Identification of Nonlinearity in Conductivity Equation via Dirichlet-to-Neumann Map^{∗}

Hyeonbae Kang^{†} School of Mathematical Sciences Seoul National University Seoul 151-747, Korea hkang@math.snu.ac.kr

Gen Nakamura^{‡} Department of Mathematics Hokkaido University Sapporo 060-0810, Japan gnaka@math.sci.hokudai.ac.jp

January 9, 2002

Abstract

We prove that the linear term and quadratic nonlinear term entering a nonlinear elliptic equation of divergence type can be uniquely identified by the Dirichlet to Neuman map. The unique identifiability is proved using the complex geometrical optics solutions and singular solutions.

Mathematics subject classification (MSC2000): 35R30 Keywords: Inverse conductivity problem, nonlinear equations, elasticity tensor, Dirichlet to Neuman map, complex geometrical optics solutions, singular solutions

1 Introduction

Let Ω be a bounded domain in R^{n} (n ≥ 2) with a C^{2} boundary ∂Ω. We consider the following nonlinear boundary value problem:

(1.1)  \{ \begin{matrix}∇ · C(x, ∇u) = 0 in Ω, \\ u|_{∂Ω} = f. \end{matrix}

The nonlinear function C takes the following form:

(1.2) C(x, q) = γq + Q(x, q) := γq + P (x, q) + R(x, q), x ∈ \overline{Ω}, q ∈ R^{n} (or C^{n}).

∗This work was completed while both authors were visiting the Mathematical Sciences Research Institute (MSRI). We would like to thank the institute for partial support and providing stimulating environment for the research.

†partly suppoted by KOSEF 98-0701-03-5 and 2000-1-10300-001-1.

‡partly supported by Grant-in-Aid for Scientific Research (C)(No.12640153).

Here γ ∈ C^{2}(\overline{Ω}), and there exists C_{0} > 0 such that

(1.3)  γ(x) ≥ C_{0} x ∈ Ω.

P (x, q) represents the quadratic nonlinearity:

(1.4)  P (x, q) = ( ∑_{1≤k≤l≤n} c_{kl}^{1}(x)q_{k}q_{l}, ⋯, ∑_{1≤k≤l≤n} c_{kl}^{n}(x)q_{k}q_{l}) .

We suppose that c_{kl}^{i} ∈ C^{1}(\overline{Ω}) are real valued functions and

(1.5)  ∥γ∥_{C^{1}(\overline{Ω})}, ∥c_{kl}^{i}∥_{C^{1}(\overline{Ω})} ≤ C_{1} (1 ≤ i ≤ n, 1 ≤ k ≤ l ≤ n)

for some C_{1} > 0. R(x, q) ∈ C^{2}(\overline{Ω} × H) with H := \{q ∈ R^{n} or C^{n} ; |q| ≤ h\} (h > 0) represents the higher order nonlinearity, namely, it satisfies

(1.6)  \{ \begin{matrix}|∂_{x_{j}}R(x, q)| ≤ C_{2}|q|^{3}, \\ |∂_{x_{j}}∂_{q_{k}}R(x, q)| ≤ C_{2}|q|^{2}, \\ |∂_{q_{j}}∂_{q_{k}}R(x, q)| ≤ C_{2}|q|^{1} \\ (1 ≤ j, k ≤ n, x ∈ \overline{Ω}, q ∈ H) \end{matrix}

for some C_{2} > 0 independent of x and q.

The following theorem is probably well-known. However, since we are not able to find a proper reference, we give a short proof of it based on the contraction mapping principle in the Appendix at the end of this paper. For the elasticity equation, a proof using the implicit function theorem can be found in [4].

Theorem 1.1 Let n < p < ∞. There exist ϵ and δ < h/2 such that for any f ∈ W^{2−1/p,p}(∂Ω) satisfying ∥f∥_{W^{2−1/p,p}(∂Ω)} < ϵ, (1.1) admits a unique solution u such that ∥u∥_{W^{2,p}(Ω)} < δ. Moreover there exists C > 0 independent of f such that

(1.7)  ∥u∥_{W^{2,p}(Ω)} ≤ C∥f∥_{W^{2−1/p,p}(∂Ω)}.

We define the Dirichlet-to-Neumann (DN) map Λ_{C}(f) for f with ∥f∥_{W^{2−1/p,p}(∂Ω)} < ϵ to be

(1.8)  Λ_{C}(f) := C(x, ∇u)|_{∂Ω} · ν ∈ W^{1−1/p,p}(∂Ω)

where u is the unique solution of (1.1) such that ∥u∥_{W^{2,p}(Ω)} < δ.

In this paper we consider the inverse boundary value problem to identify C by means of the DN map Λ_{C}. We are particularly interested in finding the linear term γ and the second order nonlinearity P (x, q).

This inverse problem has interest in it’s own right as the one to find conductivity distribution when the conductivity varies depending on the currents. It may also considered as a simplified model for the nonlinear elasticity equation. In elasticity γ(x) corresponds to the Lamé moduli and c_{kl}^{l}(x) can be thought as the third order elasticity tensor. In acousto-elasticity, this higher elasticity tensor is important [5].

We obtain the following unique identifiability theorem for γ and P.

Theorem 1.2 Suppose that n ≥ 3. Let C^{(j)}(x, q) = γ_{j}q + Q^{(j)}, j = 1, 2, where Q^{(j)} = P^{(j)}(x, q) + R^{(j)}(x, q),

P^{(j)}(x, q) = ( ∑_{1≤k≤l≤n} c_{kl}^{(j)1}(x)q_{k}q_{l}, ⋯, ∑_{1≤k≤l≤n} c_{kl}^{(j)n}(x)q_{k}q_{l}) ,

and R^{(j)} satisfy (1.3), (1.5), and (1.6), respectively, for the same constants C_{0}, C_{1}, C_{2} and h. If

(1.9)  Λ_{C^{(1)}}(f) = Λ_{C^{(2)}}(f) for all complex-valued small f ∈ W^{2−1/p,p}(∂Ω),

then

(1.10)  γ_{1}(x) = γ_{2}(x) and c_{kl}^{(1)j}(x) = c_{kl}^{(2)j}(x), x ∈ Ω

for 1 ≤ j ≤ n and 1 ≤ k ≤ l ≤ n.

The dimensional restriction n ≥ 3 is imposed since we are using the complex geometrical optics solutions of Sylvester-Uhlmann [13] to prove the Theorem.

In most models of nonlinear elasticity, the higher order tensors do not depend on x. So we consider the same inverse problem when the coefficients c_{kl}^{j} are constants. In this case we obtain the following uniqueness theorem including the two dimensions. One thing to be noted is the condition (1.9) (and (1.11) below). Since the equation considered in this paper is nonlinear, Λ_{C^{(1)}}(f) = Λ_{C^{(2)}}(f) for all realvalued small f ∈ W^{2−1/p,p}(∂Ω) does not imply the same for complex-valued small f ∈ W^{2−1/p,p}(∂Ω). Therefore, in the situation where the data Λ_{C}(f) for only realvalued small f are available, Theorem 1.2 may not be applied. The following theorem uses only real-valued Dirichlet data.

Theorem 1.3 Suppose that n ≥ 2. Let C^{(j)}(x, q) as in Theorem 1.2 except that the coefficients c_{kl}^{(j)i} are constants. If

(1.11)  Λ_{C^{(1)}}(f) = Λ_{C^{(2)}}(f) for all real-valued small f ∈ W^{2−1/p,p}(∂Ω),

then

(1.12)  γ_{1}(x) = γ_{2}(x) and c_{kl}^{(1)j} = c_{kl}^{(2)j}

for 1 ≤ j ≤ n and 1 ≤ k ≤ l ≤ n.

There have been some related works on the identification of nonlinear terms entering partial differential equations. Sun proved the global uniqueness for the nonlinear conductivity γ(x, u) when the Dirichlet data are complex valued and small [11]. Nakamura and Sun considered the similar problem for nonlinear elasticity model of St.Venant-Kirchhoff [10]. They proved that even in the presence of nonlinearity, the linear term can be identified uniquely by means of DN map. Sun and Uhlmann proved the global uniqueness up to diffeomorphism fixing the boundary for the two dimensional nonlinear anisotropic conductivity A(x, u), and also the same result for the 3 dimensional analytic conductivity [12]. Quite recently, Isakov proved the unique identifiability of the nonlinearity c(u, p) entering the quasilinear elliptic equation Δu + c(u, ∇u) = 0 (and corresponding parabolic equation) by means of DN map on ∂Ω [7]. G. Uhlmann informed us that Hervas and Sun proved a result related to ours in the two dimensional case [6]. They proved that the constant coefficients nonlinear terms with extra symmetry can be identified from the DN map defined for complex valued small Dirichlet data.

Theorem 1.2 and Theorem 1.3 are proved by investigating the first and second terms in the asymptotic expansion of Λ_{C} near 0. The first term is Λ_{γ}, the DN map corresponding to the conductivity γ. This fact was also observed in [10] and [7]. We derive certain cubic relation from the second term. This is included in Section 2. Then using the complex geometrical optics solutions of Sylvester-Uhlmann [13], we prove Theorem 1.2 in Section 3. We prove Theorem 1.3 using the singular solutions of Alessandrini [2]. Since the coefficients to be determined are constants, boundary determination is sufficient. This is the reason why the singular solutions are effectively used. The proof is included in Section 4. Appendix is to prove Theorem 1.1.

2 Asymptotics of DN map

Throughout this paper ∥ ∥_{k,p} denotes the W^{k,p}-norm on Ω, and ∥ ∥_{p} = ∥ ∥_{0,p}. Also, C > 0 denotes the general constant in estimate independent of the functions being estimated. We will use the following estimates repeatedly: if k ≥ 1, p > n, and n = 2, 3, then

(2.1)  ∥uv∥_{k,p} ≤ C∥u∥_{k,p}∥v∥_{k,p}.

This inequality holds by the Sobolev embedding theorem (see [1]). We also use the chain rule for the derivative of the composition R(x, ∇u(x)) for u ∈ W^{2,p}(Ω) and its estimate given in Chapter II, section 3 of [14].

Suppose that f ∈ W^{2−1/p,p}(∂Ω), p > n, and let t ∈ R^{1} be a small parameter. Let u^{(t)} be the solution of

\begin{cases}∇ · C(x, ∇u) = 0  in Ω, \\ u|_{∂Ω} = tf \end{cases}

such that ∥u^{(t)}∥_{2,p} ≤ δ. Then there exists a constant C > 0 such that

(2.2)  ∥u^{(t)}∥_{2,p} ≤ Ct∥f∥_{W^{2−1/p,p}(∂Ω)}.

Let u_{1} = u_{1}(f) ∈ W^{2,p}(Ω) be the solution of

\begin{cases}∇ · (γ∇u_{1}) = 0  in Ω, \\ u_{1}|_{∂Ω} = f. \end{cases}

Then by the regularity of the Dirichlet problem for elliptic equations, there exist a constant C > 0 such that we have

(2.3)  ∥u_{1}(f)∥_{2,p} ≤ C∥f∥_{W^{2−1/p,p}(∂Ω)}

for any f ∈ W^{2−1/p,p}(∂Ω). Let u_{2} be the solution of

(2.4)  \begin{cases}∇ · (γ∇u_{2}) = −∇ · P (x, ∇u_{1}) in Ω, \\ u_{2}|_{∂Ω} = 0. \end{cases}

It then follows from the regularity of elliptic equations, (2.1), and (2.3) that

∥u_{2}∥_{2,p} ≤ C∥∇ · P (x, ∇u_{1})∥_{p}
≤ C∥∇u_{1}∥_{1,p}^{2}
≤ C∥f∥_{W^{2−1/p,p}(∂Ω)}^{2}.

We now define v^{(t)} by

(2.5)  u^{(t)} = t(u_{1} + tv^{(t)}).

Then, we have from (2.2) and (2.3)

\begin{aligned}t^{2}∥v^{(t)}∥_{2,p} & ≤ ∥u^{(t)}∥_{2,p} + t∥u_{1}∥_{2,p} \\ & ≤ Ct∥f∥_{W^{2−1/p,p}(∂Ω)}. \end{aligned}

Lemma 2.1 There exist t_{0} and C_{f} depending on f, not on t, such that for all t < t_{0},

(2.6)  ∥v^{(t)} − u_{2}∥_{2,p} ≤ C_{f}t.

In particular, we have

(2.7)  ∥v^{(t)}∥_{2,p} ≤ C_{f}.

Proof. Put w^{(t)} := v^{(t)} − u_{2}. Then a straight-forward computation shows that

\begin{cases}∇ · γ∇w^{(t)} = −tg^{(t)} in Ω, \\ w^{(t)}|_{∂Ω} = 0 \end{cases}

where

g^{(t)} = t^{−1}∇ · [P (x, ∇u_{1} + t∇v^{(t)}) − P (x, ∇u_{1})] + t^{−3}∇ · R(x, t∇u_{1} + t^{2}∇v^{(t)}). It follows from (1.4), (1.6), and (2.1) that

∥g^{(t)}∥_{p} ≤ C(∥u_{1}∥_{2,p}∥v^{(t)}∥_{2,p} + t∥v^{(t)}∥_{2,p}^{2} + ∥u_{1}∥_{2,p}^{3} + t^{3}∥v^{(t)}∥_{2,p}^{3}).

It then follows from (2.6) that

\begin{aligned}∥v^{(t)} − u_{2}∥_{2,p} = ∥w^{(t)}∥_{2,p} & ≤ Ct∥g^{(t)}∥_{p} \\ & ≤ C(t∥v^{(t)}∥_{2,p} + t). \end{aligned}

Thus, if t_{0} is so small that Ct_{0} < \frac{1}{2}, then by (2.5) we obtain

∥v^{(t)}∥_{2,p} ≤ 2(∥u_{2}∥_{2,p} + 1) ≤ C.

Then (2.6) follows from (2.8). This completes the proof. □ Lemma 2.2 For a given f ∈ W^{2−1/p,p}(∂Ω), we have

(1) lim_{t→0} \frac{1}{t}[Λ_{C}(tf) − tΛ_{γ}(f)] = 0,

(2) lim_{t→0} \frac{1}{t^{2}} [Λ_{C}(tf) − tΛ_{γ}(f)] = ν · [γ∇u_{2} + P (x, ∇u_{1})].

The convergence is in the topology of W^{1−1/p,p}(∂Ω). Proof. From (2.5) we have

\begin{aligned}Λ_{C}(tf) − tΛ_{γ}(f) & = ν · [C(x, ∇u^{(t)}) − tγ∇u_{1}] \\ & = ν · [t^{2}γ∇v^{(t)} + Q(x, ∇u^{(t)})]. \end{aligned}

It follows from the trace theorem [1], (2.2) and (2.7) that

∥t^{−1}(Λ_{C}(tf) − tΛ_{γ}(f))∥_{W^{1−1/p,p}(∂Ω)} ≤ ∥tγ∇v^{(t)} + t^{−1}Q(x, ∇u^{(t)})∥_{1,p}
≤ Ct.

This proves (1).

Since Q(x, ∇u^{(t)}) = t^{2}P (x, ∇u_{1}) + t^{3}O(|∇u_{1}||∇v^{(t)}| + |∇v^{(t)}|^{2} + |∇u_{1}|^{3} + |∇v^{(t)}|^{3}),

(2) follows from (2.6). This completes the proof. □

3 Proof of Theorem 1.2

Lemma 2.2 says that we can recover Λ_{γ} and ν · [γ∇u_{2} + P (x, ∇u_{1})] on ∂Ω from Λ_{C}.

If Λ_{C^{(1)}} = Λ_{C^{(2)}}, then Λ_{γ_{1}} = Λ_{γ_{2}}. It then follows from well-known results ([3], [8, 9], [13]) that γ_{1} = γ_{2} in Ω. Put γ = γ_{1} = γ_{2}. By Lemma 2.2 (2), we have

(3.1)  ν · [γ∇u_{2}^{(1)} + P^{(1)}(x, ∇u_{1})] = ν · [γ∇u_{2}^{(2)} + P^{(2)}(x, ∇u_{1})] on ∂Ω

where u_{1} ∈ W^{2,p}(Ω) is a solution of ∇ · (γ∇u_{1}) = 0 in Ω, and u_{2}^{(j)} is the solution of (2.4) when P = P^{(j)}.

Let v ∈ W^{2,p}(Ω) be a solution of ∇ · (γ∇v) = 0 in Ω with v|_{∂Ω} = g. Since u_{2}^{(j)}|_{∂Ω} = 0, it follows from the divergence theorem that

∫_{∂Ω} ν · [γ∇u_{2}^{(j)} + P^{(j)}(x, ∇u_{1})]gdσ = ∫_{Ω}[γ∇u_{2}^{(j)} + P^{(j)}(x, ∇u_{1})] · ∇vdx
= ∫_{Ω} P^{(j)}(x, ∇u_{1}) · ∇vdx.

We thus have

(3.2)  ∫_{Ω} P^{(1)}(x, ∇u_{1}) · ∇vdx = ∫_{Ω} P^{(2)}(x, ∇u_{1}) · ∇vdx,

in other words,

(3.3) ∑_{i=1}^{n} ∑_{1≤j≤l≤n} ∫_{Ω} c_{jl}^{(1)i}(x)\frac{∂u_{1}}{∂x_{j}} \frac{∂u_{1}}{∂x_{l}} \frac{∂v}{∂x_{i}} dx = ∑_{i=1}^{n} ∑_{1≤j≤l≤n} ∫_{Ω} c_{jl}^{(2)i}(x)\frac{∂u_{1}}{∂x_{j}} \frac{∂u_{1}}{∂x_{l}} \frac{∂v}{∂x_{i}} dx

for all u_{1}, v ∈ W^{2,p}(Ω), solutions of ∇ · γ∇u = 0 in Ω. Therefore, it suffices to prove that if

(3.4)  ∑_{i=1}^{n} ∑_{1≤j≤l≤n} ∫_{Ω} c_{jl}^{i}(x)\frac{∂u_{1}}{∂x_{j}} \frac{∂u_{1}}{∂x_{l}} \frac{∂v}{∂x_{i}} dx = 0

for all u_{1}, v ∈ W^{2,p}(Ω), solutions of ∇ · γ∇u = 0, then

(3.5)  c_{jl}^{i} ≡ 0, i = 1, ⋯, n, 1 ≤ j ≤ l ≤ n.

Let u_{2} ∈ W^{2,p}(Ω) be another solution of ∇ · γ∇u = 0. Then u_{1} + u_{2} is also a solution of the equation, and hence we obtain by polarizing (3.4) that

(3.6)  ∑_{i=1}^{n} ∑_{1≤j≤l≤n} ∫_{Ω} c_{jl}^{i}(x) (\frac{∂u_{1}}{∂x_{j}} \frac{∂u_{2}}{∂x_{l}} + \frac{∂u_{1}}{∂x_{l}} \frac{∂u_{2}}{∂x_{j}} ) \frac{∂v}{∂x_{i}} dx = 0

for all u_{1}, u_{2}, v ∈ W^{2,p}(Ω), solutions of ∇ · γ∇u = 0. The arguments we made so far are true for n ≥ 2.

Let n = 3 and use complex geometrical optics solutions. Let k ∈ R^{3} and choose unit vectors ξ and η in R^{3} so that

(3.7)  k · ξ = k · η = ξ · η = 0.

Then choose t, s > 0 so that

(3.8)  t^{2} = \frac{|k|^{2}}{4} + s^{2}.

Define ρ^{(1)}, ρ^{(2)} ∈ C^{3} by

(3.9)  ρ^{(1)} = tη + i(\frac{k}{2} + sξ), ρ^{(2)} = −tη + i(\frac{k}{2} − sξ)

Then by the fundamental work of Sylvester-Uhlmann [13] there exist solutions u_{j}, j = 1, 2, of ∇ · γ∇u = 0 of the form

(3.10)  u_{j}(x) = γ^{−1/2}e^{ρ^{(j)}·x}(1 + ψ_{j}(x, ρ^{(j)})), x ∈ Ω

where ψ_{j} satisfies

(3.11)  ∥ψ_{j}∥_{L^{∞}(Ω)} ≤ \frac{C}{|s|} and ∥∇ψ_{j}∥_{L^{∞}(Ω)} ≤ C

for some C independent of t. We apply these solutions u_{j} to (3.6) and obtain

(3.12)  \begin{matrix}∑_{i=1}^{3} ∑_{1≤j≤l≤3} ∫_{Ω} γ(x)^{−1}c_{jl}^{i}(x)\frac{∂v}{∂x_{i}} (x)e^{ik·x}× \\ [(γ^{1/2}∂_{j}γ^{−1/2}(1 + ψ_{1}) + ρ_{j}^{(1)}(1 + ψ_{1}) + ∂_{j}ψ_{1})× \\ (γ^{1/2}∂_{l}γ^{−1/2}(1 + ψ_{2}) + ρ_{l}^{(2)}(1 + ψ_{2}) + ∂_{l}ψ_{2}) \\ +(γ^{1/2}∂_{l}γ^{−1/2}(1 + ψ_{1}) + ρ_{l}^{(1)}(1 + ψ_{1}) + ∂_{l}ψ_{1})× \\ (γ^{1/2}∂_{j}γ^{−1/2}(1 + ψ_{2} + ρ_{j}^{(2)}(1 + ψ_{2}) + ∂_{j}ψ_{2})]dx \\ = 0. \end{matrix}

Here ∂_{j} = ∂/∂x_{j}. Set ζ := η + iξ and note that

lim_{s→∞} \frac{ρ^{(1)}}{s} = ζ, lim_{s→∞} \frac{ρ^{(2)}}{s} = −ζ.

Therefore, by dividing the both sides of (3.12) by t, taking the limit s → ∞, and using (3.11), we get

(3.13)  ∫_{Ω} ∑_{i=1}^{3} ∑_{1≤j≤l≤3} ζ_{j}ζ_{l}c_{jl}^{i}(x)γ(x)^{−1} \frac{∂v}{∂x_{i}} (x)e^{ik·x}dx = 0.

Since (3.13) holds for all k ∈ R^{3}, we have

(3.14)  ∑_{i=1}^{3} [ ∑_{1≤j≤l≤3} ζ_{j}ζ_{l}c_{jl}^{i}(x)] \frac{∂v}{∂x_{i}} (x) = 0, x ∈ Ω.

We need the following lemma the proof of which will be given at the end of this section.

Lemma 3.1 Suppose n ≥ 2. There exist solutions v_{j} ∈ W^{2,p}(Ω), j = 1, ⋯, n, of ∇ · γ∇v_{j} = 0 in Ω such that

(3.15)  det (\frac{∂v_{j}}{∂x_{i}} (x)) ≠ 0, for almost all x ∈ Ω.

Lemma 3.1 and (3.14) yield

(3.16)  ∑_{1≤j≤l≤n} ζ_{j}ζ_{l}c_{jl}^{i}(x) = 0, x ∈ Ω, i = 1, 2, 3.

Note that (3.16) holds for all ζ in the set V := \{ζ ∈ C^{3}| ζ · ζ = 0, |ζ| = \sqrt{2}\}. If ζ = (0, z, \sqrt{−1}z) for z ∈ C with |z| = 1, then ζ ∈ V. With this ζ, (3.16) becomes

(3.17)  c_{22}^{i}(x) − c_{33}^{i}(x) + \sqrt{−1}c_{23}^{i}(x) = 0, x ∈ Ω.

Since c_{jl}^{i} are real, we have c_{22}^{i}(x) = c_{33}^{i}(x) and c_{23}^{i} = 0. By substituting ζ = (z, 0, \sqrt{−1}z) and ζ = (z, \sqrt{−1}z, 0) in order into (3.16), we obtain that

(3.18)  c_{11}^{i} = c_{22}^{i} = c_{33}^{i} and c_{jl}^{i} = 0 (j ≠ l).

Because of (3.18), (3.6) now takes the form

(3.19)  ∑_{i=1}^{3} ∑_{j=1}^{3} ∫_{Ω} c_{jj}^{i}(x)\frac{∂u_{1}}{∂x_{j}} \frac{∂u_{2}}{∂x_{j}} \frac{∂v}{∂x_{i}} dx = 0

for all u_{,}u_{2}, v. By the same argument as above (using u_{2} and v), we can conclude that

(3.20)  ∑_{j=1}^{3} [ζ_{j} ∑_{i=1}^{3} ζ_{i}c_{jj}^{i}(x)] \frac{∂u_{1}}{∂x_{j}} (x) = 0

for all u_{1} and ζ ∈ V. It then follows from Lemma 3.1 that

(3.21)  ζ_{j} ∑_{i=1}^{3} ζ_{i}c_{jj}^{i}(x) = 0, j = 1, 2, 3,

and we conclude that c_{jj}^{i} = 0, i, j = 1, 2, 3. The proof is complete. □

Proof of Lemma 3.1. We assume that n = 3. The same proof works for the two dimensional case since complex geometrical optics solutions exist in two dimensions. Choose ζ^{(1)}, ζ^{(2)}, ζ^{(3)} ∈ V so that they are linearly independent over C, e.g., ζ^{(1)} = (1, \sqrt{−1}, 0), ζ^{(2)} = (1, 0, \sqrt{−1}), ζ^{(3)} = (0, 1, \sqrt{−1}). Let

v_{j}(x) = γ^{−1/2}e^{tζ^{(j)}·x}(1 + ψ_{j}(x, tζ^{(j)})), x ∈ Ω, j = 1, 2, 3

as before. Then, we have

det (\begin{matrix}∇v_{1} \\ ∇v_{2} \\ ∇v_{3} \end{matrix}) = γ^{−3/2}e^{t(ζ^{(1)}+ζ^{(2)}+ζ^{(3)})·x}t^{3} det (\begin{matrix}ζ^{(1)}(1 + ψ_{1}) + O(t^{−1}) \\ ζ^{(2)}(1 + ψ_{2}) + O(t^{−1}) \\ ζ^{(3)}(1 + ψ_{3}) + O(t^{−1}) \end{matrix}) .

It then follows from (3.11) that if t is sufficiently large, then

det (\begin{matrix}∇v_{1}(x) \\ ∇v_{2}(x) \\ ∇v_{3}(x) \end{matrix}) ≠ 0.

This finishes the proof. □

4 Proof of Theorem 1.3

We need to prove that if

(4.1)  ∑_{i=1}^{n} ∑_{1≤k≤l≤n} c_{kl}^{i} ∫_{Ω} \frac{∂u}{∂x_{k}} \frac{∂u}{∂x_{l}} \frac{∂v}{∂x_{i}} dx = 0

for all u, v ∈ W^{2,p}(Ω) real solutions of ∇ · γ∇u = 0, then

(4.2)  c_{kl}^{i} = 0, i = 1, ⋯, n, 1 ≤ k ≤ l ≤ n.

Since (4.1) holds only for real-valued solutions u, v, we can not use the complex geometrical optics solutions. However, since the coefficients c_{kl}^{i} are assumed to be constants, we can use instead the singular solutions of Alessandrini. We only deal with the three dimensional case. Two dimensional case can be proved in the same way.

Fix j and s ≤ t. We will show that c_{st}^{j} = 0. Let e_{k}, k = 1, 2, 3, is the standard basis of R^{3}. Let α > 0, β > 0, α^{2} + β^{2} = 1. Define an orthonormal frame N, T_{1}, T_{2} by N = αe_{s} + βe_{t}, T_{2} = βe_{s} − αe_{t}, and T_{1} = e_{k} where k ≠ s, t if s ≠ t. In the case s = t, let N = e_{s}, and T_{1} and T_{2} be e_{k} and e_{l} where k ≠ l ≠ s ≠ k. For x ∈ ∂Ω, ν(x) denote the unit outward normal to ∂Ω at x. Choose x_{0} ∈ ∂Ω so that ν(x_{0}) = N. By translation if necessary, we may assume that x_{0} = 0.

Choose a solution v ∈ C^{2}(\overline{Ω}) of ∇ · γ∇v = 0 so that ∂_{i}v(0) = δ_{ij} (i = 1, 2, 3). It can be proved easily that such a function exists. Then

(4.3)  ∂_{i}v(x) = δ_{ij} + O(|x|), x → 0.

Let U be an open neighborhood of 0. Then we have from (4.1) and (4.3) that

(4.4)  | ∑_{1≤k≤l≤n} c_{kl}^{j} ∫_{Ω∩U} ∂_{k}u∂_{l}udx| ≤ C ∫_{Ω} |x||∇u|^{2}dx

for some C > 0.

By [2], for every ϵ > 0, there exists a solution u ∈ C^{2}(\overline{Ω}) such that

u(x) = \frac{1}{|x − ϵN|} + w(x)

where

(4.5)  |∇w(x)| ≤ C|x − ϵN|^{−2+α}

for some α > 0 (α < 1) and C independent of ϵ. In two dimensions we can use log |x − ϵN|. Put Γ_{ϵ}(x) := |x − ϵN|^{−1} to make notations short. Substituting this solution u to (4.4), we get

(4.6)  |∑_{k≤l} c_{kl}^{j} ∫_{Ω∩U} ∂_{k}Γ_{ϵ}∂_{l}Γ_{ϵ}dx| ≤ C ∫_{Ω} |x − ϵN|^{−4+α}dx.

For convenience, put y := x − ϵN. From (4.6), we have

(4.7)  |∑_{k≤l} c_{kl}^{j} ∫_{Ω∩U} \frac{y_{k}y_{l}}{|y|^{6}} dx| ≤ C ∫_{Ω} |x − ϵN|^{−4+α}dx.

Suppose ∂Ω near 0 is given by

∂Ω ∩ U = { ξ_{1}T_{1} + ξ_{2}T_{2} − φ(ξ_{1}, ξ_{2})N | |ξ_{1}|^{2} + |ξ_{2}|^{2} < δ^{2} }

where φ ∈ C^{2}, φ(0) = 0, and ∇φ(0) = 0, for a fixed small δ > 0. Let D := { ξ_{1}T_{1} + ξ_{2}T_{2} − ξ_{3}N | |ξ_{1}|^{2} + |ξ_{2}|^{2} < δ^{2}, 0 < ξ_{3} < δ } and define Φ : D → Ω ∩ U by

Φ(ξ_{1}T_{1} + ξ_{2}T_{2} − ξ_{3}N) := ξ_{1}T_{1} + ξ_{2}T_{2} − (ξ_{3} + φ(ξ_{1}, ξ_{2}))N.

By shrinking U if necessary, we may assume that Φ(D) = Ω ∩ U. Then we have after a coordinate change, (4.7) takes the form

(4.8)  |∑_{k≤l} c_{kl}^{j} ∫_{D} \frac{z_{k}z_{l}}{|z|^{6}} dξ| ≤ C ∫_{Ω} |x − ϵN|^{−4+α}dx.

where z = Φ(ξ) − ϵN, ξ = ξ_{1}T_{1} + ξ_{2}T_{2} − ξ_{3}N ∈ D. We now scale η → ξ = ϵη to obtain

(4.9)  ∫_{D} \frac{z_{k}z_{l}}{|z|^{6}} dξ = ϵ^{−1} ∫_{|η_{1}|^{2}+|η_{2}|^{2}<(δ/ϵ)^{2}} ∫_{η_{3}=0}^{δ/ϵ} \frac{ζ_{k}ζ_{l}}{|ζ|^{6}} dη

where ζ = η_{1}T_{1} + η_{2}T_{2} − (η_{3} + ϵ^{−1}φ(ϵη_{1}, ϵη_{2}) + 1)N. It is easy to prove, by scaling, that

(4.10)  lim_{ϵ→0} ϵ ∫_{Ω} |x − ϵN|^{−4+α}dx = 0.

Let η′ := (η_{1}, η_{2}). Since

ϵ^{−1}|φ(ϵη′)| ≤ Cϵ|η′|^{2} ≤ Cδ|η′|, for |η′| < δ/ϵ,

one can see that

\frac{ζ_{k}ζ_{l}}{|ζ|^{6}} ≤ C \frac{1}{|η|^{4} + 1}

for some C independent of ϵ. Therefore, we obtain from the dominated convergence theorem

(4.11)  lim_{ϵ→0} ∫_{|η′|<δ/ϵ} ∫_{0}^{δ/ϵ} \frac{ζ_{k}ζ_{l}}{|ζ|^{6}} dη = ∫_{R_{+}^{3}} \frac{w_{k}w_{l}}{|w|^{6}} dη

where w = η_{1}T_{1} + η_{2}T_{2} − (η_{3} + 1)N and R_{+}^{3} is the upper half space. Let

I_{kl}^{st} := ∫_{R_{+}^{3}} \frac{w_{k}w_{l}}{|w|^{6}} dη.

It follows from (4.8), (4.9), (4.10) and (4.11) that

(4.12)  ∑_{k≤l} c_{kl}^{j}I_{kl}^{st} = 0.

Suppose that s = t. In this case N was chosen to be N = e_{s}. Thus w_{k} = η_{k} if k ≠ s and w_{s} = −(η_{s} + 1). Therefore easy computations show that

I_{kl}^{ss} = \{ \begin{matrix}0 & if k ≠ l, \\ \frac{π}{4} & if k = l ≠ s, \\ \frac{π}{2} & if k = l = s. \end{matrix}

Thus by (4.12) we get

(4.13)  2c_{ss}^{j} + ∑_{k≠s} c_{kk}^{j} = 0, s = 1, 2, 3.

Hence we conclude that

(4.14)  c_{kk}^{j} = 0, j, k = 1, 2, 3,

and (4.12) takes the form

(4.15)  ∑_{k<l} c_{kl}^{j}I_{kl}^{st} = 0.

Suppose that s < t. If k ≠ s, t, then T_{1} = e_{k} and hence w_{k} = η_{1}, w_{s} = βη_{2}−α(η_{3}+1), and w_{t} = −αη_{2} − β(η_{3} + 1). Again by simple computations one can see that

I_{kl}^{st} = \{ \begin{matrix}0 & if (k, l) ≠ (s, t), k < l, \\ αβ \frac{π}{4} & if (k, l) = (s, t). \end{matrix}

It then follows from (4.15) that for s < t

(4.16)  c_{st}^{j} = 0, j = 1, 2, 3.

By (4.14) and (4.16), the proof is complete. □

5 Appendix- Proof of Theorem 1.1

Replace R(x, q) by R̃(x, q) := χ(q)R(x, q) + (1 − χ(q))|q|^{3} with χ ∈ C_{0}^{∞}(R^{n} or C^{n}) satisfying 0 ≤ χ(q) ≤ 1, χ(q) = 1 (|q| ≤ h/2), 0 (|q| ≥ h). Here, for the complex vector q ∈ C^{n} is identified with (Req, Imq) ∈ R^{2n} and χ(q) is considered as a function of variables Req, Imq. Then, (1.6) holds for any q ∈ R^{n} or C^{n} with some new C_{2} determined by h, χ(q) and the old C_{2}. Suppose that ∥f∥_{W^{2−1/p,p(∂Ω)}} ≤ ϵ (ϵ to be determined later). Let u_{0} be the solution of

(5.1)  \begin{cases}∇ · γ∇u_{0} = 0, in Ω, \\ u_{0}|_{∂Ω} = f. \end{cases}

If u is a solution of (1.1), then v defined by u = u_{0} + v satisfies

(5.2)  \begin{cases}∇ · γ∇v + ∇ · Q(x, ∇u_{0} + ∇v) = 0, in Ω, \\ v|_{∂Ω} = 0. \end{cases}

Let L_{γ}^{−1} be the solution operator for the problem

(5.3)  \begin{cases}∇ · γ∇u = g ∈ L^{p}(Ω), in Ω, \\ u|_{∂Ω} = 0, \end{cases}

namely, L_{γ}^{−1}g is the solution of (5.3). Then L_{γ}^{−1} is linear and there exists C > 0 independent of g such that

∥L_{γ}^{−1}g∥_{2,p} ≤ C∥g∥_{p}.

Let X := {φ ∈ W^{2,p}(Ω) | φ|_{∂Ω} = 0, ∥φ∥_{2,p} ≤ δ } (δ to be determined later) and define an operator A on X by

A(φ)(x) := L_{γ}^{−1}∇ · Q(x, ∇u_{0} + ∇φ).

Then, it follows from (1.4) and (1.6) that

(5.4)  \begin{aligned}∥A(φ)∥_{2,p} & ≤ C∥Q(∇u_{0} + ∇φ)∥_{1,p} \\ & ≤ C(∥u_{0}∥_{2,p}^{2} + ∥φ∥_{2,p}^{2} + ∥u_{0}∥_{2,p}^{3} + ∥φ∥_{2,p}^{3}) \\ & ≤ C(ϵ^{2} + δ^{2} + ϵ^{3} + δ^{3}), \end{aligned}

and (5.5)

∥A(φ_{1}) − A(φ_{2})∥_{2,p} ≤ C∥Q(x, ∇u_{0} + ∇φ_{1}) − Q(x, ∇u_{0} + ∇φ_{2})∥_{1,p}
≤ C(∥u_{0} + θφ_{1} + (1 − θ)φ_{2}∥_{2,p} + ∥u_{0} + θφ_{1} + (1 − θ)φ_{2}∥_{2,p}^{2})
∥∇(φ_{1} − φ_{2})∥_{1,p}
≤ C(ϵ + δ + ϵ^{2} + δ^{2})∥φ_{1} − φ_{2}∥_{2,p}

for some 0 < θ < 1. Therefore, if C(ϵ^{2} + δ^{2} + ϵ^{3} + δ^{3}) ≤ δ and C(ϵ + δ + ϵ^{2} + δ^{2}) < 1, then A is a contraction on X.

By the contraction mapping principle, there exists a unique fixed point of A, say v. Then v is the solution of (5.3). The estimate (1.7) follows from the equation and regularity of the linear elliptic equation.

Finally, we take ϵ and δ more smaller if necessary so that they satisfy C_{3}ϵ + δ ≤ h/2. Then, u = u_{0} + v is the solution of (1.1).

Acknowledgement. The authors thank Prof. Kubota Koji for a useful comment.

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