Hölder stability in determining elastic coefficients of Biot's system in poroelastic media

In this paper, we investigate an inverse problem of determining the four spatially varying elastic coefficients of Biot’s system simultaneously, i.e., the two Lamé parameters, the dilatational coupling factor and the bulk modulus, by a single measurement of data on a neighbourhood of the boundary. Following the idea of the B-K method, we prove the Hölder stability estimate of this inverse problem based on Carleman estimates.


Introduction
It is an important topic to determine coefficients of wave equations from measurements of boundary observations in inverse problems. In this research topic, there is a considerable number of papers dealing with the uniqueness and stability, for example, see [1][2][3][4][5][6] and the references therein. The analysis is based on the Carleman estimate near the boundary. A Carleman estimate, originally proposed by Carleman [7], is an inequality for the solution to a partial differential equation with the weighted L 2 -norm. Since [7], the theory of Carleman estimates has been studied extensively (e.g., [8][9][10][11]). Since the Carleman estimate depends essentially on the type of differential equation, many difficulties arise in particular for boundary value problems of wave equations.
For works on Carleman estimates for scalar wave equations, we refer to [12][13][14][15][16][17]. In [17], Imanuvilov and Yamamoto modified the Bukhgeim-Klibanov (B-K) method [8] to determine a coefficient in the acoustic equation. This method has been applied to prove the uniqueness and stability in determining coefficients in elastic wave equations by Isakov, Wang and Yamamoto [18]. The Carleman estimate for Lamé system has also been investigated well (e.g., [19][20][21][22][23]). In [20], the uniqueness in inverse problems for the isotropic Lameé system by three measurements is given. In [22], the authors proved the uniqueness and stability in determination three coefficients using two measurements. In [21], Imanuvilov and Yamamoto proved it with a single measurement. In [19], the authors proved a logarithmic stability estimate for determining spatially varying density and two Lamé coefficients by a single measurement of data on an arbitrarily given subboundary.
The poroplastic model is a more real model describing wave propagation in fluid saturated porous media than the acoustic model and the elastic model. And Biot's linear theory has long been regarded as its basis [24,25]. Let Ω be an open and bounded domain in  3 with ¥ C boundary G = ¶W. Then the wave equations in poroelastic media (i.e., Biot's system) in the source-free case can be written as ( ) and D m l , is the elliptic linear differential operator given by m m l l m ( ) denote respectively the solid frame and fluid phase displacement vectors at the location x and the time t, where T · denotes the transpose of matrices. In (1.1), m l q , , and r are the four elastic coefficients. More specifically, m x ( ) and l x ( ) are the Lamé parameters, q(x) is the dilatational coupling factor and r(x) is the bulk modulus. And r r , 11 12 and r 22 are the density parameters which can be expressed in terms of the solid and fluid densities, the porosity and the tortuosity. We will assume the Lamé parameters μ and λ satisfy We can prove (e.g., [26]) that the system (1.
, , is the usual function space (e.g., [27]). For given initial data Φ and Ψ, the suitable boundary condition and the coefficients r r r , , 11 12 22 , we denote the solution to the system (1. In this paper, we study the stability of the inverse problem for determining the four elastic coefficients simultaneously. To our best knowledge, the work on the inverse problem for the system (1.1) is not enough. In [28], Bellassoued and Yamamoto established a Hölder stability estimate for the source inverse problem rather than the coefficients inverse problem based on a new Carleman estimate for system (1.1). In [29], Bellassoued and Riahi established a Carleman estimate for Biotʼs consolidation model [30] and proved the Lipschitz stability and the uniqueness in determining the coefficients. In [31], the authors proved the logarithmic stability of an inverse problem for Biotʼs consolidation model in poro-elasticity for determining all coefficients simultaneously provided that initial data satisfy some nondegenerate condition. Both the Biotʼs consolidation model and Biotʼs system (1.1) involve poro-elasticity. However, they are different. The obvious difference is that there are the scalar temperature function and the consolidation effects in Biotʼs consolidation model while they are not appeared in (1.1). In addition, the fluid phase displacement vector u f in (1.1) also makes a difference, since we can only control u div f in the existing Carleman estimate. Let w Ì W be an arbitrary given subdomain such that w ¶ É ¶W and w w =´-T T , T ( ). The purpose of this article is to determine m l x x q x r x , , , ( ) ( ) ( ) ( ) by one measurement m l u q r , , , ( )in w T . We will follow the idea used in scale isotropic non-stationary Lamé system in [32] to prove the stability of this inverse problem. The main achievement of this paper is that we prove a Hölder stability estimate for simultaneously determining the four elastic coefficients with a single observation (i.e., theorem 2.3). Moreover, a novel sufficient condition for the initial data is proposed originally.

The main result
In order to state the main result of this paper, we first introduce some assumptions. For Î W  x 0 3 ⧹ , we define the following set of the scalar coefficients for given constants > m 0 and q Î 0, 1 ( ): 2.1 Now we introduce the following four assumptions A.1-A.4.
Assumption A.1. We assume that the coefficients Here we need the extra information of coefficients under consideration in a neighbourhood of ¶W to guarantee the condition in lemma 3.3.
In order to simultaneously determine the four elastic coefficients, we choose the initial data F Y , satisfying the following assumption A.4.
and assume that the eigenvalues as well as the corresponding normalized eigenvectors of e F s ( ) and e Y s ( )have bounded derivatives with respect to x. That is to say, if the scalar functions a x b x , ( ) ( ) and the vector functions y z and z x ( ) with respect to x exist and are bounded. Moreover, we assume that We remark that the matrices in (2.5)-(2.6) are both 2×2 matrices, which is more simple in comparison with the previous condition in [19] (PP.1331). If the similar condition in [19] for the initial data is applied here, we need to introduce a 18×13 full rank matrix.
Remark 2.2. If a symmetric matrix z P ( ) is a continuously differentiable function of single variable ζ, then its eigenvalues are also continuously differentiable with respect to ζ, for which one can refer to [33] and theorem 6.8 in [34]. However, for a matrix with multiple variables, there exist some counter examples. Fortunately, we still can choose the initial data satisfying the assumption A. 4 ( ) , then the assumption A.4 is satisfied.
The purpose of this paper is to analyze the stability in the inverse problem of determining the four spatially varying elastic coefficients m x ( ), l x ( ), q(x), r(x) in (1.1) with a single measurement of data on a neighbour of the boundary. The main result is given by theorem 2.3, which shows that a single measurement on a neighbour of the boundary is enough to determine the four spatially varying elastic coefficients.
The remainder of the paper is to prove theorem 2.3.

Proof of theorem 2.3
In this section, we will prove theorem 2.3. There are two subsections in this section. In section 3.1, we present some necessary lemmas. Then in section 3.2, we show detail proof of theorem 2.3.
j j Let a a a a = , , , n For arbitrary fixed Ï W x 0 , let J W   : given by x t 2 , where κ is a chosen constant [28] and b > 0. Moreover, set . Thus for given h > 0, we can choose small δ such that    ( ( )) and = Î G G G H Q , x is a real symmetric matrix, and its eigenvalues and the corresponding normalized eigenvectors have bounded derivatives with respect to x, then there exists a positive Taking inner product on both sides of (3.6) with d k , we obtain and by the assumption of the existence of the derivatives of l k , we can rewrite (3.7) as by the Green formula. So there exists b On the other hand, we have Combining (3.8) and (3.9), we obtain (3.5). , Hence there exist constants t 0 and > Combining (3.12) and (3.13), we obtain ò ò If we multiply a function Î W u H 2 3 ( ( )) by a cut-off function c Î W ¥ C 0 ( ) satisfying c   0 1 and c = 1 in w W⧹ , we immediately obtain the following corollary 3.5.
Lemma 3.6. There exists a positive constant > C 0 such that the following estimate holds Proof. We refer to lemma 3.4 in the paper by Bellassoued and Yamamoto [28], which says: and substituting z x t , ( )for tj z x t e ,  Next we turn to estimate the weighted H 2 -norm of m l q r , , , * * * * in Ω. Note that