Simultaneous Identification of the Diffusion Coefficient and the   Potential for the Schr\"odinger Operator with only one Observation

Laure Cardoulis (MIP); Patricia Gaitan (LATP)

arXiv:0911.3300·math.AP·February 1, 2010

Simultaneous Identification of the Diffusion Coefficient and the Potential for the Schr\"odinger Operator with only one Observation

Laure Cardoulis (MIP), Patricia Gaitan (LATP)

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TL;DR

This paper establishes a stability result for simultaneously identifying the diffusion coefficient and potential in a Schrödinger operator within an unbounded strip, using minimal boundary and temporal data.

Contribution

It provides the first stability estimate for jointly recovering two coefficients of a Schrödinger operator with limited boundary observations.

Findings

01

Proves stability for coefficient identification with one boundary observation

02

Demonstrates the feasibility of coefficient recovery with minimal data

03

Extends previous results to unbounded domains

Abstract

This article is devoted to prove a stability result for two independent coefficients for a Schr\"odinger operator in an unbounded strip. The result is obtained with only one observation on an unbounded subset of the boundary and the data of the solution at a fixed time on the whole domain.

Equations135

\left\{\begin{array}[]{ll}Hq:=i\partial_{t}q+a\Delta q+bq=0\;\mbox{ in }\;\Omega\times(0,T),\\ q(x,t)=F(x,t)\;\mbox{ on }\;\partial\Omega\times(0,T),\\ q(x,0)=q_{0}(x)\;\mbox{ in }\;\Omega,\end{array}\right.

\left\{\begin{array}[]{ll}Hq:=i\partial_{t}q+a\Delta q+bq=0\;\mbox{ in }\;\Omega\times(0,T),\\ q(x,t)=F(x,t)\;\mbox{ on }\;\partial\Omega\times(0,T),\\ q(x,0)=q_{0}(x)\;\mbox{ in }\;\Omega,\end{array}\right.

∥ a - a ∥_{L^{2} (Ω)}^{2} + ∥ b - b ∥_{L^{2} (Ω)}^{2}

∥ a - a ∥_{L^{2} (Ω)}^{2} + ∥ b - b ∥_{L^{2} (Ω)}^{2}

i \partial_{t} q + a Δ q + b q = f .

i \partial_{t} q + a Δ q + b q = f .

D^{2}\widetilde{\beta}=\left(\begin{array}[]{cc}\partial_{x_{1}}(a^{2}\partial_{x_{1}}\widetilde{\beta})&\partial_{x_{1}}(a^{2}\partial_{x_{2}}\widetilde{\beta})\\ \partial_{x_{2}}(a^{2}\partial_{x_{1}}\widetilde{\beta})&\partial_{x_{2}}(a^{2}\partial_{x_{2}}\widetilde{\beta})\end{array}\right).

D^{2}\widetilde{\beta}=\left(\begin{array}[]{cc}\partial_{x_{1}}(a^{2}\partial_{x_{1}}\widetilde{\beta})&\partial_{x_{1}}(a^{2}\partial_{x_{2}}\widetilde{\beta})\\ \partial_{x_{2}}(a^{2}\partial_{x_{1}}\widetilde{\beta})&\partial_{x_{2}}(a^{2}\partial_{x_{2}}\widetilde{\beta})\end{array}\right).

A = 2 \partial_{x_{2}} (a^{2} \partial_{x_{2}} \tilde{β}) - \partial_{x_{2}} a \partial_{x_{2}} \tilde{β} + 2 a^{2} (\partial_{x_{2}} \tilde{β})^{2} \geq cs t > 0

A = 2 \partial_{x_{2}} (a^{2} \partial_{x_{2}} \tilde{β}) - \partial_{x_{2}} a \partial_{x_{2}} \tilde{β} + 2 a^{2} (\partial_{x_{2}} \tilde{β})^{2} \geq cs t > 0

- \frac{( \partial _{x_{1}} ( a ^{2} \partial _{x_{2}} β ~ ) ) ^{2}}{A} - \partial_{x_{2}} a \partial_{x_{2}} \tilde{β} \geq cs t > 0.

- \frac{( \partial _{x_{1}} ( a ^{2} \partial _{x_{2}} β ~ ) ) ^{2}}{A} - \partial_{x_{2}} a \partial_{x_{2}} \tilde{β} \geq cs t > 0.

φ (x, t) = \frac{e ^{λ β (x)}}{( T + t ) ( T - t )}, η (x, t) = \frac{e ^{2 λ K} - e ^{λ β (x)}}{( T + t ) ( T - t )} .

φ (x, t) = \frac{e ^{λ β (x)}}{( T + t ) ( T - t )}, η (x, t) = \frac{e ^{2 λ K} - e ^{λ β (x)}}{( T + t ) ( T - t )} .

H q := i \partial_{t} q + a Δ q + b q \mbox in Ω \times (- T, T) .

H q := i \partial_{t} q + a Δ q + b q \mbox in Ω \times (- T, T) .

M_{1} ψ

M_{1} ψ

M_{2} ψ

\int_{- T}^{T} \int_{Ω} ∣ M ψ ∣^{2} d x d t

\int_{- T}^{T} \int_{Ω} ∣ M ψ ∣^{2} d x d t

s^{3} λ^{4} \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ q ∣^{2} d x d t + s λ \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣\nabla q ∣^{2} d x d t + ∥ M_{1} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2}

s^{3} λ^{4} \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ q ∣^{2} d x d t + s λ \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣\nabla q ∣^{2} d x d t + ∥ M_{1} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2}

+ ∥ M_{2} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2} \leq C s λ \int_{- T}^{T} \int_{Γ^{+}} e^{- 2 sη} ∣ \partial_{ν} q ∣^{2} \partial_{ν} β d σ d t

+ ∥ M_{2} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2} \leq C s λ \int_{- T}^{T} \int_{Γ^{+}} e^{- 2 sη} ∣ \partial_{ν} q ∣^{2} \partial_{ν} β d σ d t

+ \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ H q ∣^{2} d x d t,

s^{3} λ^{4} \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ q ∣^{2} d x d t + s λ \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣\nabla q ∣^{2} d x d t + ∥ M_{1} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2}

s^{3} λ^{4} \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ q ∣^{2} d x d t + s λ \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣\nabla q ∣^{2} d x d t + ∥ M_{1} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2}

+ ∥ M_{2} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2} + s^{- 1} λ^{- 1} \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ i \partial_{t} q + a Δ q ∣^{2} d x d t

+ ∥ M_{2} (e^{- sη} q) ∥_{L^{2} (Ω \times (- T, T))}^{2} + s^{- 1} λ^{- 1} \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ i \partial_{t} q + a Δ q ∣^{2} d x d t

\leq C [s λ \int_{- T}^{T} \int_{Γ^{+}} e^{- 2 sη} ∣ \partial_{ν} q ∣^{2} \partial_{ν} β d σ d t + \int_{- T}^{T} \int_{Ω} e^{- 2 sη} ∣ H q ∣^{2} d x d t] .

ℜ (\int_{- T}^{T} \int_{Ω} M_{1} ψ \overline{M_{2} ψ} d x d t) = i = 1 \sum 4 j = 1 \sum 3 I_{ij}

ℜ (\int_{- T}^{T} \int_{Ω} M_{1} ψ \overline{M_{2} ψ} d x d t) = i = 1 \sum 4 j = 1 \sum 3 I_{ij}

I_{11} = ℜ (\int_{- T}^{T} \int_{Ω} (i \partial_{t} ψ) (- i s \partial_{t} η \overline{ψ}) d x d t), I_{12} = ℜ (\int_{- T}^{T} \int_{Ω} (i \partial_{t} ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t),

I_{11} = ℜ (\int_{- T}^{T} \int_{Ω} (i \partial_{t} ψ) (- i s \partial_{t} η \overline{ψ}) d x d t), I_{12} = ℜ (\int_{- T}^{T} \int_{Ω} (i \partial_{t} ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t),

I_{13} = ℜ (\int_{- T}^{T} \int_{Ω} (i \partial_{t} ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t), I_{21} = ℜ (\int_{- T}^{T} \int_{Ω} (a Δ ψ) (- i s \partial_{t} η \overline{ψ}) d x d t),

I_{13} = ℜ (\int_{- T}^{T} \int_{Ω} (i \partial_{t} ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t), I_{21} = ℜ (\int_{- T}^{T} \int_{Ω} (a Δ ψ) (- i s \partial_{t} η \overline{ψ}) d x d t),

I_{22} = ℜ (\int_{- T}^{T} \int_{Ω} (a Δ ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t), I_{23} = ℜ (\int_{- T}^{T} \int_{Ω} (a Δ ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t),

I_{22} = ℜ (\int_{- T}^{T} \int_{Ω} (a Δ ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t), I_{23} = ℜ (\int_{- T}^{T} \int_{Ω} (a Δ ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t),

I_{31} = ℜ (\int_{- T}^{T} \int_{Ω} (s^{2} a ∣\nabla η ∣^{2} ψ) (- i s \partial_{t} η \overline{ψ}) d x d t), I_{32} = ℜ (\int_{- T}^{T} \int_{Ω} (s^{2} a ∣\nabla η ∣^{2} ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t),

I_{31} = ℜ (\int_{- T}^{T} \int_{Ω} (s^{2} a ∣\nabla η ∣^{2} ψ) (- i s \partial_{t} η \overline{ψ}) d x d t), I_{32} = ℜ (\int_{- T}^{T} \int_{Ω} (s^{2} a ∣\nabla η ∣^{2} ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t),

I_{33} = ℜ (\int_{- T}^{T} \int_{Ω} (s^{2} a ∣\nabla η ∣^{2} ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t), I_{41} = ℜ (\int_{- T}^{T} \int_{Ω} ((b - s \nabla η \cdot \nabla a) ψ) (- i s \partial_{t} η \overline{ψ}) d x d t),

I_{33} = ℜ (\int_{- T}^{T} \int_{Ω} (s^{2} a ∣\nabla η ∣^{2} ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t), I_{41} = ℜ (\int_{- T}^{T} \int_{Ω} ((b - s \nabla η \cdot \nabla a) ψ) (- i s \partial_{t} η \overline{ψ}) d x d t),

I_{42} = ℜ (\int_{- T}^{T} \int_{Ω} ((b - s \nabla η \cdot \nabla a) ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t), I_{43} = ℜ (\int_{- T}^{T} \int_{Ω} ((b - s \nabla η \cdot \nabla a) ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t) .

I_{42} = ℜ (\int_{- T}^{T} \int_{Ω} ((b - s \nabla η \cdot \nabla a) ψ) (2 a s \nabla η \cdot \nabla \overline{ψ}) d x d t), I_{43} = ℜ (\int_{- T}^{T} \int_{Ω} ((b - s \nabla η \cdot \nabla a) ψ) (s \nabla \cdot (a \nabla η) \overline{ψ}) d x d t) .

i \partial_{t} q + a Δ q = M_{1} q - s^{2} a ∣\nabla η ∣^{2} q + (b - s \nabla η \cdot \nabla a) q .

i \partial_{t} q + a Δ q = M_{1} q - s^{2} a ∣\nabla η ∣^{2} q + (b - s \nabla η \cdot \nabla a) q .

i \partial_{t} q + a Δ q

i \partial_{t} q + a Δ q

\int_{- T}^{T} \int_{Ω} \int_{0}^{t} q (x, ξ) d ξ^{2} e^{- 2 sη} d x d t \leq \frac{κ}{s} \int_{- T}^{T} \int_{Ω} ∣ q (x, t) ∣^{2} e^{- 2 sη} d x d t,

\int_{- T}^{T} \int_{Ω} \int_{0}^{t} q (x, ξ) d ξ^{2} e^{- 2 sη} d x d t \leq \frac{κ}{s} \int_{- T}^{T} \int_{Ω} ∣ q (x, t) ∣^{2} e^{- 2 sη} d x d t,

\int_{- T}^{T} \int_{Ω} \int_{0}^{t} q (x, ξ) d ξ^{2} e^{- 2 sη} d x d t \leq \int_{Ω} \int_{- T}^{T} ∣ t ∣ \int_{0}^{t} ∣ q (x, ξ) ∣^{2} d ξ e^{- 2 sη} d x d t

\int_{- T}^{T} \int_{Ω} \int_{0}^{t} q (x, ξ) d ξ^{2} e^{- 2 sη} d x d t \leq \int_{Ω} \int_{- T}^{T} ∣ t ∣ \int_{0}^{t} ∣ q (x, ξ) ∣^{2} d ξ e^{- 2 sη} d x d t

\leq \int_{Ω} \int_{0}^{T} t (\int_{0}^{t} ∣ q (x, ξ) ∣^{2} d ξ) e^{- 2 sη} d x d t + \int_{Ω} \int_{- T}^{0} (- t) (\int_{t}^{0} ∣ q (x, ξ) ∣^{2} d ξ) e^{- 2 sη} d x d t .

\leq \int_{Ω} \int_{0}^{T} t (\int_{0}^{t} ∣ q (x, ξ) ∣^{2} d ξ) e^{- 2 sη} d x d t + \int_{Ω} \int_{- T}^{0} (- t) (\int_{t}^{0} ∣ q (x, ξ) ∣^{2} d ξ) e^{- 2 sη} d x d t .

\partial_{t} (e^{- 2 sη (x, t)}) = - 2 s (e^{2 λ K} - e^{λ β (x)}) \frac{2 t}{( T ^{2} - t ^{2} ) ^{2}} e^{- 2 sη (x, t)} .

\partial_{t} (e^{- 2 sη (x, t)}) = - 2 s (e^{2 λ K} - e^{λ β (x)}) \frac{2 t}{( T ^{2} - t ^{2} ) ^{2}} e^{- 2 sη (x, t)} .

t e^{- 2 sη (x, t)} = - \frac{( T ^{2} - t ^{2} ) ^{2}}{4 s α ( x )} \partial_{t} (e^{- 2 sη (x, t)}) .

t e^{- 2 sη (x, t)} = - \frac{( T ^{2} - t ^{2} ) ^{2}}{4 s α ( x )} \partial_{t} (e^{- 2 sη (x, t)}) .

\int_{Ω} \int_{0}^{T} t (\int_{0}^{t} ∣ q (x, ξ) ∣^{2} d ξ) e^{- 2 sη} d x d t = \int_{Ω} \int_{0}^{T} (\int_{0}^{t} ∣ q (x, ξ) ∣^{2} d ξ) \frac{( T ^{2} - t ^{2} ) ^{2}}{- 4 s α ( x )} \partial_{t} (e^{- 2 sη}) d t d x

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TopicsNumerical methods in inverse problems · Advanced Mathematical Modeling in Engineering · Stability and Controllability of Differential Equations

Full text

Simultaneous Identification of the Diffusion Coefficient and the

Potential for the Schrödinger Operator with only one Observation

Laure Cardoulis and Patricia Gaitan Université de Toulouse, UT1 CEREMATH, CNRS, Institut de Mathématiques de Toulouse, UMR 5219, 21 All es de Brienne 31042 Toulouse, France Université d’Aix-Marseille, IUT Aix-en-Provence Avenue Gaston Berger 413 av Gaston Berger, 13625 Aix-en-Provence et LATP, UMR CNRS 6632, 39, rue Joliot Curie, 13453 Marseille Cedex 13, France

Abstract

This article is devoted to prove a stability result for two independent coefficients for a Schrödinger operator in an unbounded strip. The result is obtained with only one observation on an unbounded subset of the boundary and the data of the solution at a fixed time on the whole domain.

1 Introduction

Let $\Omega=\mathbb{R}\times(d,2d)$ be an unbounded strip of $\mathbb{R}^{2}$ with a fixed width $d>0$ . Let $\nu$ be the outward unit normal to $\Omega$ on $\Gamma=\partial\Omega$ . We denote $x=(x_{1},x_{2})$ and $\Gamma=\Gamma^{+}\cup\Gamma^{-}$ , where $\Gamma^{+}=\{x\in\Gamma;\;x_{2}=2d\}$ and $\Gamma^{-}=\{x\in\Gamma;\;x_{2}=d\}$ . We consider the following Schrödinger equation

[TABLE]

where $a$ and $b$ are real-valued functions such that $a\in\mathcal{C}^{3}(\overline{\Omega})$ , $b\in\mathcal{C}^{2}(\overline{\Omega})$ and $a(x)\geq a_{min}>0$ . Moreover, we assume that $a$ is bounded and $b$ and all its derivatives up to order two are bounded. If we assume that $q_{0}$ belongs to $H^{4}(\Omega)$ and $F\in H^{2}(0,T,H^{2}(\partial\Omega))\cap H^{1}(0,T,H^{4}(\partial\Omega))\cap H^{3}(0,T,L^{2}(\partial\Omega))$ , then (1.1) admits a solution in $H^{1}(0,T,H^{2}(\Omega))\cap H^{2}(0,T,L^{2}(\Omega))$ .

Our problem can be stated as follows:

Is it possible to determine the coefficients $a$ and $b$ from the measurement of $\partial_{\nu}(\partial_{t}^{2}q)$ on $\Gamma^{+}$ ?

Let $q$ (resp. $\widetilde{q}$ ) be a solution of (1.1) associated with ( $a$ , $b$ , $F$ , $q_{0}$ ) (resp. ( $\widetilde{a}$ , $\widetilde{b}$ , $F$ , $q_{0}$ )). We assume that $q_{0}$ is a real valued function.

Our main result is

[TABLE]

where $C$ is a positive constant which depends on $(\Omega,\Gamma,T)$ and where the above norms are weighted Sobolev norms.

This paper is an improvement of the work [10] in the sense that we simultaneously determine with only one observation, two independent coefficients, the diffusion coefficient and the potential. We use for that two important tools: Carleman estimate (2.5) and Lemma 2.4.

Carleman inequalities constitute a very efficient tool to derive observability estimates. The method of Carleman estimates has been introduced in the field of inverse problems by Bukhgeim and Klibanov (see [5], [6], [13], [14]). Carleman estimates techniques are presented in [15] for standard coefficients inverse problems for both linear and non-linear partial differential equations. These methods give a local Lipschitz stability around a single known solution.

A lot of works using the same strategy concern the wave equation (see [16], [3], [2]) and the heat equation (see [18], [12], [4]). For the determination of a time-independent potential in Schrödinger evolution equation, we can refer to [1] for bounded domains and [10] for unbounded domains. We can also cite [17] where the authors use weight functions satisfying a relaxed pseudo-convexity condition which allows to prove Carleman inequalities with less restrictive boundary observations.

Up to our knowledge, there are few results concerning the simultaneous identification of two coefficients with only one observation. In [11] a stability result is given for the particular case where each coefficient only depends on one variable ( $a=a(x_{2})$ and $b=b(x_{1})$ ) for the operator $i\partial_{t}q+\nabla\cdot(a\nabla q)+bq$ in an unbounded strip of $\mathbb{R}^{2}.$ The authors give a stability result for the diffusion coefficient $a$ and the potential $b$ with only one observation in an unbounded part of the boundary.

A physical background could be the reconstruction of the diffusion coefficient and the potential in a strip in geophysics. There are also applications in quantum mechanics: inverse problems associated with curved quantum guides (see [7], [8], [9]).

This paper is organized as follows. Section $2$ is devoted to some usefull estimates. We first give an adapted global Carleman estimate for the operator $H$ . We then recall the crucial Lemma given in [15]. In Section $3$ we state and prove our main result.

2 Some Usefull Estimates

2.1 Global Carleman Inequality

Let $a$ be a real-valued function in $\mathcal{C}^{3}(\overline{\Omega})$ and $b$ be a real-valued function in $\mathcal{C}^{2}(\overline{\Omega})$ such that

Assumption 2.1.

•

$a\geq a_{min}>0$ , $a$ and all its derivatives up to order three are bounded,

•

$b$ * and its derivatives up to order two are bounded.*

Let $q(x,t)$ be a function equals to zero on $\partial\Omega\times(-T,T)$ and solution of the Schrödinger equation

[TABLE]

We prove here a global Carleman-type estimate for $q$ with a single observation acting on a part $\Gamma^{+}$ of the boundary $\Gamma$ in the right-hand side of the estimate.

Note that this estimate is quite similar to the one obtained in [10], but the computations are different. Indeed, the weigth function $\beta$ does not satisfy the same pseudo-convexity assumptions (see Assumption 2.2) and the decomposition of the operator $H$ is different (see (2.3)).

Let $\widetilde{\beta}$ be a $\mathcal{C}^{4}(\overline{\Omega})$ positive function such that there exists positive constants $C_{0},C_{pc}$ which satisfy

Assumption 2.2.

•

$|\nabla\widetilde{\beta}|\geq C_{0}>0\;\;\mbox{ in }\;\;\Omega,\;\;{\partial}_{\nu}{\widetilde{\beta}}\leq 0\;\;\mbox{on}\;\;\Gamma^{-}$ ,

•

$\widetilde{\beta}$ * and all its derivatives up to order four are bounded in $\overline{\Omega}$ ,*

•

$2\Re(D^{2}\widetilde{\beta}(\zeta,\bar{\zeta}))-\nabla a\cdot\nabla\widetilde{\beta}|\zeta|^{2}+2a^{2}|\nabla\widetilde{\beta}\cdot\zeta|^{2}\geq C_{pc}|\zeta|^{2}$ , for all $\zeta\in\mathbb{C}$

where

[TABLE]

Note that the last assertion of Assumption 2.2 expresses the pseudo-convexity condition for the function $\beta$ . This Assumption imposes restrictive conditions for the choice of the diffusion coefficient $a$ in connection with the function $\widetilde{\beta}$ as in [10].

Note that there exist functions satisfying such assumptions. Indeed if we assume that $\tilde{\beta}(x):=\tilde{\beta}(x_{2})$ , these conditions can be written in the following form:

[TABLE]

and

[TABLE]

For example $\tilde{\beta}(x)=e^{-x_{2}}$ with $a(x)=\frac{1}{2}(x_{2}^{2}+5)$ satisfy the previous conditions (with $x_{2}\in(d,2d)$ ).

Then, we define $\beta=\widetilde{\beta}+K$ with $K=m\|\widetilde{\beta}\|_{\infty}$ and $m>1$ . For $\lambda>0$ and $t\in(-T,T)$ , we define the following weight functions

[TABLE]

We set $\psi=e^{-s\eta}q$ , $M\psi=e^{-s\eta}H(e^{s\eta}\psi)$ for $s>0$ . Let $H$ be the operator defined by

[TABLE]

Following [1], we introduce the operators :

[TABLE]

Then

[TABLE]

where $\overline{z}$ is the conjugate of $z$ , $\Re\ (z)$ its real part and $\Im\ (z)$ its imaginary part. Then the following result holds.

Theorem 2.3.

Let $H$ , $M_{1}$ , $M_{2}$ be the operators defined respectively by (2.2), (2.3). We assume that Assumptions 2.1 and 2.2 are satisfied. Then there exist $\lambda_{0}>0$ , $s_{0}>0$ and a positive constant $C=C(\Omega,\Gamma,T)$ such that, for any $\lambda\geq\lambda_{0}$ and any $s\geq s_{0}$ , the next inequality holds:

[TABLE]

for all $q$ satisfying $q\in L^{2}(-T,T;H^{1}_{0}(\Omega)\cap H^{2}(\Omega))\cap H^{1}(-T,T;L^{2}(\Omega)),$ $\partial_{\nu}q\in L^{2}(-T,T;L^{2}(\Gamma)).$ Moreover we have

[TABLE]

**Proof:

**We have to estimate the scalar product

[TABLE]

with

[TABLE]

Following [1], using integrations by part and Young estimates, we get (2.4). Moreover from (2.3) we have:

[TABLE]

So

[TABLE]

And we deduce (2.5) from (2.4).

2.2 The Crucial Lemma

We recall in this section the proof of a very important lemma proved by Klibanov and Timonov (see for example [14], [15]).

Lemma 2.4.

There exists a positive constant $\kappa$ such that

[TABLE]

for all $s>0$ .

**Proof :

**By the Cauchy-Schwartz inequality, we have

[TABLE]

Note that

[TABLE]

So, if we denote by $\alpha(x)=e^{2\lambda K}-e^{\lambda\beta(x)}$ , we have

[TABLE]

For the first integral of the right hand side of (2.6), by integration by parts we have

[TABLE]

Here we used $\alpha(x)>0$ for all $x\in\overline{\Omega}$ and we obtain

[TABLE]

Similarly for the second integral of the right hand side of (2.6)

[TABLE]

Thus the proof of Lemma 2.4 is completed.

3 Stability result

In this section, we establish a stability inequality for the diffusion coefficient $a$ and the potential $b$ .

Let $q\in\mathcal{C}^{2}(\Omega\times(0,T))$ be solution of

[TABLE]

and $\widetilde{q}\in\mathcal{C}^{2}(\Omega\times(0,T))$ be solution of

[TABLE]

where $(a,b)$ and $(\widetilde{a},\widetilde{b})$ both satisfy Assumption 2.1.

Assumption 3.1.

•

All the time-derivatives up to order three and the space-derivatives up to order four for $\tilde{q}$ exist and are bounded.

•

There exists a positive constant $C>0$ such that $|\tilde{q}|\geq C$ , $\displaystyle{|\partial_{t}(\frac{\Delta\tilde{q}}{\tilde{q}})|}\geq C,$ $|\Delta\tilde{q}|\geq C,$ $\displaystyle{|\partial_{t}(\frac{\tilde{q}}{\Delta\tilde{q}})|}\geq C.$

•

$q_{0}$ * is a real-valued function.*

Since $q_{0}$ is a real-valued function, we can extend the function $q$ (resp. $\widetilde{q}$ ) on $\Omega\times(-T,T)$ by the formula $q(x,t)=\overline{q}(x,-t)$ for every $(x,t)\in\Omega\times(-T,0).$ Note that this extension satisfies the previous Carleman estimate. Our main stability result is

Theorem 3.2.

Let $q$ and $\widetilde{q}$ be solutions of (1.1) in $\mathcal{C}^{2}(\Omega\times(0,T))$ such that $q-\widetilde{q}\in H^{2}((-T,T);H^{2}(\Omega))$ . We assume that Assumptions 2.1, 2.2, 3.1 are satisfied. Then there exists a positive constant $C=C(\Omega,\Gamma,T)$ such that for $s$ and $\lambda$ large enough,

[TABLE]

Therefore

[TABLE]

where the previous norms are weighted Sobolev norms.

**Proof:

**We denote by $u=q-\tilde{q},$ $\alpha=\tilde{a}-a$ and $\gamma=\tilde{b}-b$ , so we get:

[TABLE]

The proof will be done in two steps: in a first step we prove an estimation for $\alpha$ and in a second step for $\gamma.$

First step: We set $\displaystyle{u_{1}=\frac{u}{\tilde{q}}}.$ Then from (3.7) $u_{1}$ is solution of

[TABLE]

where $\displaystyle{A_{11}=i\frac{\partial_{t}\tilde{q}}{\tilde{q}}+a\frac{\Delta\tilde{q}}{\tilde{q}}}$ and $\displaystyle{B_{11}=\frac{2a}{\tilde{q}}\nabla\tilde{q}}.$

Then defining $u_{2}=\partial_{t}u_{1}$ we get that $u_{2}$ satisfies

[TABLE]

where $A_{12}=\partial_{t}A_{11},\ A_{22}=A_{11},\ B_{12}=\partial_{t}B_{11},\ B_{12}=B_{11}.$

Now let $u_{3}=\displaystyle{\frac{u_{2}}{\partial_{t}(\frac{\Delta\tilde{q}}{\tilde{q}})}},$ then $u_{3}$ is solution of

[TABLE]

where $A_{i3}$ and $B_{i3}$ are bounded functions.

If we denote by $\displaystyle{g=\partial_{t}(\frac{\Delta\tilde{q}}{\tilde{q}})}$ , then

[TABLE]

At last we define $u_{4}=\partial_{t}u_{3}$ and $u_{4}$ satisfies

[TABLE]

where $A_{i4}$ and $B_{i4}$ are still bounded functions. Note that $A_{14}=\partial_{t}A_{13},\ A_{24}=\partial_{t}A_{23}+A_{13},\ A_{34}=\partial_{t}A_{33}+A_{23}\partial_{t}g+B_{23}\cdot\nabla(\partial_{t}g),\ A_{44}=A_{23}g+A_{33}+B_{23}\cdot\nabla g,\ B_{14}=\partial_{t}B_{13},\ B_{24}=\partial_{t}B_{23}+B_{13},\ B_{34}=\partial_{t}B_{33}+\partial_{t}gB_{23},\ B_{44}=B_{33}+gB_{23}.$

Applying the Carleman inequality (2.5) for $u_{4}$ we obtain (for $s$ and $\lambda$ sufficiently large):

[TABLE]

Note that $\int_{-T}^{T}\int_{\Omega}e^{-2s\eta}|u_{1}|^{2}\ dx\ dt=\int_{-T}^{T}\int_{\Omega}e^{-2s\eta}|\int_{0}^{t}\partial_{t}u_{1}|^{2}\ dx\ dt$ , so from Lemma 2.4 we get

[TABLE]

By the same way, we have

[TABLE]

So (3.9) becomes

[TABLE]

Furthermore from (3.8) we have (with $C$ a positive constant)

[TABLE]

Therefore for $s$ sufficiently large, from Lemma 2.4

[TABLE]

Using (3.10) we get

[TABLE]

and then

[TABLE]

Second step: By the same way we obtain an estimation of $\gamma$ . We set

[TABLE]

Following the same methodology as in the first step, we obtain:

[TABLE]

From (3.11) and (3.12) we can conclude.

Remark 3.3.

Note that the following function $\tilde{q}(x,t)=e^{-it}+x_{2}^{2}+5$ with $\displaystyle{\tilde{a}(x)=\frac{x_{2}^{2}+5}{2}},$ $\tilde{b}(x)=-1$ satisfies Assumption 3.1. 2. 2.

This method works for the Schrödinger operator in the divergential form:

[TABLE]

We still obtain a similar stability result but with more restrictive hypotheses on the regularity of the function $\widetilde{q}$ .

Acknowledgment: We dedicate this paper to the memory of our friend and colleague Pierre Duclos, Professor at the University of Toulon in France.

References

[1] L. Baudouin and J.P. Puel, 2002, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 18, 1537–1554.
[2] L. Baudouin, A. Mercado and A. Osses, 2007, A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem, Inverse Problems, 23, 257–278.
[3] M. Bellassoued, 2004, Uniqueness and stability in determining the speed of propagation of second order hyperbolic equation with variable coefficient, Appl. An., 83, 983–1014.
[4] A. Benabdallah, P. Gaitan and J. Le Rousseau, 2007, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., 46, 1849–1881.
[5] A.L. Bukhgeim, 1999, Volterra Equations and Inverse Problems, Inverse and Ill-Posed Problems Series, VSP, Utrecht.
[6] A.L. Bukhgeim and M.V. Klibanov, 1981, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Dokl., 17, 244-247.
[7] B. Chenaud, P. Duclos, P. Freitas and D. Krejcirik, Geometrically induced discrete spectrum in curved tubes, Diff. Geom. Appl. 23 (2005), no2, 95-105.
[8] P. Duclos and P. Exner, Curvature-Induced Bound States in Quantum Waveguides in Two and Three Dimensions, Rev. Math. Phys. 7, (1995), 73-102.
[9] P. Duclos, P. Exner and D. Krejcirik, Bound States in Curved Quantum Layers, Comm. Math. Phys. 223 (2001), 13-28.
[10] L. Cardoulis, M. Cristofol and P. Gaitan, 2008, Inverse problem for the Schrödinger operator in an unbounded strip, J. Inverse and Ill-posed Problems, 16, no 2, 127–146.
[11] L. Cardoulis and P. Gaitan, 2009, Identification of two independent coefficients with one observation for the Schrödinger operator in an unbounded strip, to appear in Comptes Rendus Académie des Sciences.
[12] O. Yu. Immanuvilov and M. Yamamoto, 2001, Global uniqueness and stability in determining coefficients of wave equations, Comm. Partial Diff. Equat., 26, 1409–1425.
[13] M.V. Klibanov, 1984, Inverse problems in the large and Carleman bounds, Differential Equations, 20, 755–760.
[14] M.V. Klibanov, 1992, Inverse problems and Carleman estimates, Inverse Problems, 8, 575–596.
[15] M.V. Klibanov and A. Timonov, 2004, Carleman estimates for coefficient inverse problems and numerical applications, Inverse and Ill-posed series,VSP, Utrecht.
[16] I. Lasiecka, R. Triggiani and P.F. Yao, 1999, Inverse/observability estimates for second order hyperbolic equations with variable coefficients, J. Math. Anal. Appl., 235, 13–57.
[17] A. Mercado, A. Osses and L. Rosier, 2008, Inverse problems for the Schrödinger equations via Carleman inequalities with degenerate weights, Inverse Problems, 24, 015017.
[18] J.P. Puel and M. Yamamoto, 1997, Generic well-posedness in a multidimensional hyperbolic inverse problem, J. Inverse Ill-Posed Probl., 1, 53–83.

Bibliography18

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[1] L. Baudouin and J.P. Puel , 2002, Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems , 18, 1537–1554.
2[2] L. Baudouin, A. Mercado and A. Osses , 2007, A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem, Inverse Problems , 23, 257–278.
3[3] M. Bellassoued , 2004, Uniqueness and stability in determining the speed of propagation of second order hyperbolic equation with variable coefficient, Appl. An. , 83, 983–1014.
4[4] A. Benabdallah, P. Gaitan and J. Le Rousseau , 2007, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim. , 46, 1849–1881.
5[5] A.L. Bukhgeim , 1999, Volterra Equations and Inverse Problems, Inverse and Ill-Posed Problems Series, VSP, Utrecht .
6[6] A.L. Bukhgeim and M.V. Klibanov , 1981, Uniqueness in the large of a class of multidimensional inverse problems, Soviet Math. Dokl. , 17, 244-247.
7[7] B. Chenaud, P. Duclos, P. Freitas and D. Krejcirik , Geometrically induced discrete spectrum in curved tubes , Diff. Geom. Appl. 23 (2005), no 2, 95-105.
8[8] P. Duclos and P. Exner , Curvature-Induced Bound States in Quantum Waveguides in Two and Three Dimensions , Rev. Math. Phys. 7, (1995), 73-102.

TL;DR

Contribution

Findings

Abstract

Peer Reviews

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Taxonomy

Simultaneous Identification of the Diffusion Coefficient and the

Abstract

1 Introduction

2 Some Usefull Estimates

2.1 Global Carleman Inequality

Assumption 2.1**.**

Assumption 2.2**.**

Theorem 2.3**.**

2.2 The Crucial Lemma

Lemma 2.4**.**

3 Stability result

Assumption 3.1**.**

Theorem 3.2**.**

Remark 3.3**.**

References

Assumption 2.1.

Assumption 2.2.

Theorem 2.3.

Lemma 2.4.

Assumption 3.1.

Theorem 3.2.

Remark 3.3.