The Buckling Operator: Inverse Boundary Value Problem

Abstract

In this paper, we consider a zeroth-order perturbation $q\left(x\right)$ of the buckling operator ${\Delta }^2-\kappa \Delta$, which can be uniquely determined by measuring the Dirichlet-to-Neumann data on the boundary. We extend the conclusion of the biharmonic operator to the buckling operator, but the Dirichlet-to-Neumann map given in this study is more meaningful and general.

1. Introduction

Let $\mathsf{\Omega }\subset {\mathbb{R}}^n(n\ge 3)$ be a bounded domain with $C^{\infty }$ boundary $\partial \mathsf{\Omega }$. Let us consider the following buckling operator ${\Delta }^2-\kappa \Delta$ with zeroth-order perturbation:

$${\mathcal{L}}_q(x,\tau )={\Delta }^2-\kappa \Delta +q\left(x\right).$$

Here, parameter $\kappa >0$, $\Delta$ denotes the Laplacian, ${\Delta }^2$ is the biharmonic operator, and the electric potential $q\in L^{\infty }(\mathsf{\Omega },\mathbb{C})$ is a complex-valued function. The buckling operator ${\Delta }^2-\kappa \Delta$, equipped with the domain:

$$H^4\left(\mathsf{\Omega }\right)\cap H_0^2\left(\mathsf{\Omega }\right),H_0^2\left(\mathsf{\Omega }\right)=\{u\in H^2{\left(\mathsf{\Omega }\right):u|}_{\partial \mathsf{\Omega }}={\partial }_{\nu }u{|}_{\partial \mathsf{\Omega }}=0\}$$

is closed, and its spectrum is discrete. Here and in what follows, $H^s\left(\mathsf{\Omega }\right),s\in \mathbb{R}$ is the standard Sobolev space on $\mathsf{\Omega }$. We define the Dirichlet trace of u by $(u,\frac{\partial u}{\partial \nu }){|}_{\partial \mathsf{\Omega }}$, $\forall u\in H^4\left(\mathsf{\Omega }\right)$, where $\nu$ is the unit outer normal to the boundary $\partial \mathsf{\Omega }$. Let us make the assumption that 0 is not an eigenvalue of ${\mathcal{L}}_q(x,\kappa ):H^4\left(\mathsf{\Omega }\right)\cap H_0^2\left(\mathsf{\Omega }\right)\to L^2\left(\mathsf{\Omega }\right)$. Then, for any $g=(g_0,g_1)\in H^{7/2}(\partial \mathsf{\Omega })\times H^{5/2}(\partial \mathsf{\Omega })$, the following boundary value problem:

$$\left\{\begin{array}{cc}{\mathcal{L}}_q(x,\kappa )u=0\hfill & \mathrm{in} \mathsf{\Omega },\hfill \\ u=g_0\hfill & \mathrm{on} \partial \mathsf{\Omega },\hfill \\ \frac{\partial u}{\partial \nu }=g_1\hfill & \mathrm{on} \partial \mathsf{\Omega },\hfill \end{array}\right.$$

(1)

has a unique solution $u\in H^4\left(\mathsf{\Omega }\right)$. We also define the Neumann trace of u by

$$\left((1-\iota )\frac{{\partial }^{2}u}{\partial {\nu }^2}+\iota \Delta u,\kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\right){|}_{\partial \mathsf{\Omega }}\forall u\in H^4\left(\mathsf{\Omega }\right),$$

where Poisson’s ratio $\iota \in (-1/(n-1),1)$ and $P_{\partial \mathsf{\Omega }}$ projects a vector V at a point x on $\partial \mathsf{\Omega }$ onto the tangent space at x on $\partial \mathsf{\Omega }$. The ${\mathrm{div}}_{\partial \mathsf{\Omega }}$ represents the surface divergence, and $D^{2}u$ is the Hessian matrix. The above boundary conditions are called natural or free boundary conditions with nonzero Poisson’s ratio [1,2]. Physically, in two dimensions, the function u is a transverse vibrational mode of the plate, $\mathsf{\Omega }$ is a homogeneous, isotropic plate, and the parameter $\kappa$ describes the ratio of the lateral tension to the flexural rigidity of the plate. Positive $\kappa$ corresponds to a plate under tension, while negative $\kappa$ gives us a plate under compression.

We then define the Dirichlet-to-Neumann map associated with the perturbed buckling operator ${\mathcal{N}}_q:H^{7/2}(\partial \mathsf{\Omega })\times H^{5/2}(\partial \mathsf{\Omega })\to H^{3/2}(\partial \mathsf{\Omega })\times H^{1/2}(\partial \mathsf{\Omega })$ by

$${\mathcal{N}}_q(g_0,g_1)=\left((1-\iota )\frac{{\partial }^{2}u}{\partial {\nu }^2}+\iota \Delta u,\kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\right){|}_{\partial \mathsf{\Omega }},$$

(2)

where $u\in H^4\left(\mathsf{\Omega }\right)$ is the solution to (1). Let us also define the set of Cauchy data ${\mathcal{C}}_q$:

$$\begin{array}{cc}\hfill {\mathcal{C}}_q=\{& \left(u,\frac{\partial u}{\partial \nu },(1-\iota )\frac{{\partial }^{2}u}{\partial {\nu }^2}+\iota \Delta u,\kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\right){|}_{\partial \mathsf{\Omega }}:\hfill \\ \hfill & u\in H^4\left(\mathsf{\Omega }\right),{\mathcal{L}}_{q}u=0\mathrm{in} \mathsf{\Omega }\}.\hfill \end{array}$$

(3)

Notice that, when $\iota =1,\kappa =0$, the Cauchy data for the perturbed buckling operator becomes

$${\left\{\right(u|}_{\partial \mathsf{\Omega }},\frac{\partial u}{\partial \nu }{|}_{\partial \mathsf{\Omega }}{,\Delta u|}_{\partial \mathsf{\Omega }},\frac{\partial \Delta u}{\partial \nu }{|}_{\partial \mathsf{\Omega }}):u\in H^4\left(\mathsf{\Omega }\right),{\mathcal{L}}_{q}u=0\mathrm{in} \mathsf{\Omega }\}$$

For this case, one can see [3].

Higher-order operators appear in the fields of physics and geometry, for example the study of the Paneitz–Branson operator in conformal geometry and the study of the Kirchoff plate equation in the theory of elasticity; see [4]. Thus, the inverse scattering problems for the biharmonic, or more generally polyharmonic, operator are therefore natural to consider. It addresses the recovery of the coefficients from measurements taken on the boundary [5,6,7,8,9], the lower regularity of the coefficients [10,11,12], stability issues [13,14], and the Weyl-type asymptotic formula [15,16,17] from the boundary Dirichlet-to-Neumann data. Apart from works already mentioned, for, the positive problem of elliptic equations we refer the reader to [18,19,20,21,22].

In this paper, we study the inverse problem of recovering the potential $q\left(x\right)$ of the buckling operator ${\Delta }^2-\kappa \Delta$ from the Dirichlet-to-Neumann map, in particular the Neumann trace of u given by the free boundary conditions with nonzero Poisson’s ratio. Thus, the Dirichlet-to-Neumann map given in this study is more meaningful and general. This work provides a fresh look in order to determine the perturbation $q\left(x\right)$, which we do not see often. Our main result is as follows.

Theorem 1. 

Let $\mathsf{\Omega }\subset {\mathbb{R}}^n(n\ge 3)$ be a bounded domain with the $C^{\infty }$ boundary, and let $q^{\left(1\right)},q^{\left(2\right)}\in L^{\infty }(\mathsf{\Omega },\mathbb{C})$. If

$${\mathcal{N}}_{q^{\left(1\right)}}(g_0,g_1)={\mathcal{N}}_{q^{\left(2\right)}}(g_0,g_1)forall(g_0,g_1)\in H^{7/2}(\partial \mathsf{\Omega })\times H^{5/2}(\partial \mathsf{\Omega }),$$

then we have $q^{\left(1\right)}=q^{\left(2\right)}$ in Ω.

In addition, if ${\mathcal{C}}_{q^{\left(1\right)}}={\mathcal{C}}_{q^{\left(2\right)}}$, then ${\mathcal{N}}_{q^{\left(1\right)}}(g_0,g_1)={\mathcal{N}}_{q^{\left(2\right)}}(g_0,g_1)$. Therefore, we can easily obtain the following corollary from Theorem 1.

Corollary 1. 

Under the assumptions of Theorem 1. If ${\mathcal{C}}_{q^{\left(1\right)}}={\mathcal{C}}_{q^{\left(2\right)}}$, then we have $q^{\left(1\right)}=q^{\left(2\right)}$ in Ω.

Remark 1. 

If we consider a similar problem with the following natural boundary conditions:

$$\begin{array}{cc}{\displaystyle \kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-{\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)=f_0}\hfill & on \partial \mathsf{\Omega },\hfill \\ {\displaystyle \frac{{\partial }^{2}u}{\partial {\nu }^2}=f_1}\hfill & on \partial \mathsf{\Omega },\hfill \end{array}$$

where the symbols have the same meanings as above (see [23]), that is $\iota =0$, the Dirichlet-to-Neumann map ${\mathcal{N}}_q:H^{7/2}(\partial \mathsf{\Omega })\times H^{5/2}(\partial \mathsf{\Omega })\to H^{3/2}(\partial \mathsf{\Omega })\times H^{1/2}(\partial \mathsf{\Omega })$ becomes

$${\mathcal{N}}_q(g_0,g_1)=\left(\frac{{\partial }^{2}u}{\partial {\nu }^2},\kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-{\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\right){|}_{\partial \mathsf{\Omega }},$$

where $u\in H^4\left(\mathsf{\Omega }\right)$ is the solution to the problem (1). The perturbation $q\left(x\right)$ also can be uniquely determined by the Dirichlet-to-Neumann data.

The key step in the proof of Theorem 1 is to construct a special class of complex geometric optics (CGO) solutions for the operator ${\mathcal{L}}_q$. The idea of constructing such CGO solutions to the Schrödinger operator goes back to the fundamental paper by Sylvester and Uhlmann [24], which has been extensively used to show the unique recovery of coefficients in many inverse problems, e.g., conductivity equation [25,26,27], fractional Laplacian [28], and the Navier–Stokes equation [29]. We also use the Carleman estimate, which is based on the corresponding method for the Laplacian.

The paper is organized as follows. In Section 2, we construct complex geometric optics solutions on a bounded domain. Section 3 is devoted to the main integral identity for the perturbed buckling operator. The proof of Theorem 1 is given in Section 4.

2. Complex Geometric Optics Solutions

In this section, we shall construct some complex geometric optics solutions to the equation ${\mathcal{L}}_{q}u=0$ on a bounded domain. For the construction of various complex geometric optics solutions and their applications, one can see [30,31,32].

Let $\mathsf{\Omega }\subset {\mathbb{R}}^n(n\ge 3)$ be a bounded domain with the $C^{\infty }$ boundary. We will look for the solutions of ${\mathcal{L}}_{q}u=0$ in $\mathsf{\Omega }$ having the form:

$$u(x,h)=e^{\frac{ix·\rho }{h}}(a(x,\rho )+hr(x,\rho ,h)).$$

(4)

Here, $\rho \in {\mathbb{C}}^n$ satisfies $\rho ·\rho =0$, amplitude $a\in C^{\infty }\left(\overline{\mathsf{\Omega }}\right)$, correction term r satisfies ${\parallel r\parallel }_{H_{scl}^1}=\mathcal{O}\left(1\right)$, and $h>0$ is a small number.

In this paper, we shall consider $\rho ={\rho }^{\left(0\right)}+{\rho }^{\left(1\right)}$ with ${\rho }^{\left(0\right)}$ independent of h, ${\rho }^{\left(1\right)}=\mathcal{O}\left(h\right)$, and $|\mathrm{Re}{\rho }^{\left(0\right)}|=|\mathrm{Im}{\rho }^{\left(0\right)}|=1$. Consider the following conjugated operator:

$$e^{\frac{-ix·\rho }{h}}{h}^4{\mathcal{L}}_q{e}^{\frac{ix·\rho }{h}}={(-h^2\Delta -2i\rho ·h\nabla )}^2+\kappa h^2(-h^2\Delta -2i\rho ·h\nabla )+h^{4}q.$$

In order to get rid of the lowest-order term containing h, we need

$$e^{\frac{-ix·\rho }{h}}{h}^4{\mathcal{L}}_q\left(e^{\frac{ix·\rho }{h}}a\right)=\mathcal{O}\left(h^3\right)$$

in $L^2\left(\mathsf{\Omega }\right)$. Choose $a\in C^{\infty }\left(\overline{\mathsf{\Omega }}\right)$ such that

$${({\rho }^{\left(0\right)}·\nabla )}^{2}a=0\mathrm{in}\mathsf{\Omega }$$

(5)

holds. As ${\rho }^{\left(0\right)}·\nabla$ is a $\overline{\partial }$-operator, so the above equation has a solution $a=a(x,{\rho }^{\left(0\right)})\in C^{\infty }\left(\overline{\mathsf{\Omega }}\right)$.

Definition 1 

(P. 470 of [32]). A real smooth function φ on an open set $\tilde{\mathsf{\Omega }}$ is called a limiting Carleman weight if $\nabla \phi \ne 0$ in $\tilde{\mathsf{\Omega }}$, and the Poisson bracket of $\mathrm{Re}{p}_{\phi }$ and $\mathrm{Im}{p}_{\phi }$ satisfies

$$\{\mathrm{Re}{p}_{\phi },\mathrm{Im}{p}_{\phi }\}(x,\xi )=0when{p}_{\phi }(x,\xi )=0,(x,\xi )\in \tilde{\mathsf{\Omega }}\times {\mathbb{R}}^n,$$

where $p_{\phi }$ is the semiclassical principal symbol of $P_{\phi }=e^{\frac{\phi }{h}}(-h^2\Delta )e^{-\frac{\phi }{h}}$, satisfying

$$p_{\phi }(x,\xi )={\xi }^2+2i\nabla \phi ·\xi -{|\nabla \phi |}^{2}x\in \tilde{\mathsf{\Omega }},\xi \in {\mathbb{R}}^n.$$

In this paper, we shall choose the linear weights $\phi \left(x\right)=\alpha ·x,\alpha \in {\mathbb{R}}^n,\left|\alpha \right|=1$.

Proposition 1 

(Lemma 2.1 of [33]). Let φ be a limiting Carleman weight for the semiclassical Laplacian on $\tilde{\mathsf{\Omega }}$. Then, for all $h>0$ small enough, the Carleman estimate

$$\parallel e^{\frac{\phi }{h}}(-h^2\Delta )e^{-\frac{\phi }{h}}{u\parallel }_{H_{scl}^s}\ge \frac{h}{C_{s,\mathsf{\Omega }}}{\parallel u\parallel }_{H_{scl}^{s+1}},$$

(6)

holds, for all $u\in C_0^{\infty }\left(\mathsf{\Omega }\right)$.

Here and in what follows, we define the norm ${\parallel u\parallel }_{H_{scl}^s}={\left|\right|⟨hD⟩}^{s}u{\left|\right|}_{L^2}$ with $⟨\xi ⟩={(1+|\xi |}^2{)}^{1/2}$ on the semiclassical Sobolev spaces $H_{scl}^s\left({\mathbb{R}}^n\right)(s\in \mathbb{R})$. Since it is a semiclassical space, the parameter h goes to zero, but then, it is no longer a Sobolev space. We set

$${\mathcal{L}}_{\phi }=e^{\frac{\phi }{h}}{h}^4{\mathcal{L}}_q{e}^{-\frac{\phi }{h}}.$$

Proposition 2. 

Assume $q\in L^{\infty }(\mathsf{\Omega },\mathbb{C})$ and that φ is a limiting Carleman weight for the semiclassical Laplacian on $\tilde{\mathsf{\Omega }}$. If $-2\le s\le 0$, then we have

$$\parallel e^{\frac{\phi }{h}}{h}^4{\mathcal{L}}_q{e}^{-\frac{\phi }{h}}{u\parallel }_{H_{scl}^s}\ge \frac{h^2}{C_{s,\mathsf{\Omega },\kappa }}{\parallel u\parallel }_{H_{scl}^{s+2}},$$

(7)

for all $u\in C_0^{\infty }\left(\mathsf{\Omega }\right)$ and $h>0$ small enough.

Proof. 

Notice that $H_{scl}^{s+1}\to H_{scl}^s$, and we have

$${\parallel u\parallel }_{H_{scl}^s}\le \mathcal{O}\left(1\right){\parallel u\parallel }_{H_{scl}^{s+1}}.$$

Due to the Proposition 1, there exists a constant $C_{s,\kappa ,\mathsf{\Omega }}$ such that

$$\parallel e^{\frac{\phi }{h}}(-h^2(\Delta -\kappa ))e^{-\frac{\phi }{h}}{u\parallel }_{H_{scl}^s}\ge \frac{h}{C_{s,\kappa ,\mathsf{\Omega }}}{\parallel u\parallel }_{H_{scl}^{s+1}},$$

for all $u\in C_0^{\infty }\left(\mathsf{\Omega }\right)$ and $h>0$ small enough. Note that we have the following factorization:

$${\Delta }^2-\kappa \Delta =\Delta (\Delta -\kappa ).$$

Applying the Proposition 1 again, we obtain

$$\begin{array}{cc}\hfill \parallel e^{\frac{\phi }{h}}(& -h^2\Delta (-h^2(\Delta -\kappa )){)e^{-\frac{\phi }{h}}u\parallel }_{H_{scl}^s}\hfill \\ \hfill & \ge \frac{h}{C_{s,\mathsf{\Omega }}}\parallel e^{\frac{\phi }{h}}(-h^2(\Delta -\kappa ))e^{-\frac{\phi }{h}}{u\parallel }_{H_{scl}^{s+1}}\ge \frac{h^2}{C_{s,\kappa ,\mathsf{\Omega }}}{\parallel u\parallel }_{H_{scl}^{s+2}}.\hfill \end{array}$$

(8)

Next, we deal with the perturbation $h^{4}q$:

$${\parallel qu\parallel }_{H_{scl}^s}\le {\parallel qu\parallel }_{L^2}\le {\parallel q\parallel }_{L^{\infty }}{\parallel u\parallel }_{L^2}\le {\parallel q\parallel }_{L^{\infty }}{\parallel u\parallel }_{H_{scl}^{s+2}}.$$

for $-2\le s\le 0$. Thus, we obtain the desired result. □

We denote by $H_{scl}^1\left(\mathsf{\Omega }\right)$ the semiclassical Sobolev space of order one on $\mathsf{\Omega }$, equipped with the norm ${\parallel u\parallel }_{H_{scl}^1\left(\mathsf{\Omega }\right)}^2={\parallel u\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2+{\parallel h\nabla u\parallel }_{L^2\left(\mathsf{\Omega }\right)}^2$. The following solvability result is an immediate consequence of the above Carleman estimate.

Proposition 3. 

Assume $q\in L^{\infty }(\mathsf{\Omega },\mathbb{C})$ and φ is a limiting Carleman weight for the semiclassical Laplacian on $\tilde{\mathsf{\Omega }}$. Then, for any $v\in L^2\left(\mathsf{\Omega }\right)$, there exists a solution $u\in H^2\left(\mathsf{\Omega }\right)$ of the equation:

$${\mathcal{L}}_{\phi }u=vin\mathsf{\Omega },$$

such that

$${\parallel u\parallel }_{H_{scl}^2}\le \frac{C}{h^2}{\parallel v\parallel }_{L^2},$$

for all $h>0$ small enough.

 Proof. 

We extend $q\in L^{\infty }(\mathsf{\Omega },\mathbb{C})$ and v to be zero in ${\mathbb{R}}^n\backslash \mathsf{\Omega }$, solve the equation in ${\mathbb{R}}^n$, and set the formal $L^2$-adjoint of ${\mathcal{L}}_{\phi }$:

$${\mathcal{L}}_{\phi }^{*}:=e^{-\frac{\phi }{h}}{h}^4{\mathcal{L}}_{\overline{q}}{e}^{\frac{\phi }{h}}.$$

Define the following complex linear functional:

$$L:{\mathcal{L}}_{\phi }^{*}{C}_0^{\infty }\left(\mathsf{\Omega }\right)\to \mathbb{C},{\mathcal{L}}_{\phi }^{*}w\to {(w,v)}_{L^2}.$$

The map L is well defined from Proposition 2 with ${\mathcal{L}}_{\phi }$ replaced by ${\mathcal{L}}_{\phi }^{*}$. For any $w\in C_0^{\infty }\left(\mathsf{\Omega }\right)$, we have

$$|L\left({\mathcal{L}}_{\phi }^{*}w\right)|=|{(w,v)}_{L^2}{|\le \parallel w\parallel }_{L^2}{\parallel v\parallel }_{L^2}\le \frac{C}{h^2}\parallel {\mathcal{L}}_{\phi }^{*}{w\parallel }_{H_{scl}^{-2}}{\parallel v\parallel }_{L^2}.$$

Thus, L is bounded in the $H^{-2}$-norm. Now, by the Hahn–Banach theorem, we extend L to $\tilde{L}$, which is a linear continuous functional on $H^{-2}\left({\mathbb{R}}^n\right)$. Applying the Riesz representation theorem, there exists $u\in H^2\left({\mathbb{R}}^n\right)$ such that

$$\tilde{L}\left(w\right)={(w,u)}_{(H^{-2},H^2)}{\mathrm{and}\parallel u\parallel }_{H_{scl}^2}\le \frac{C}{h^2}{\parallel v\parallel }_{L^2},$$

for all $w\in H^{-2}\left({\mathbb{R}}^n\right)$. Here, ${(·,·)}_{(H^{-2},H^2)}$ represents for the usual $L^2$-duality. □

Now, we construct the complex geometric optics solution of Equation (4). From (5), we have

$$e^{\frac{-ix·\rho }{h}}{h}^4{\mathcal{L}}_q{e}^{\frac{ix·\rho }{h}}a=\mathcal{O}\left(h^3\right).$$

Applying Proposition 3, we can obtain a solution $r\in H^1\left(\mathsf{\Omega }\right)$ such that

$$e^{\frac{-ix·\rho }{h}}{h}^4{\mathcal{L}}_q{e}^{\frac{ix·\rho }{h}}\left(hr\right)=-e^{\frac{-ix·\rho }{h}}{h}^4{\mathcal{L}}_q{e}^{\frac{ix·\rho }{h}}a\mathrm{in}\mathsf{\Omega },$$

for $h>0$ small enough.

Summing up, we have the following result.

Proposition 4. 

Assume $q\in L^{\infty }(\mathsf{\Omega },\mathbb{C})$ and $\rho \in {\mathbb{C}}^n$ such that $\rho ·\rho =0,\left|\mathrm{Re}\rho \right|=\left|\mathrm{Im}\rho \right|=1$, then there exist solutions $u(x,\rho ;h)\in H^1\left(\mathsf{\Omega }\right)$ to the equation ${\mathcal{L}}_{q}u=0$ in Ω, having the following form:

$$u(x,\rho ;h)=e^{\frac{ix·\rho }{h}}(a(x,\rho )+hr(x,\rho ,h)),$$

for all $h>0$ small enough. Here, $a(·,\rho )\in C^{\infty }\left(\overline{\mathsf{\Omega }}\right)$ satisfies (5) and ${\parallel r\parallel }_{H_{scl}^1}=\mathcal{O}\left(1\right)$.

Remark 2. 

By elliptic regularity, we know that the complex geometric optics solutions belong to $H^4\left(\mathsf{\Omega }\right)$.

3. An Integral Identity

In this section, we give Green’s formula for the buckling operator. Now, multiplying ${\mathcal{L}}_{q}u=0$ by $\varphi$ and integrating yield:

$${\int }_{\mathsf{\Omega }}[(1-\iota )D^{2}u:D^2\varphi +\iota \Delta u\Delta \varphi +\kappa Du·D\varphi +qu\varphi ]dx=0,$$

where $\varphi \in H^2\left(\mathsf{\Omega }\right)$ is a test function and

$$D^{2}u:D^2\varphi =\sum _{i,j=1}^n\frac{{\partial }^{2}u}{\partial x_i\partial x_j}\frac{{\partial }^2\varphi }{\partial x_i\partial x_j},$$

with ${\left\{x_i\right\}}_{1\le i\le n}$ the Cartesian coordinates of ${\mathbb{R}}^n$, and $dx,dS$ are volume densities on $\mathsf{\Omega }$ and $\partial \mathsf{\Omega }$, respectively. Applying the divergence theorem twice:

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\iota \Delta u·\Delta \varphi dx& =\iota {\int }_{\partial \mathsf{\Omega }}\Delta u\frac{\partial \varphi }{\partial \nu }dS-\iota {\int }_{\mathsf{\Omega }}D(\Delta u)·D\varphi dx\hfill \\ \hfill & =\iota {\int }_{\partial \mathsf{\Omega }}[\Delta u\frac{\partial \varphi }{\partial \nu }-\varphi \frac{\partial (\Delta u)}{\partial \nu }]dS+\iota {\int }_{\mathsf{\Omega }}{\Delta }^{2}u·\varphi dx.\hfill \end{array}$$

Thus, the Hessian term becomes

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}(1-\iota )D^{2}u:D^2\varphi dx& =(1-\iota )\sum _{i,j=1}^n{\int }_{\mathsf{\Omega }}\frac{{\partial }^{2}u}{\partial x_i\partial x_j}\frac{{\partial }^2\varphi }{\partial x_i\partial x_j}dx\hfill \\ \hfill & =(1-\iota )\sum _{j=1}^n{\int }_{\mathsf{\Omega }}D\left(u_{x_j}\right)·D\left({\varphi }_{x_j}\right)dx\hfill \\ \hfill & =(1-\iota )\left[\sum _{j=1}^n{\int }_{\partial \mathsf{\Omega }}{\varphi }_{x_j}\frac{\partial \left(u_{x_j}\right)}{\partial \nu }dS-\sum _{j=1}^n{\int }_{\mathsf{\Omega }}\Delta \left(u_{x_j}\right)·{\varphi }_{x_j}dx\right]\hfill \\ \hfill & =(1-\iota )\left\{{\int }_{\partial \mathsf{\Omega }}[D\varphi ·(D^{2}u·\nu )-\varphi \frac{\partial (\Delta u)}{\partial \nu }]dS+{\int }_{\mathsf{\Omega }}\varphi ·{\Delta }^{2}udx\right\}.\hfill \end{array}$$

Note that the gradient ${\mathrm{grad}}_{\partial \mathsf{\Omega }}$ equals $D-\nu {\partial }_{\nu }$ on a neighborhood of the boundary. By the divergence theorem on $\partial \mathsf{\Omega }$,

$$\begin{array}{cc}\hfill & {\int }_{\partial \mathsf{\Omega }}D\varphi ·(\left(D^{2}u\right)·\nu )dS\hfill \\ \hfill & ={\int }_{\partial \mathsf{\Omega }}(\nu \frac{\partial \varphi }{\partial \nu }+{\mathrm{grad}}_{\partial \mathsf{\Omega }}\varphi )·(\nu \frac{{\partial }^{2}u}{\partial {\nu }^2}+P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right])dS\hfill \\ \hfill & ={\int }_{\partial \mathsf{\Omega }}\frac{\partial \varphi }{\partial \nu }\frac{{\partial }^{2}u}{\partial {\nu }^2}+⟨{\mathrm{grad}}_{\partial \mathsf{\Omega }}\varphi ,P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]⟩dS\hfill \\ \hfill & ={\int }_{\partial \mathsf{\Omega }}\frac{\partial \varphi }{\partial \nu }\frac{{\partial }^{2}u}{\partial {\nu }^2}-\varphi {\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)dS.\hfill \end{array}$$

Thus, the Hessian term equals

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}(1-\iota )D^{2}u:D^2\varphi dx=& (1-\iota )\{{\int }_{\partial \mathsf{\Omega }}[\frac{\partial \varphi }{\partial \nu }\frac{{\partial }^{2}u}{\partial {\nu }^2}-\varphi {\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\hfill \\ \hfill & -\varphi \frac{\partial (\Delta u)}{\partial \nu }]dS+{\int }_{\mathsf{\Omega }}\varphi ·{\Delta }^{2}udx\}.\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}({\Delta }^{2}u& -\kappa \Delta u+qu)\varphi dx\hfill \\ \hfill =& -(1-\iota ){\int }_{\partial \mathsf{\Omega }}[\frac{\partial \varphi }{\partial \nu }\frac{{\partial }^{2}u}{\partial {\nu }^2}-\varphi {\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)-\varphi \frac{\partial (\Delta u)}{\partial \nu }]dS\hfill \\ \hfill & -\iota {\int }_{\partial \mathsf{\Omega }}[\Delta u\frac{\partial \varphi }{\partial \nu }-\varphi \frac{\partial (\Delta u)}{\partial \nu }]dS-\kappa {\int }_{\partial \mathsf{\Omega }}\varphi \frac{\partial u}{\partial \nu }dS\hfill \\ \hfill =& -{\int }_{\partial \mathsf{\Omega }}\left[\kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\right]\varphi dS\hfill \\ \hfill & -{\int }_{\partial \mathsf{\Omega }}\left[(1-\iota )\frac{{\partial }^{2}u}{\partial {\nu }^2}+\iota \Delta u\right]\frac{\partial \varphi }{\partial \nu }dS.\hfill \end{array}$$

Summing up, Green’s formula of the buckling operator gives

$$\begin{array}{cc}\hfill {({\mathcal{L}}_{q}u,v)}_{L^2\left(\mathsf{\Omega }\right)}& -{(u,{\mathcal{L}}_q^{*}v)}_{L^2\left(\mathsf{\Omega }\right)}\hfill \\ \hfill =& -{\int }_{\partial \mathsf{\Omega }}\left[\kappa \frac{\partial u}{\partial \nu }-\frac{\partial \Delta u}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}u\right)\nu \right]\right)\right]vdS\hfill \\ \hfill & -{\int }_{\partial \mathsf{\Omega }}\left[(1-\iota )\frac{{\partial }^{2}u}{\partial {\nu }^2}+\iota \Delta u\right]\frac{\partial v}{\partial \nu }dS\hfill \\ \hfill & +{\int }_{\partial \mathsf{\Omega }}\left[\kappa \frac{\partial v}{\partial \nu }-\frac{\partial \Delta v}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \mathsf{\Omega }}\left[\left(D^{2}v\right)\nu \right]\right)\right]udS\hfill \\ \hfill & +{\int }_{\partial \mathsf{\Omega }}\left[(1-\iota )\frac{{\partial }^{2}v}{\partial {\nu }^2}+\iota \Delta v\right]\frac{\partial u}{\partial \nu }dS,\hfill \end{array}$$

for all $u,v\in H^4\left(\mathsf{\Omega }\right)$.

4. Proof of Theorem 1

In this section, we shall prove Theorem 1. We start with the following result.

Proposition 5. 

Let $\mathsf{\Omega }\subset \subset \tilde{\mathsf{\Omega }}$ be two bounded domains in ${\mathbb{R}}^n$ with smooth boundaries. Assume that ${\tilde{q}}^{\left(1\right)},{\tilde{q}}^{\left(2\right)}\in L^{\infty }(\tilde{\mathsf{\Omega }},\mathbb{C})$ satisfy ${\tilde{q}}^{\left(1\right)}={\tilde{q}}^{\left(2\right)}$ in $\tilde{\mathsf{\Omega }}\backslash \mathsf{\Omega }$. If $C_{{\tilde{q}}^{\left(1\right)}}^{\mathsf{\Omega }}=C_{{\tilde{q}}^{\left(2\right)}}^{\mathsf{\Omega }}$, then we have $C_{{\tilde{q}}^{\left(1\right)}}^{\tilde{\mathsf{\Omega }}}=C_{{\tilde{q}}^{\left(2\right)}}^{\tilde{\mathsf{\Omega }}}$.

Proof. 

Let $\tilde{u}\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ be a solution of ${\mathcal{L}}_{{\tilde{q}}^{\left(1\right)}}\tilde{u}=0$ in $\tilde{\mathsf{\Omega }}$. Since $C_{{\tilde{q}}^{\left(1\right)}}^{\mathsf{\Omega }}=C_{{\tilde{q}}^{\left(2\right)}}^{\mathsf{\Omega }}$, there exists $v\in H^4\left(\mathsf{\Omega }\right)$, solving ${\mathcal{L}}_{{\tilde{q}}^{\left(2\right)}}v=0$ in $\mathsf{\Omega }$ and satisfying

$$(v,\frac{\partial v}{\partial \nu }){|}_{\partial \mathsf{\Omega }}=(\tilde{u},\frac{\partial \tilde{u}}{\partial \nu }){|}_{\partial \mathsf{\Omega }},$$

and

$$\begin{array}{cc}\hfill & \left((1-\iota )\frac{{\partial }^{2}v}{\partial {\nu }^2}+\iota \Delta v,\kappa \frac{\partial v}{\partial \nu }-\frac{\partial \Delta v}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \nu }\left[\left(D^{2}v\right)\nu \right]\right)\right){|}_{\partial \mathsf{\Omega }}\hfill \\ \hfill =& \left((1-\iota )\frac{{\partial }^2\tilde{u}}{\partial {\nu }^2}+\iota \Delta \tilde{u},\kappa \frac{\partial \tilde{u}}{\partial \nu }-\frac{\partial \Delta \tilde{u}}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \nu }\left[\left(D^2\tilde{u}\right)\nu \right]\right)\right){|}_{\partial \mathsf{\Omega }}.\hfill \end{array}$$

Setting

$$\tilde{v}=\left\{\begin{array}{c}v\mathrm{in}\mathsf{\Omega },\hfill \\ \tilde{u}\mathrm{in}\tilde{\mathsf{\Omega }}\backslash \mathsf{\Omega },\hfill \end{array}\right.$$

we obtain $\tilde{v}\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ and ${\mathcal{L}}_{{\tilde{q}}^{\left(2\right)}}\tilde{v}=0$ in $\tilde{\mathsf{\Omega }}$. Thus, $C_{{\tilde{q}}^{\left(1\right)}}^{\tilde{\mathsf{\Omega }}}\subset C_{{\tilde{q}}^{\left(2\right)}}^{\tilde{\mathsf{\Omega }}}$. Another direction with the same argument shows this claim. □

Let $\tilde{\mathsf{\Omega }}$ be an open domain in ${\mathbb{R}}^n$, such that $\mathsf{\Omega }\subset \subset \tilde{\mathsf{\Omega }}$. We extend $q^{\left(j\right)}\in L^{\infty }\left(\mathsf{\Omega }\right)(j=1,2)$ to be zero in $\tilde{\mathsf{\Omega }}\backslash \mathsf{\Omega }$ and denote the extensions ${\tilde{q}}^{\left(j\right)}$. Thanks to Proposition 5, we have $C_{{\tilde{q}}^{\left(1\right)}}^{\tilde{\mathsf{\Omega }}}=C_{{\tilde{q}}^{\left(2\right)}}^{\tilde{\mathsf{\Omega }}}$. Let ${\tilde{u}}_1\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ satisfy the following equation:

$$\left\{\begin{array}{cc}{\mathcal{L}}_{{\tilde{q}}^{\left(1\right)}}(x,\kappa )u=0\hfill & \mathrm{in}\tilde{\mathsf{\Omega }},\hfill \\ u=g_0\hfill & \mathrm{on}\partial \tilde{\mathsf{\Omega }},\hfill \\ \frac{\partial u}{\partial \nu }=g_1\hfill & \mathrm{on}\partial \tilde{\mathsf{\Omega }},\hfill \end{array}\right.$$

Since $C_{{\tilde{q}}^{\left(1\right)}}^{\tilde{\mathsf{\Omega }}}=C_{{\tilde{q}}^{\left(2\right)}}^{\tilde{\mathsf{\Omega }}}$, then there exists a solution ${\tilde{u}}_2\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ that satisfies

$$\begin{array}{cc}\hfill {\mathcal{L}}_{{\tilde{q}}^{\left(2\right)}}& (x,\kappa )u=0\mathrm{in}\tilde{\mathsf{\Omega }},\hfill \\ \hfill {\tilde{u}}_2={\tilde{u}}_1& \frac{\partial {\tilde{u}}_2}{\partial \nu }=\frac{\partial {\tilde{u}}_1}{\partial \nu }\mathrm{on}\partial \tilde{\mathsf{\Omega }},\hfill \end{array}$$

$$(1-\iota )\frac{{\partial }^2{\tilde{u}}_2}{\partial {\nu }^2}+\iota \Delta {\tilde{u}}_2=(1-\iota )\frac{{\partial }^2{\tilde{u}}_2}{\partial {\nu }^2}+\iota \Delta {\tilde{u}}_1\mathrm{on}\partial \tilde{\mathsf{\Omega }},$$

$$\begin{array}{cc}\hfill & \kappa \frac{\partial {\tilde{u}}_1}{\partial \nu }-\frac{\partial \Delta {\tilde{u}}_1}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \nu }\left[\left(D^2{\tilde{u}}_2\right)\nu \right]\right)\hfill \\ \hfill =& \kappa \frac{\partial {\tilde{u}}_1}{\partial \nu }-\frac{\partial \Delta {\tilde{u}}_1}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \nu }\left[\left(D^2{\tilde{u}}_1\right)\nu \right]\right)\mathrm{on}\partial \tilde{\mathsf{\Omega }}.\hfill \end{array}$$

Setting $w:={\tilde{u}}_1-{\tilde{u}}_2$, we have

$${\mathcal{L}}_{{\tilde{q}}^{\left(1\right)}}\left(w\right)=({\tilde{q}}^{\left(2\right)}-{\tilde{q}}^{\left(1\right)}){\tilde{u}}_2\mathrm{in}\tilde{\mathsf{\Omega }}.$$

and

$$w=\frac{\partial w}{\partial \nu }=(1-\iota )\frac{{\partial }^{2}w}{\partial {\nu }^2}+\iota \Delta w=0\mathrm{on} \partial \tilde{\mathsf{\Omega }}$$

$$\kappa \frac{\partial w}{\partial \nu }-\frac{\partial \Delta w}{\partial \nu }-(1-\iota ){\mathrm{div}}_{\partial \mathsf{\Omega }}\left(P_{\partial \nu }\left[\left(D^{2}w\right)\nu \right]\right)=0\mathrm{on} \partial \tilde{\mathsf{\Omega }}$$

Therefore, we have the following Green’s formula:

$${({\mathcal{L}}_{\tilde{q}}\tilde{u},\tilde{v})}_{L^2\left(\tilde{\mathsf{\Omega }}\right)}={(\tilde{u},{\mathcal{L}}_{\tilde{q}}^{*}\tilde{v})}_{L^2\left(\tilde{\mathsf{\Omega }}\right)}\tilde{u},\tilde{v}\in H^4\left(\tilde{\mathsf{\Omega }}\right).$$

(9)

Let $\tilde{v}\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ be a solution of

$${\mathcal{L}}_q^{*}\tilde{v}=0\mathrm{in}\tilde{\mathsf{\Omega }}.$$

Applying Green’s formula (9) over $\tilde{\mathsf{\Omega }}$, we obtain

$${\int }_{\tilde{\mathsf{\Omega }}}({\tilde{q}}^{\left(2\right)}-{\tilde{q}}^{\left(1\right)}){\tilde{u}}_2\overline{\tilde{v}}dx=0.$$

(10)

Next, we shall construct the appropriate complex geometric optics solutions to show ${\tilde{q}}^{\left(2\right)}={\tilde{q}}^{\left(1\right)}$. Let $\xi ,{\mu }_1,{\mu }_2\in {\mathbb{R}}^n$ be such that $|{\mu }_1|=|{\mu }_2|=1$ and ${\mu }_1·{\mu }_2={\mu }_1·\xi ={\mu }_2·\xi =0$. Similarly, in [34], we let $h>0$ small enough and set

$${\rho }_1=-\frac{h\xi }{2}+\sqrt{1-h^2\frac{{\left|\xi \right|}^2}{4}}{\mu }_1-i{\mu }_2,{\rho }_2=\frac{h\xi }{2}+\sqrt{1-h^2\frac{{\left|\xi \right|}^2}{4}}{\mu }_1+i{\mu }_2,$$

such that ${\rho }_j·{\rho }_j=0,j=1,2$ and ${\rho }_2-\overline{{\rho }_1}=h\xi$. Then, by Proposition 4, there exist solutions ${\tilde{u}}_2(x,{\rho }_2;h)\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ and $\tilde{v}(x,{\rho }_1;h)\in H^4\left(\tilde{\mathsf{\Omega }}\right)$ to the equations ${\mathcal{L}}_{{\tilde{q}}^{\left(2\right)}}{u}_2=0$ and ${\mathcal{L}}_{{\tilde{q}}^{\left(1\right)}}^{*}v=0$ in $\tilde{\mathsf{\Omega }}$, respectively, of the form:

$${\tilde{u}}_2(x,{\rho }_2;h)=e^{\frac{ix·{\rho }_2}{h}}(a_2(x,{\mu }_1+i{\mu }_2)+h{r}_2(x,{\rho }_2,h)),$$

(11)

$$\tilde{v}(x,{\rho }_1;h)=e^{\frac{ix·{\rho }_1}{h}}(a_1(x,{\mu }_1-i{\mu }_2)+h{r}_1(x,{\rho }_1,h)),$$

(12)

for $h>0$ small enough. Here, the amplitudes $a_1(·,{\mu }_1-i{\mu }_2),a_2(·,{\mu }_1+i{\mu }_2)\in C^{\infty }\left(\overline{\tilde{\mathsf{\Omega }}}\right)$ satisfy the following transport equations:

$${(({\mu }_1+i{\mu }_2)·\nabla )}^2{a}_2(x,{\mu }_1+i{\mu }_2)=0\mathrm{in}\tilde{\mathsf{\Omega }},$$

$${(({\mu }_1-i{\mu }_2)·\nabla )}^2{a}_1(x,{\mu }_1-i{\mu }_2)=0\mathrm{in}\tilde{\mathsf{\Omega }},$$

and

$$\parallel r_j{\parallel }_{H_{scl}^1}=\mathcal{O}\left(1\right),j=1,2.$$

(13)

Substituting ${\tilde{u}}_2$ and $\tilde{v}$, given by (11) and (12), into (10), we obtain

$${\int }_{\tilde{\mathsf{\Omega }}}({\tilde{q}}^{\left(2\right)}-{\tilde{q}}^{\left(1\right)})e^{ix·\xi }(a_2+h{r}_2)(\overline{a_1}+h\overline{r_1})dx=0.$$

Letting $h\to 0^{+}$, we obtain

$${\int }_{\tilde{\mathsf{\Omega }}}({\tilde{q}}^{\left(2\right)}-{\tilde{q}}^{\left(1\right)})e^{ix·\xi }{a}_2\overline{a_1}dx=0.$$

(14)

Here, we use (13) and the fact $a_1(·,{\mu }_1-i{\mu }_2),a_2(·,{\mu }_1+i{\mu }_2)\in C^{\infty }\left(\overline{\tilde{\mathsf{\Omega }}}\right)$, to conclude that

$$\left|{\int }_{\tilde{\mathsf{\Omega }}}{r}_2\overline{r_1}dx\right|\le {\left({\int }_{\tilde{\mathsf{\Omega }}}{r}_2^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\tilde{\mathsf{\Omega }}}{r}_1^{2}dx\right)}^{\frac{1}{2}}\le \mathcal{O}\left(1\right),$$

$$|{\int }_{\tilde{\mathsf{\Omega }}}{a}_2\overline{r_1}dx|\le {\left({\int }_{\tilde{\mathsf{\Omega }}}{a}_2^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\tilde{\mathsf{\Omega }}}{r}_1^{2}dx\right)}^{\frac{1}{2}}\le \mathcal{O}\left(1\right),$$

and

$$|{\int }_{\tilde{\mathsf{\Omega }}}{r}_2\overline{a_1}dx|\le {\left({\int }_{\tilde{\mathsf{\Omega }}}{r}_2^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_{\tilde{\mathsf{\Omega }}}{a}_1^{2}dx\right)}^{\frac{1}{2}}\le \mathcal{O}\left(1\right).$$

Obviously, $a_1=a_2=1$ satisfies the transport equations. Thus, we insert $a_1=a_2=1$ in (14), and we have

$${\int }_{\tilde{\mathsf{\Omega }}}({\tilde{q}}^{\left(2\right)}-{\tilde{q}}^{\left(1\right)})e^{ix·\xi }dx=0,$$

for all $\xi \in {\mathbb{R}}^n$. By the uniqueness of the inverse Fourier transformation ${\tilde{q}}^{\left(2\right)}-{\tilde{q}}^{\left(1\right)}=0$ in $\tilde{\mathsf{\Omega }}$, hence $q^{\left(2\right)}=q^{\left(1\right)}$ in $\mathsf{\Omega }$. This completes the proof of Theorem 1.