( q_{1} - q_{2} )$. Then for r ⩾ 0 and $\eta \in \mathbb {R}^{n}$ with |η| = 1 the following statements hold: if 0 ⩽ r ⩽ a₀k²M then

$$\begin{equation} | \mathcal {F} \widetilde{q} ( r \eta ) | \le \frac{C \Vert \chi \Vert _{H^{s} ( \Omega )}}{a_{0}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} + Ck^{2} \exp ( C a_{0} k^{2} M )\mathop {\mathrm{dist}}(\mathcal {C}_{q_1},\mathcal {C}_{q_2}) \end{equation} \tag{ 3.1 }$$

holds; if r ⩾ C₁k²M then

$$\begin{equation} | \mathcal {F} \widetilde{q} ( r \eta ) | \le \frac{C M k^{2} \Vert \chi \Vert _{H^{s} ( \Omega )}}{r} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} + {C}{k^{2}} \exp ( C r ) \mathop {\mathrm{dist}}(\mathcal {C}_{q_1},\mathcal {C}_{q_2}) \end{equation} \tag{ 3.2 }$$

holds, where C > 0 depends only on n, s, M and Ω.

Proof. In the following proof, the letter C stands for a general constant depending only on n, s and Ω. By proposition 2.3, we can construct CGO solutions u_l(x) to the equation (1.2) with q = q_l having the form of

$$\begin{equation*} u_{l} (x) = \exp \left( \frac{\xi _{l}}{2} \cdot x \right) (1 + \psi _{l} (x)) \end{equation*}$$

for l = 1, 2, and we have

$$\begin{eqnarray} &&\fl \left| \int _{\Omega } ( q_{2} - q_{1} ) \exp \left( \frac{1}{2} ( \xi _{1} + \xi _{2} ) \cdot x \right) ( 1 + \psi _{1} + \psi _{2} + \psi _{1} \psi _{2} ) \, \mathrm{d}x \right| \nonumber \\ &&\le \frac{1}{k^{2}} \left\Vert \left( u_{1} , \frac{\partial u_1}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \cdot \mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2} ) \cdot \left\Vert \left( u_{2} , \frac{\partial u_2}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \end{eqnarray} \tag{ 3.3 }$$

from proposition 3.1, where ψ_l satisfies

$$\begin{equation*} \Vert \psi _{l} \Vert _{H^{s} ( \Omega )} \le \frac{C k^{2}}{| \xi _{l} |} \Vert q_{l} \Vert _{H^{s} ( \Omega )} \end{equation*}$$

if $\xi _{l} \in \mathbb {C}^{n}$ satisfies ξ_l · ξ_l = 0, |ξ_l| ⩾ 1 and

$$\begin{equation} | \xi _{l} | \ge C_{1} k^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )} . \end{equation} \tag{ 3.4 }$$

We remark that $\Vert \psi _{l} \Vert _{H^{s} ( \Omega )} \le C$ also holds. Indeed, we have

$$\begin{equation*} \Vert \psi _{l} \Vert _{H^{s} ( \Omega )} \le \frac{C k^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )}}{| \xi _{l} |} \le \frac{C k^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )}}{C_{1} k^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )}} = \frac{C}{C_{1}} = C. \end{equation*}$$

Now, let r ⩾ 0, and $\eta \in \mathbb {R}^{n}$ satisfy |η| = 1. We assume that $\alpha , \zeta \in \mathbb {R}^{n}$ satisfy

$$\begin{equation} \alpha \cdot \eta = \alpha \cdot \zeta = \eta \cdot \zeta = 0 \qquad \mbox{and}\qquad | \zeta |^{2} = | \alpha |^{2} + r^{2}. \end{equation} \tag{ 3.5 }$$

Define ξ₁ and ξ₂ as

$$\begin{equation*} \xi _{1} = \zeta + \mathrm{i}\alpha - \mathrm{i}r \eta \qquad \mbox{and} \qquad \xi _{2} = - \zeta - \mathrm{i}\alpha - \mathrm{i}r \eta . \end{equation*}$$

Then, we have

$$\begin{equation*} \xi _{l} \cdot \xi _{l} = 0, \qquad | \xi _{l} |^{2} = | \zeta |^{2} + | \alpha |^{2} + r^{2} = 2 | \zeta |^{2} \qquad ( l = 1, 2 ) \end{equation*}$$

and (ξ₁ + ξ₂)/2 = −irη.

Hence by (3.3), we immediately obtain that

$$\begin{eqnarray} &&\fl | \mathcal {F} \widetilde{q} ( r \eta ) | \le \left| \int _{\Omega } ( q_{2} - q_{1} ) \exp ( - \mathrm{i}r \eta \cdot x ) ( \psi _{1} + \psi _{2} + \psi _{1} \psi _{2} ) \, \mathrm{d}x \right| \nonumber \\ &&\mbox{} + \frac{1}{k^{2}} \left\Vert \left( u_{1} , \frac{\partial u_1}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \cdot \mathop {\mathrm{dist}}( \mathcal {C}_{q_1} , \mathcal {C}_{q_2} ) \cdot \left\Vert \left( u_{2} , \frac{\partial u_2}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \hspace{-12.75pt} \end{eqnarray} \tag{ 3.6 }$$

provided |ξ_l| ⩾ 1 and (3.4) are satisfied. We first estimate the first term on the right-hand side of (3.6) by

$$\begin{eqnarray*} &&\fl \left| \int _{\Omega } ( q_{2} - q_{1} ) \exp ( - \mathrm{i}r \eta \cdot x ) ( \psi _{1} + \psi _{2} + \psi _{1} \psi _{2} )\, \mathrm{d}x \right| \\ &&= \left| \int _{\Omega } ( q_{2} - q_{1} ) \exp ( - \mathrm{i}r \eta \cdot x ) \chi ( \psi _{1} + \psi _{2} + \psi _{1} \psi _{2} )\, \mathrm{d}x \right| \\ &&\le \Vert q_{2} - q_{1} \Vert _{H^{-s} ( \Omega )} \bigl \Vert \chi ( \psi _{1} + \psi _{2} + \psi _{1} \psi _{2} ) \bigr \Vert _{H^{s} ( \Omega )}\\ &&\le \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} \Vert \chi \Vert _{H^{s} ( \Omega )} \bigl ( \Vert \psi _{1} \Vert _{H^{s} ( \Omega )} + \Vert \psi _{2} \Vert _{H^{s} ( \Omega )} + \Vert \psi _{1} \Vert _{H^{s} ( \Omega )} \Vert \psi _{2} \Vert _{H^{s} ( \Omega )} \bigr ) \\ &&\le \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} \Vert \chi \Vert _{H^{s} ( \Omega )} \left( \frac{C k^{2}}{\sqrt{2} | \zeta |} + \frac{C k^{2}}{\sqrt{2} | \zeta |} + C \frac{C k^{2}}{\sqrt{2} | \zeta |} \right) \sum _{l = 1}^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )} \\ &&= \frac{C k^{2} \Vert \chi \Vert _{H^{s} ( \Omega )}}{| \zeta |} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} \sum _{l = 1}^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )} \end{eqnarray*}$$

since χ(ψ₁ + ψ₂ + ψ₁ψ₂) ∈ H^s₀(Ω) and s > n/2.

Now we assume that $\Vert q_{l} \Vert _{H^{s} ( \Omega )} \le M$ and k² ⩾ 1/C₁M. Then we can see that

$$\begin{equation*} \left\Vert \frac{\partial u_l}{\partial \nu } \right\Vert _{H^{-1/2}(\partial \Omega )} \le C_{M} k^2 \Vert u_l \Vert _{H^1(\Omega )} \end{equation*}$$

where C_M depends only on n, s, M and Ω since u_l ∈ H¹(Ω) is a solution of Δu_l + k²q_lu_l = 0 in Ω. Consequently, we obtain that

$$\begin{equation} \left\Vert \left( u_l , \frac{\partial u_l}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \le C_{M} k^2 \Vert u_l \Vert _{H^1(\Omega )}. \end{equation} \tag{ 3.7 }$$

Now by taking R large enough such that Ω⊂B_R(0), we have

$$\begin{eqnarray*} \Vert u_{l} \Vert _{L^{2} (\Omega )} & \le | \Omega |^{1/2} \Vert u_{l} \Vert _{C^{0} ( \Omega )} \le | \Omega |^{1/2} \exp \left( \frac{| \mathop {\mathrm{Re}}\xi _{l} |}{2} R \right) ( 1 + \Vert \psi _{l} \Vert _{L^{\infty } ( \Omega )}) \\ & \le C \exp \left( \frac{| \mathop {\mathrm{Re}}\xi _{l} |}{2} R \right)( 1 + \Vert \psi _{l} \Vert _{H^{s} ( \Omega )}) \\ & \le C \exp \left( \frac{| \mathop {\mathrm{Re}}\xi _{l} |}{2} R \right) ( 1 + C ) = C \exp \left( \frac{| \zeta |}{2} R \right). \end{eqnarray*}$$

Likewise, we can obtain that

$$\begin{eqnarray*} \Vert \nabla u_{l} \Vert _{L^{2} ( \Omega )} & = \left\Vert \frac{\xi _{l}}{2} u_{l} + \exp \left( \frac{\xi _{l}}{2} \cdot \bullet \right) ( \nabla \psi _{l} ) \right\Vert _{L^{2} (\Omega )} \\ & \le \frac{\sqrt{2} | \zeta |}{2} C \exp \left( \frac{| \zeta |}{2} R \right) + | \Omega |^{1/2} \exp \left( \frac{| \zeta |}{2} R \right) \Vert \nabla \psi _{l} \Vert _{C^{0} ( \Omega )} \\ & \le C | \zeta | \exp \left( \frac{| \zeta |}{2} R \right) + C \exp \left( \frac{| \zeta |}{2} R \right) \Vert \nabla \psi _{l} \Vert _{H^{s-1} ( \Omega )} \\ & \le C | \zeta | \exp \left( \frac{| \zeta |}{2} R \right) + C \exp \left( \frac{| \zeta |}{2} R \right) \Vert \psi _{l} \Vert _{H^{s} ( \Omega )} \\ & \le C \exp ( C | \zeta | ) \end{eqnarray*}$$

since s − 1 > n/2. Combining above estimates and (3.7) yields

$$\begin{equation*} \left\Vert \left( u_l , \frac{\partial u_l}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \le C_{M} k^2 \exp ( C | \zeta | ). \end{equation*}$$

Therefore, we can estimate the second term of the right-hand side of (3.6) by

$$\begin{eqnarray*} &&\fl \frac{1}{k^{2}} \left\Vert \left( u_{1} , \frac{\partial u_1}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}} \cdot \mathop {\mathrm{dist}}( \mathcal {C}_{q_1}, \mathcal {C}_{q_2}) \cdot \left\Vert \left( u_{2} , \frac{\partial u_2}{\partial \nu } \right) \right\Vert _{H^{1/2} \oplus H^{-1/2}}\\ &&\le C_{M} k^2 \exp ( C | \zeta | ) \mathop {\mathrm{dist}}( \mathcal {C}_{q_1} , \mathcal {C}_{q_2}). \end{eqnarray*}$$

Summing up, we have shown that for r > 0 and for $\eta \in \mathbb {R}^{n}$ with |η| = 1 if we take α and ζ satisfying the conditions (3.5), |ζ| ⩾ 2^−1/2 and

$$\begin{equation} | \zeta | \ge 2^{-1/2} C_{1} k^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )} \end{equation} \tag{ 3.8 }$$

then

$$\begin{equation} \fl | \mathcal {F} \widetilde{q} ( r \eta ) | \le \frac{C k^{2} \Vert \chi \Vert _{H^{s} ( \Omega )}}{| \zeta |} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )} \sum _{l=1}^{2} \Vert q_{l} \Vert _{H^{s} ( \Omega )} + C_{M} k^{2} \exp ( C | \zeta | ) \mathop {\mathrm{dist}}( \mathcal {C}_{q_1} , \mathcal {C}_{q_2} ) \end{equation} \tag{ 3.9 }$$

holds.

Now, if

$$\begin{equation} | \zeta | \ge C_{1} k^{2} M \end{equation} \tag{ 3.10 }$$

holds, then (3.8) and |ζ| ⩾ 2^−1/2 are satisfied. Pick a₀ ⩾ C₁. We first consider the case where 0 ⩽ r ⩽ a₀k²M. By choosing α and ζ satisfying

$$\begin{eqnarray*} &&\fl \alpha \cdot \eta = \alpha \cdot \zeta = \eta \cdot \zeta = 0, \qquad | \zeta | = a_{0} k^{2} M ( \ge r ) \qquad \mbox{and} \qquad | \alpha | = \sqrt{( a_{0} k^{2} M )^{2} - r^{2}} \end{eqnarray*}$$

both (3.5) and (3.10) are then satisfied since a₀ ⩾ C₁. Hence we obtain (3.9), that is (3.1). On the other hand, when r ⩾ C₁k²M, we can choose α = 0, η · ζ = 0 and |ζ| = r. Then (3.5), (3.10) are satisfied and thus (3.9) holds and consequently (3.2) is valid. □

Now we prove our main result.

Proof. As above, C denotes a general constant depending only on n, s, M and Ω. Written in polar coordinates, we have

$$\begin{eqnarray} \fl \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} &=& C \int _{0}^{\infty } \int _{| \eta | = 1} | \mathcal {F} \widetilde{q} ( r \eta ) |^{2} ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}\eta \, \mathrm{d}r \nonumber \\ \fl &=& C \biggl ( \int _{0}^{a_{0} k^{2} M} \int _{| \eta | = 1} | \mathcal {F} \widetilde{q} ( r \eta ) |^{2} ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}\eta \, \mathrm{d}r \nonumber \\ \fl &&+\, \int _{a_{0} k^{2} M}^{T} \int _{| \eta | = 1} | \mathcal {F} \widetilde{q} ( r \eta ) |^{2} ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}\eta \, \mathrm{d}r \nonumber \\ \fl &&+\, \int _{T}^{\infty } \int _{| \eta | = 1} | \mathcal {F} \widetilde{q} ( r \eta ) |^{2} ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}\eta \, \mathrm{d}r \biggr ) \nonumber \\ \fl &=:& C( I_{1} + I_{2} + I_{3} ), \end{eqnarray} \tag{ 3.11 }$$

where a₀ ⩾ C₁ and T ⩾ a₀k²M are parameters which will be chosen later. Here C₁ is the constant given in lemma 3.2. From now on, we take k² ⩾ 1/(C₁M).

Our task now is to estimate each integral separately. We begin with I₃. Since $| \mathcal {F} \widetilde{q} ( r \eta ) | \le C \Vert q_{1} - q_{2} \Vert _{L^{2} ( \Omega )}$, q₁ − q₂ ∈ H^s₀(Ω) and s > n/2, we have that

$$\begin{eqnarray} I_{3} & \le C \int _{T}^{\infty } \Vert q_{1} - q_{2} \Vert _{L^{2} ( \Omega )}^{2} ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}r \le C T^{- m} \Vert q_{1} - q_{2} \Vert _{L^{2} ( \Omega )}^{2} \nonumber \\ & \le C T^{- m} \left( \varepsilon \Vert q_{1} - q_{2} \Vert _{H^{-s} ( \Omega )}^{2} + \frac{C}{\varepsilon } \Vert q_{1} - q_{2} \Vert _{H^{s} ( \Omega )}^{2} \right) \nonumber \\ & \le C T^{- m} \left( \varepsilon \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} + \frac{M^{2}}{\varepsilon } \right) \end{eqnarray} \tag{ 3.12 }$$

for ε > 0, where m := 2s − n.

On the other hand, by lemma 3.2, we can estimate

$$\begin{eqnarray} \fl I_{1} &\le& C \int _{0}^{a_{0} k^{2} M} ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}r \left[ \frac{\Vert \chi \Vert _{H^{s} ( \Omega )}^{2}}{a_{0}^{2}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} + k^4\,{\exp ( 2 C a_{0} k^{2} M )}\mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^2 \right] \nonumber \\ \fl &\le& C \int _{0}^{\infty } ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}r \left[ \frac{C_{\chi }^{2}}{a_{0}^{2}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} + k^4\,{\exp ( C a_{0} k^{2} M )}\mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^2 \right] \nonumber \\ \fl &=& \frac{C C_{\chi }^{2}}{a_{0}^{2}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} + k^4{C \exp ( C a_{0} k^{2} M )}\mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2} , \end{eqnarray} \tag{ 3.13 }$$

where χ ∈ C^∞₀(Ω) satisfies χ ≡ 1 near $\mathop {\mathrm{supp}}( q_{2} - q_{1} )$ and $C_{\chi } := \Vert \chi \Vert _{H^{s} ( \Omega )}$. In view of

$$\begin{eqnarray*} \int _{a_{0} k^{2} M}^{T} ( 1 + r^{2} )^{-s} r^{n-3} \mathrm{d}r & \le \int _{a_{0} k^{2} M}^{T} r^{- 2 s + n-3} \mathrm{d}r \le C ( a_{0} k^{2} M )^{- 2 s + n - 2} \\ & \le C ( a_{0} k^{2} M )^{-2} ( C_{1} k^{2} M )^{- m} \le \frac{C}{a_{0}^{2} k^{4} M^{2}} \end{eqnarray*}$$

and

$$\begin{eqnarray*} \int _{a_{0} k^{2} M}^{T} \exp ( C r ) ( 1 + r^{2} )^{-s} r^{n-1} \mathrm{d}r & \le \exp ( C T ) \int _{a_{0} k^{2} M}^{T} ( 1 + r^{2} )^{-s} r^{n-1} \mathrm{d}r \\ & \le \exp ( C T ) \int _{0}^{\infty } ( 1 + r^{2} )^{-s} r^{n-1} \mathrm{d}r \\ & \le C \exp ( C T ), \end{eqnarray*}$$

we have that

$$\begin{eqnarray} \fl I_{2} &\le& C M^{2} k^{4} \Vert \chi \Vert _{H^{s} ( \Omega )}^{2} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} \int _{a_{0} k^{2} M}^{T} ( 1 + r^{2} )^{-s} r^{n-3} \, \mathrm{d}r \nonumber \\ \fl &&+\, {C}{k^{4}} \mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^2 \int _{a_{0} k^{2} M}^{T} \exp ( C r ) ( 1 + r^{2} )^{-s} r^{n-1} \, \mathrm{d}r \nonumber \\ \fl &\le& \frac{C C_{\chi }^{2}}{a_{0}^{2}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} +{C}{k^{4}} \exp ( C T )\mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2}. \end{eqnarray} \tag{ 3.14 }$$

Combining (3.11)–(3.14) gives

$$\begin{eqnarray*} \fl \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} &\le& C ( I_{1} + I_{2} + I_{3} ) \\ \fl &\le& \frac{C C_{\chi }^{2}}{a_{0}^{2}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} +{Ck^4 \exp ( C a_{0} k^{2} M )}\mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2}+ \frac{C C_{\chi }^{2}}{a_{0}^{2}} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} \\ \fl &&+ {C}{k^{4}} \exp ( C T )\mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2}+\, C T^{- m} \left( \varepsilon \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} + \frac{M^{2}}{\varepsilon } \right) \\ \fl &=& \left( \frac{C_{2}^{2} C_{\chi }^{2}}{a_{0}^{2}} + C_{3} T^{- m} \varepsilon \right) \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} \\ \fl &&+\, {C}{k^{4}} \left( \exp ( C a_{0} k^{2} M ) + \exp ( C T ) \right) \mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2} + \frac{C M^{2}}{\varepsilon } T^{- m}, \end{eqnarray*}$$

where positive constants C₂ and C₃ depend only on n, s and Ω.

Now we pick a₀ and ε as

$$\begin{equation*} a_{0} = 2 C_{2} C_{\chi } \ge C_{1} \mbox{ and } \varepsilon = \frac{T^{m}}{4 C_{3}} \end{equation*}$$

(if needed, we take C₂ large enough). We then obtain that

$$\begin{eqnarray} \fl \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} &\le& {C}{k^{4}} \left[ \exp ( 2 C_{2} C C_{\chi } k^{2} M ) + \exp ( C T ) \right] \mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2} + C T^{- 2 m} M^{2} \nonumber \\ \fl &=& {C}{k^{4}} \exp ( C a k^{2} ) A + C \Phi (T) \end{eqnarray} \tag{ 3.15 }$$

for T ⩾ a₀k²M = 2C₂C_χk²M = ak², where

$$\begin{equation*} \Phi (T) := {k^{4}} \exp ( C_{4} T ) A + M^{2} T^{- 2 m}, \end{equation*}$$

$A := \mathop {\mathrm{dist}}(\mathcal{C}_{q_1},\mathcal{C}_{q_2})^{2}$, a := 2C₂C_χM and C₄ > 0 depends only on n, s and Ω.

To continue, we consider two cases

$$\begin{equation} a k^{2} \le p \log \frac{1}{A} \end{equation} \tag{ 3.16 }$$

and

$$\begin{equation} a k^{2} \ge p \log \frac{1}{A}, \end{equation} \tag{ 3.17 }$$

where p will be determined later (see (3.26)).

For the first case (3.16), our aim is to show that there exists T ⩾ ak² such that

$$\begin{equation} \Phi (T) \le 2 C_{5} \left( k^{2} + \log \frac{1}{A} \right)^{- 2 m}. \end{equation} \tag{ 3.18 }$$

Substituting (3.18) into (3.15) clearly implies (1.3). Now to derive (3.18), it is enough to prove that

$$\begin{equation} {k^{4}} \exp ( C_{4} T ) A \le C_{5} \left( k^{2} + \log \frac{1}{A} \right)^{- 2 m} \end{equation} \tag{ 3.19 }$$

and

$$\begin{equation} M^{2} T^{- 2 m} \le C_{5} \left( k^{2} + \log \frac{1}{A} \right)^{- 2 m}. \end{equation} \tag{ 3.20 }$$

Remark that (3.20) is equivalent to

$$\begin{equation*} T \ge C_{5}^{- 1 / 2 m} M^{1 / m} \left( k^{2} + \log \frac{1}{A} \right), \end{equation*}$$

which holds if

$$\begin{equation} T \ge C_{5}^{- 1 / 2 m} M^{1/m} \left( 1 + \frac{p}{a} \right) \log \frac{1}{A} \end{equation} \tag{ 3.21 }$$

because of (3.16). Setting T = plog (1/A) (⩾ak² by (3.16)), then (3.21) holds provided

$$\begin{equation} p \ge C_{5}^{- 1 / 2 m} M^{1/m} \left( 1 + \frac{p}{a} \right). \end{equation} \tag{ 3.22 }$$

Now we turn to (3.19). It is clear that (3.19) is equivalent to

$$\begin{equation} C_{4} p \log \frac{1}{A} \le \log C_{5} - 2 \log k^{2} + \log \frac{1}{A} - 2 m \log \left( k^{2} + \log \frac{1}{A} \right) \end{equation} \tag{ 3.23 }$$

since T = plog (1/A). It follows from (3.16) that

$$\begin{equation*} \log \left( k^{2} + \log \frac{1}{A} \right) \le \log \left( \frac{p}{a} \log \frac{1}{A} + \log \frac{1}{A} \right) = \log \left( \frac{p}{a} + 1 \right) + \log \log \frac{1}{A} . \end{equation*}$$

Hence (3.23) is verified if we can show that

$$\begin{equation} \fl C_{4} p \log \frac{1}{A} \le \log C_{5} - 2 \log \left(\frac{p}{a}\log \frac{1}{A}\right) + \log \frac{1}{A} - 2 m \left( \log \left( \frac{p}{a} + 1 \right) + \log \log \frac{1}{A} \right) \end{equation} \tag{ 3.24 }$$

for log (1/A) ⩾ 1. To obtain (3.24), it suffices to prove

$$\begin{equation*} C_{4} p \log \frac{1}{A}\le \log C_{5} + \log \frac{1}{A} - 2 (m+1) \left( \log \left( \frac{p}{a} + 1 \right) + \log \log \frac{1}{A} \right), \end{equation*}$$

i.e.

$$\begin{equation} \fl ( 1 - C_{4} p ) \log \frac{1}{A} - 2 (m+1) \log \log \frac{1}{A} + \log C_{5}- 2 (m+1) \log \left( \frac{p}{a} + 1 \right) \ge 0 \end{equation} \tag{ 3.25 }$$

for log (1/A) ⩾ 1. Now we choose

$$\begin{equation} p = \frac{1}{2 C_{4}}. \end{equation} \tag{ 3.26 }$$

Then (3.25) becomes

$$\begin{equation} \log \frac{1}{A} - 4 (m+1) \log \log \frac{1}{A} + 2 \log C_{5} - 4 (m+1) \log \left( \frac{p}{a} + 1 \right) \ge 0. \end{equation} \tag{ 3.27 }$$

Note that

$$\begin{eqnarray*} \inf _{0 < A \le 1 / e} \left( \log \frac{1}{A} - 4 (m+1) \log \log \frac{1}{A} \right) & = \inf _{z \ge 1} ( z - 4 (m+1) \log z ) \\ & \ge \inf _{z > 0} ( z - 4 (m+1) \log z )\\ & = 4 (m+1) \log \frac{e}{4 (m+1)} . \end{eqnarray*}$$

Hence, if we choose C₅ such that

$$\begin{equation} C_{5} \ge ( M C_{1} )^{2} \left( \frac{p}{a} + 1 \right)^{2 (m+1)} \left( \frac{4 (m+1)}{e} \right)^{2 (m+1)} \end{equation} \tag{ 3.28 }$$

then (3.27) follows. Finally, we take

$$\begin{equation*} C_{5} := \max \left\lbrace C_{1}^{2} \left( \frac{4 (m+1)}{e} \right)^{2 (m+1)} , \ p^{- 2 (m+1)} \right\rbrace M^{2} \left( 1 + \frac{p}{a} \right)^{2 (m+1)}, \end{equation*}$$

which depends only on n, Ω, s, M and χ. With such a choice of C₅, the conditions (3.28) and (3.22) hold, and thus estimate (3.18) is satisfied.

Next we consider the second case (3.17). By (3.15) with T = ak², we obtain that

$$\begin{eqnarray*} \Vert \widetilde{q} \Vert _{H^{-s} ( \mathbb {R}^{n} )}^{2} & \le {C}{k^{4}} \exp ( C a k^{2} ) A + {C}{k^{4}} \exp ( C_{4} a k^{2} ) A + C M^{2} ( a k^{2} )^{- 2 m} \\ & \le {C}{k^{4}} \exp ( C a k^{2} ) A + C M^{2} a^{- 2 m} k^{- 4 m}. \end{eqnarray*}$$

Hence, it remains to show that

$$\begin{equation*} k^{- 4 m} \le C_{6} \left( k^{2} + \log \frac{1}{A} \right)^{- 2 m}, \end{equation*}$$

i.e.

$$\begin{equation} k^{2} \ge C_{6}^{- 1 / 2 m} \left( k^{2} + \log \frac{1}{A} \right). \end{equation} \tag{ 3.29 }$$

Since

$$\begin{equation*} k^{2} + \log \frac{1}{A} \le \left( 1 + \frac{a}{p} \right) k^{2} \end{equation*}$$

by (3.17), we have (3.29) if we take C₆ large enough so that

$$\begin{equation*} C_{6} \ge \left( 1 + \frac{a}{p} \right)^{2 m}. \end{equation*}$$

The proof is completed. □

SN was partially supported by Grant-in-Aid for Young Scientists (B). GU was partly supported by NSF and a Visiting Distinguished Rothschild Fellowship at the Isaac Newton Institute. J-NW was partially supported by the National Science Council of Taiwan. We would also like to thank P Stefanov for helpful discussions.