SEMIGROUP-THEORETIC APPROACH TO IDENTIFICATION OF LINEAR DIFFUSION COEFFICIENTS

. Let X be a complex Banach space and A : D ( A ) → X a quasi-m -sectorial operator in X . This paper is concerned with the identiﬁcation of diﬀusion coeﬃcients ν > 0 in the initial-value problem: ( d/dt ) u ( t ) + νAu ( t ) = 0 , t ∈ (0 ,T ) , u (0) = x ∈ X, with additional condition (cid:107) u ( T ) (cid:107) = ρ , where ρ > 0 is known. Except for the additional condition, the solution to the initial-value problem is given by u ( t ) := e − tνA x ∈ C ([0 ,T ]; X ) ∩ C 1 ((0 ,T ]; X ). Therefore, the identiﬁcation of ν is reduced to solving the equation (cid:107) e − νTA x (cid:107) = ρ . It will be shown that the unique root ν = ν ( x,ρ ) depends on ( x,ρ ) locally Lipschitz continuously if the datum ( x,ρ ) fulﬁlls the restriction (cid:107) x (cid:107) > ρ . This extends those results in Mola[6](2011).

which depends continuously on the initial datum x, and whose energy u(t) is nonincreasing on the trajectories (here, · is the norm in X). This result holds for all positive values of the diffusion coefficient ν, which, for the sake of our investigation, will be always considered as a mere constant quantity.
The sectoriality (or sectorial-valuedness) was first introduced in the Hilbert space framework (see Kato [4,Section V.3.10]) as an extension of the notion of nonnegative selfadjointness. Then the notion was generalized to the Banach space case. Let F be the (single-valued) duality mapping on X with uniformly convex adjoint X * and f, g X,X * denote the pairing between f ∈ X and g ∈ X * . We define a sector by S(tan θ) := {z ∈ C; |Im z| ≤ (tan θ)Re z} if 0 < θ < π/2, Then a linear operator A with domain D(A) and range R(A) in X is said to be sectorial of type S(tan θ) if Au, F (u) X,X * ∈ S(tan θ), θ ∈ [0, π/2), for u ∈ D(A).
Note that an accretive operator can be regarded as sectorial of type S(tan π/2).
In particular, a sectorial operator A is m-sectorial (or regularly m-accretive) if, additionally, R(1 + A) = X. It should be noted that a linear operator A is the generator of an analytic contraction semigroup {e −tA ; t ≥ 0} on X if and only if −A is an m-sectorial operator in (reflexive) X. In this case, if t > 0, then Ae −tA is bounded on X (smoothing effect). Back to our problem, we stress that in the basic case when A = −∆ on the domain D(A) = W 2,p (Ω) ∩ W 1,p 0 (Ω) in L p (Ω) (1 < p < ∞) (that is, A is the L prealization of the Laplace operator on a domain Ω ⊂ R N ), then, as it is well-known, A is m-sectorial of type S(c p ) in L p , as a consequence of the inequality Im −∆u, |u| p−2 u L p ,L p ≤ c p Re (−∆ − α)u, |u| p−2 u L p ,L p which holds for all u ∈ D(A), where α is the first eigenvalue of A and we have defined c p := |p − 2|/(2 √ p − 1) (see, e.g., Okazawa [7] and Voigt [10]). Moreover, in the Hilbert case p = 2, from c 2 = 0 it follows that A = −∆ on the domain D(A) = H 2 (Ω) ∩ H 1 0 (Ω) is nonnegative and selfadjoint in L 2 (Ω). The coefficient ν accounts for important features of the diffusion phenomena associated with the Cauchy problem (1.1). For example (see [9]), in the case where u represents the concentration of a chemical substance subjected to diffusion, then (1.1) may be regarded as Fick's second law, and ν reflects the mobility of the diffusing species in the given environment (accordingly, it is expected to assume larger values in gases, smaller ones in liquids, and extremely small ones in solids). Nevertheless, as it is understood, the measurement of such an intrinsic quantity as ν can be extremely difficult to be performed. Thus, it is natural to think of ν as an unknown of our problem, as well as the "solution" u. Therefore, in order to recast a proper determination, it is necessary to feed the problem with additional measurements on the accessible parameters (overdeterminating conditions). Among all the possible choices, we shall study the inverse problem or identification of the diffusion constant ν under the "final-time" energy measurement where ρ is a given positive constant, to be considered from now on as a datum of our problem, together with the initial value x ∈ X.
The inverse problem we presented has been studied by the first author in [6], under the assumption that A is positive-definite and self-adjoint. The main idea therein introduced is the one to use the fundamental energy identity at the final to deduce the following relationship between the unknowns where A 1/2 denotes the square root of A. In other words, the unknown constant ν can be computed in terms of u as a nonlinear functional N (u) which, consequently, entails a modification of equation (1.1) as It is important to stress that, strictly speaking, (1.3) is not a differential (forward or backward) equation, due to the nonlocal and noncausal nature of the functional N , which requires the knowledge of the global dynamics on the whole interval [0, T ]. Thus, usual abstract techniques for quasilinear equations cannot be applied to our case. The results provided in [6] concern existence and uniqueness of a weak solution (u, ν) as well as its continuous dependence on the data (x, ρ), and have been achieved by adapting a finite-dimensional Faedo-Galerkin approximation scheme to the inverse problem. This requires a deep use of the real spectral decomposition, which restricts the application of the abstract result to the realization of secondorder differential operators only on bounded domains, and can be performed only in the Hilbert case. The aim of the present paper is to study the same inverse problem as in [6] by means of deeper techniques in nonlinear analysis in Banach spaces (with uniformly convex adjoint). The main advantages of this alternative approach are twofold. First, for the sake of applications, such an approach may apply to the realization of second-order differential operators also on unbounded domains, and extends the previous results even to complex Banach spaces. Second, and more importantly, such a formulation opens new scenarios towards more general equations (i.e. fully nonlinear) in which the identification problem can be stated. Those features will be object of further investigations in the next future.

2.
Preliminaries. Let F be the duality mapping on a general Banach space X to its adjoint X * : Here we denote by v, f X,X * the pairing between v ∈ X and f ∈ X * . Assume that X * is strictly convex. Then F is single-valued. Now we consider the "tangent functional" of the unit ball in X: (see, e.g., Miyadera [5, Section 1, Chapter 2]). In other words, ϕ is Gâteaux differentiable. Thus ϕ is Fréchet differentiable, with Fréchet gradient F (x) ∈ X * = L(X, C) if F : X → X * is continuous on the whole of X (see Zeidler [11,Proposition 4.8 (c)]). Assume further that X * is uniformly convex. Then F is uniformly continuous on bounded subsets of X (see Zeidler [12,Proposition 32.22 (e)]). This implies that ϕ ∈ C 1 (X; R) with ϕ = I (the identity) if X is a Hilbert space (cf. Section 3) or with ϕ = F if X has the uniformly convex X * (cf. Section 4).
As employed by Goldstein [3, Definition 1.5.8] (see also Ouhabaz [8,p. 97]), an important role in semigroup theory is played by the notion of sectorial operator in a general Banach space (more precisely, in the case where F is not single-valued). That is, a linear operator A in X is said to be sectorial of type S(tan θ) if for every Now let A be a linear quasi-m-accretive operator in a Banach space X, that is, where α > 0 is a constant. In other words, A − α is m-accretive in X. A quasi-msectorial operator in X satisfies the stronger estimate: implies that the semigroup generated by −A is of negative type: On account to the functional setting above introduced, we can state the abstract formulation of the inverse problem we aim at investigating.
Problem P . Given (x, ρ) ∈ H × R + , find a vector-valued function u : [0, T ] → H and a real number ν > 0 fulfilling the equation with the initial datum

6)
and the additional constraint First of all, we know that for every ν > 0, u(t) = e −νAT x is a unique solution to the initial-value problem (2.5)-(2.6) of suitable regularity (see [1,Chapter 7]). So it remains to verify additional condition (2.7), that is, the one-to-one correspondence between ρ and ν when x is fixed. To this end, for ν ≥ 0 put where f ν is an arbitrary element in F (e −νAT x) (see [5,Lemma 2.8]). Thus we see from the intermediate theorem that for every ρ ∈ (0, x ) there exists a unique ν > 0 satisfying (2.7). Therefore, choosing ν > 0 as the unique root of φ(ν) − ρ = 0, u(t) = e −νAT x is a unique solution to Problem P . Moreover, uniqueness of the root immediately shows that the map (x, ρ) → (ν, u) is continuous.
Generally speaking, ν is expected to be determined by the pair (x, ρ) as above and depends continuously on the data (x, ρ). One can expect more regularity of the solution map. In order to establish this assertion we shall use the Banach space version of the implicit function theorem applied to the map Remark 2.1. Recalling Feller's renorming trick, the above-mentioned assertion can be extended to general C 0 -semigroups of negative type, at the expense of enlarging the lower bound on x 0 . In fact, replacing the norm by the Feller norm Then the condition x > ρ has to be replaced with x > M ρ.
3. Identification of diffusion coefficients in Hilbert spaces. Let H be a Hilbert space with inner product (·, ·). Then the duality mapping F is the identity. In order to differentiate e −νT A x with respect to ν, we first consider an approximate problem, that is, a problem with initial value x ∈ H replaced with x ∈ D(A) (we need this process only when A is quasi-m-accretive). To this end let D(A) be a Hilbert space with inner product and norm: We are now in a position to state the main result in the Hilbert space case.
To solve Problem (P ) D(A) we apply the implicit function theorem to the function of three variables: The ρ-dependence of ψ is not so complicated. To simplify the notation we also use the function of two variables: (ii) (x-derivative). Let x ∈ D(A). Then one has Here A * is the adjoint operator of A.
(ii) Let x ∈ D(A). Since ψ(x, ρ, ν) = φ(x, ν) − ρ 2 /2, we can show that φ is Fréchet differentiable with respect to x: In fact, we can write as Now, put δ := 2 ε. If y − x < δ, then we have the implication of (3.1): It remains to show that ∇ x φ(·, ν) in (3.1) is the Fréchet derivative of φ with respect to x. Since A − α is m-accretive in H if and only if so is A * − α, e −νT A * makes sense, with the equality e −νT A * = e −νT A . Therefore it follows from (3.1) that ∇ x φ(·, ν) : D(A) → X is a bounded linear operator, with norm-estimate note that X = L(X, C) is identified as X.

Proof of Theorem 3.1. Part (I) For every pair of (x, ρ) there is a unique function
Noting that for x 1 , x 2 ≥ ρ 0 and ρ 1 , ρ 2 ≥ ρ 0 for some fixed ρ 0 > 0. Here dist D(A)\B(ρ0) (x 1 , x 2 ) denotes the geodesic distance from x 1 to x 2 ; note that for sufficiently large n, m. Therefore for every x ∈ H \ B(ρ 0 ) we may define Since ρ 0 is arbitrary, we may define ν ∈ C((H \ B(ρ)) × R + ), satisfying Note that in the latter non-parabolic case, the unique solution u to the initial value problem (2.5)-(2.6) loses the differentiability. Thus, solutions are to be intended in a weak sense as in [6]. 4. Identification of diffusion coefficients in Banach space X with uniformly convex X * . Let X be a Banach space with uniformly convex adjoint X * , with duality mapping F : A typical example is the Lebesgue space L p (1 < p < ∞) with F : L p → L p given by In other words, one has As in Section 3 we denote by G (θ, −α), (θ, α) ∈ [0, π/2] × R + , the set of all linear operators A such that A − α are m-sectorial of type S(tan θ) in X. For example, when θ ∈ (0, π/2], then we have note that if A ∈ G (θ, −α), then D(A) is dense in X. The set of admissible data is also defined as in Section 3: where Y = X or D(A). Now we consider the further problems Problem (P ) D(A) (non-parabolic case). Given A ∈ G (θ, −α) and (x, ρ) ∈ A (D(A)) find the pair (ν, u) ∈ R + × C 1 ([0, T ]; X) fulfilling equation (2.5) (with t ≥ 0), initial datum (2.6) and additional condition (2.7).
We are in a position to state the main result in this section.
Proof. It suffices to prove (ii) and (iii).
(ii) Put Therefore e −νT A * makes sense, and the next equality holds In the ε-δ argument the assertion is expressed as To see this we can employ (4.1) with u := e −νT A x and v := e −νT A y, that is, if This is nothing but (4.4), with ∇ x φ(x, ν) given by (4.3). It remains to show that ∇ x φ(·, ν) in (4.4) is the Fréchet derivative of φ with respect to x. But, it follows from (4.3) that ∇ x φ(·, ν) : X → X * is a bounded linear operator, with norm-estimate This completes the proof of ψ ∈ C 1 (Q(ρ 0 )).
Using Lemma 4.3 instead of Lemma 3.2, we can prove Theorem 4.2 in the same way as in Section 3. Of course Theorem 3.1 is contained in Theorem 4.2.

5.
Applications in L p -space. In this last section we shall display concrete applications of the abstract results stated in the previous sections to a family of initial-boundary value problems. We stress the fact that all the function spaces we mention can be complex-valued, thus extending analogous results in [6], where the underlying theory was only real. We recall that, in this case −∆ − α is m-sectorial in L p and the dissipation constant α can be chosen to be the first eigenvalue of A. Thus, the first inverse problem we want to study reads Problem 1. Find the function u : Ω × [0, T ] → C and the real constant ν > 0 such that the following parabolic initial-boundary value problem is satisfied : where u 0 ∈ L p (Ω) and ρ > 0 are given.
Then the abstract Theorem 4.2 yields such that solving Problem 1. In particular, if p = 2, then

5.2.
Realization on unbounded domains. Now, denote by ∆ = N i=1 (∂/∂x i ) 2 the Laplace operator in the space L p = L p (R N ) of the complex-valued p-th-powersummable functions. Then we define in L p the realization of negative Laplace operator: We recall that, in this case α − ∆ is m-sectorial, with dissipation constant α > 0.
Thus, the second inverse problem we want to study reads Problem 2. Find the function u : R N × [0, T ] → C and the real constant ν > 0 such that the following parabolic initial value problem is satisfied : where u 0 ∈ L p (R N ) and ρ > 0 are given.
Once more, abstract Theorem 4.2 yields Then there exists a unique pair solving Problem 2. In particular, if p = 2, then Appendix. Let X := L p = L p (Ω) = L p (Ω; C), where Ω ⊂ R N is a domain (1 < p < ∞, N ∈ N), with norm · L p . Then F (u) := u 2−p L p |u| p−2 u is the (normalized) duality mapping on L p (Ω) to its adjoint L p (Ω), p −1 + p −1 = 1, with Hereby, we shall show that ϕ(u) := (1/2) u 2 L p (p ≥ 4) is Fréchet differentiable, with Fréchet derivative F (u) by means of a direct computation because, as it appears, no direct proof is displayed in the literature.
Proof of Lemma A2. Put Setting v := u + h, we see from (A3) that L p , that is, ∇ u p −1 u p L p = |u| p−2 u. Taking (A1) into account, we have finished the proof of Proposition A1.