Strong well-posedness and inverse identification problem of a non-local phase field tumor model with degenerate mobilities

We extend previous weak well-posedness results obtained in Frigeri et al. (2017) concerning a non-local variant of a diffuse interface tumor model proposed by Hawkins-Daarud et al. (2012). The model consists of a non-local Cahn--Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction-diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumor distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.


Introduction
Mathematical modeling for tumor growth dynamics has undergone a swift development in the last decades (see for instance pioneering works such as [14,15,64]). Even now, the full complexity of the tumor disease is far from being understood, and through mathematical modeling, scientists and medical practitioners now possess a powerful tool to predict and analyse tumor growth behaviour without inflicting serious harm to the patients.
In this contribution, we address the issue of well-posedness for a certain continuum model for tumor growth. The original model, derived in Hawkins-Daruud et al. [49] (see also [48,50]), is based on the well-known phase field methodology that has seen increased applications in tumor growth, and takes the form where the primary variables (ϕ, µ, σ) denote the phase field, the associated chemical potential, and the nutrient concentration, respectively. The phase field ϕ serves as an indicator on the location of tumor and non-tumor cells, which are separated by a thin interfacial layer whose thickness is related to the positive constants A and B, while the non-negative functions m(ϕ) and n(ϕ) correspond to the cellular and nutrient mobilities, respectively. The function F ′ (ϕ) is the derivative of a potential F (ϕ), which is a characteristic feature of phase field models. Lastly, the non-negative constant χ is a chemotactic sensitivity of the nutrient and P (ϕ) denotes a proliferation function, see [42,49] for more details.
The mathematical and numerical analysis, and optimal control for the above model of Hawkins-Daarud et al. and its variants have been performed by many authors, of which we mention [6, 9-13, 32, 50, 57-61, 65]. Such intensive study and broad range of results are possible thanks to the Lyapunov structure of the model, where in a bounded domain Ω ⊂ R d , under no-flux boundary conditions ∂ n ϕ = m(ϕ)∂ n µ = n(ϕ)∂ n σ = 0 (∂ n f = ∇f ·n is the normal derivative) sufficiently smooth solutions satisfy d dt E(ϕ(t), σ(t)) + t 0 m 1/2 (ϕ)∇µ 2 + n 1/2 (ϕ)∇(σ − χϕ) 2 with the free energy function In the above, L(ϕ) is the Ginzburg-Landau energy function, which is responsible for phase separation and surface tension effects in the context of phase field models. In our current context of tumor growth, L(ϕ) is associated with cell-to-cell adhesion, where tumor cells prefer to adhere to each other rather than to non-tumor cells.
In [36], the existence of weak solutions to (1.2) for a wide range of non-degenerate mobility functions m, n, proliferating function P , and potential F has been established by the first and second authors of this work. Continuous dependence on initial data (and hence uniqueness of weak solutions) can be achieved under the additional requirement that χ = 0 and n = 1. Furthermore, by adapting the method introduced in [24] the authors in [36] were able to establish weak well-posedness of the non-local tumor model ( On the other hand, the presence of these degenerate/singular terms limit the analytical investigations of (1.2) to the class of weak solutions, and to the best of the authors' knowledge, numerical analysis and optimal control involving the non-local model (1.2) with degenerate mobility and singular potentials have not received much attention in the literature. Therefore, the purpose of this work is to prove the well-posedness of strong solutions in order to facilitate future investigations.
In the following, we consider a bounded domain Ω ⊂ R d , d ∈ {2, 3} with Lipschitz boundary Γ := ∂Ω. For a fixed but arbitrary constant T > 0 we denote the parabolic cylinder and its boundary by Q t := Ω × (0, t) and Σ t := Γ × (0, t) for all t ∈ (0, T ), with Q := Q T and Σ := Σ T . In light of previous results in [36] we switch off the chemotaxis mechanism by setting χ = 0, and owing to the degeneracy of the mobility m the gradient of the chemical potential µ which appears in equation (1.2a) cannot be controlled in any Lebesgue space. Thus, following [4,24,29,33,39,46,47], we introduce the auxiliary function (1.4) which exhibits the following useful relations (1.6b) For boundary conditions, we take the no-flux conditions m(ϕ)∂ n µ = n(ϕ)∂ n σ = 0, which translate to and for initial conditions, we prescribe Since the weak well-posedness to (1.6)-(1.8), which we collectively call (P), is a direct consequence of the main results of [36], the main focus of this work is to show the existence of strong solutions using techniques inspired by [29] for the non-local Cahn-Hilliard-Navier-Stokes system. In our setting, this involves a bootstrapping argument in which we first improve the regularity of ϕ by fixing σ and employing a time discretisation of (1.6a), and then we improve the regularity of σ with the help of new regularities for ϕ. Under suitable assumptions detailed in the next section, our main results are H 1 (0, T ; L 2 (Ω))∩L ∞ (0, T ; H 1 (Ω))∩L 2 (0, T ; H 2 (Ω))-regularities for ϕ and σ (see Theorem 3.1) for d ∈ {2, 3} and W 1,∞ (0, T ; L 2 (Ω))∩H 1 (0, T ; H 1 (Ω))∩L ∞ (0, T ; H 2 (Ω))-regularities for ϕ and σ with general assumptions for d = 2, whereas for d = 3 solely under the additional requirements that the nutrient mobility n = 1 and λ is a positive constant (see Theorem 3.2). In turn, these regularities lead to continuous dependence on initial data in stronger norms (see Theorems 3.3 and 3.5) compared to those established in [36]. It is also worth mentioning the arguments of [29] do not apply directly to our model due to the presence of the proliferation term P (ϕ)(σ − AF ′ (ϕ) + BJ ⋆ ϕ) in (1.6a), and some crucial parts of the argument have to be modified in order for the analysis to go through.
As an application of the new solution regularities, we study an inverse problem relevant to tumor growth, which involves identifying the initial tumor distribution encoded by ϕ 0 based on measurements of the phase field variable at terminal time ϕ(T ). Thanks to the well-posedness of (P) we can introduce the notion of a solution operator S : ϕ 0 → ϕ(T ). Then, given a measurement ϕ Ω : Ω → R of the phase field variable, the inverse problem can be formulated as: Find ϕ 0 such that S(ϕ 0 ) = ϕ Ω a.e. in Ω. (1.9) Due to the compactness of the solution operator S : H 1 (Ω) → H 1 (Ω), the inverse problem is ill-posed [25,Chap. 10]. To overcome this, we employ Tikhonov regularisation and formulate the resulting problem as a constrained minimisation problem. More precisely, we employ optimal control methods treating ϕ 0 as the optimal control to the problem where U denotes a suitable set of admissible controls and α > 0 is a regularisation parameter. Our main results for (1.10) are (i) the existence of a solution ϕ α 0 ∈ U for any α > 0, (ii) how to obtain a solution to the inverse problem (1.9) from {ϕ α 0 } α>0 as α → 0 (provided the solution set of (1.9) is non-empty), and (iii) the derivation of first-order optimality conditions for ϕ α 0 . The precise formulation can be found in Theorem 4.4. In particular, thanks to the new solution regularities to (P), practitioners interested in solving the inverse identification problem (1.9) that involve the non-local tumor model (1.6) with degenerate mobility and singular potentials can first obtain numerical approximations of {ϕ α 0 } α>0 by solving the optimality conditions, and then sending α → 0 in an appropriate way to deduce a solution to (1.9).
The paper is structured as follows: In Section 2 we state the notation and recall previous results on (P), and in Section 3 we state and prove strong well-posedness to (P). In Section 4 we study the optimal control problem (1.10) and derive desirable properties involving minimisers and the first-order optimality conditions.

Mathematical setting and previous results
In this section, we recall some useful mathematical tools and previous results on (P) established in [36]. We define H := L 2 (Ω), V := H 1 (Ω), W := H 2 (Ω), (2.1) and equip them with their standard norms. Moreover, for an arbitrary Banach space X, we indicate with · X , X * , and ·, · X its norm, its topological dual and the duality pairing between X * and X, respectively. Likewise, for every 1 ≤ p ≤ ∞, we simply use · p to denote the usual norm in L p (Ω), with · = · 2 . Furthermore, we use (·, ·) to denote the L 2 (Ω)-inner product. As (V, H, V * ) forms a Hilbert triplet, i.e., the injections V ⊂ H ≡ H * ⊂ V * are both continuous and dense, we have the following identification For u ∈ L 1 (Ω), we use the notation u = 1 |Ω| (u, 1) to denote the mean value of u. The following particular case of the Gagliardo-Nirenberg inequality in two dimensions will be repeatedly employed throughout our analysis Lastly, we also recall the Agmon's inequality in two dimensions Throughout the paper, we will use the symbol C to denote constants which depend only on structural data of the problem. On the other hand, we will sometimes stress the dependence of the appearing constant by adding a self-explanatory subscript. Moreover, Q ≥ 0 will stand for a generic monotone non-decreasing continuous function of all its arguments.
For the analysis, we make the following structural assumptions: Moreover, there exist constants ε 0 ∈ (0, 1] and α 0 > 0 such that m is non-increasing 1]) and there exist a positive constant n * such that (A4) P ∈ C 0 ([−1, 1]) is non-negative, and there exist positive constants k and ε 0 such that For convenience, we will denote with λ ∞ and P ∞ the uniform bound of λ and P , respectively.
Remark 2.1. We point out that, as a consequence of (A3), we have that and for all 1 ≤ p ≤ ∞. These estimates will be repeatedly employed.
The advantage of this condition is that it allows us to include proliferation functions of the form P (s) = P 0 (1 − s 2 ) α χ [−1,1] (s), with α = 1, once the mobility and potential are assumed as in (1.3), where P 0 denotes a non-negative constant. Notice that, given (1.3), in order to satisfy (A4) we need α ≥ 2.
Under the above assumptions, the existence of weak solutions can be obtained by employing a suitable approximation scheme that resembles the one introduced in [24]. More precisely, an approximate problem is solved at first by suitably regularizing F , P and m. Then, uniform estimates with respect to the approximating parameter are derived which allow to pass to the limit by classical weak and strong compactness arguments. The weak existence result for (P) that can derived from [36,Thm. 2.3] is formulated as follows.
Moreover, there exists a positive constant K 1 which depends only on Ω, T , and on the data of the system such that For continuous dependence on initial data (which also entails uniqueness of solutions) further assumptions are needed: (B1) m ∈ C 0,1 ([−1, 1]) and n = 1.
Theorem 2.2. Suppose that (A1)-(A4) and (B1)-(B2) are satisfied. Let [ϕ i , σ i ], for i = 1, 2, be two solutions to (P) corresponding to initial data [ϕ 0,i , σ 0,i ] satisfying (A5). Then, there exists a positive constant K 2 which depends only on Ω, T , and on the data of the system such that Remark 2.5. We point out that due to our choice of the non-local Ginzburg-Landau energy F in (1.1), in the notation of [36] we have F 2 = 0, a(x) = 0 and F 1 = F . Hence, we can simplify several assumptions for well-posedness.

Strong well-posedness
Further regularity for the weak solution to (P) can be established with a more regular convolution kernel J. For instance, the assumption J ∈ W 2,1 loc (R d ) would be sufficient from an analytical point of view. However, as pointed out in [28], this assumption excludes the physically relevant cases of Newtonian and Bessel potential kernels. A way to overcame this issue is to assume that J is admissible in the following sense: is said to be admissible if it fulfils the following conditions: • J is radially symmetric, i.e. J(x) = J(|x|) for a non-increasing function J.
For strong well-posedness, we reinforce previous assumptions by assuming that: Furthermore, if σ 0 ∈ V , and assuming that n = 1 when d = 3, it holds that Lastly, for d ∈ {2, 3} and n = 1, if σ 0 ∈ W with ∂ n σ = 0 and (C4) also hold, then We are also able to prove a stronger regularity result.

5)
and for the case d = 2, the regularity (3.3) also hold for σ without the previous restriction on the nutrient mobility n. Moreover, there exists a positive constant K 3 which depends only on Ω, T , J, and on the data of the system such that We point out that λ = mF ′′ being a constant for the assumption of Theorem 3.2 implies Λ(s) = Aα 0 s. This does not take away the combination of degenerate mobility and singular potential from the non-local model.
Next, we present two improvements of the continuous dependence results of [36] (see Theorem 2.2), where due to the improved regularity for ϕ we can consider a non-constant mobility n(ϕ) in the case d = 2. This fact is new with respect to [36], where the regularity of the weak solution confines the analysis to the case of constant mobility n = 1. The first improvement is a weak-strong continuous dependence result. , suppose in addition that λ is a constant and n = 1. Assume that initial data [ϕ 0,i , σ 0,i ], for i = 1, 2, are given such that [ϕ 0,1 , σ 0,1 ] ∈ V ×V and [ϕ 0,2 , σ 0,2 ] ∈ H ×H (with ϕ 0,1 , ϕ 0,2 satisfying also (A5)). Let [ϕ 1 , σ 1 ], and [ϕ 2 , σ 2 ] be the corresponding solutions, given by Theorem 3.1, and by Theorem 2.1, respectively. Then, there exists a positive constant K 4 which depends only on Ω, T , J, and on the data of the system such that Theorem 2.2 and Theorem 3.3 entail uniqueness of the solution to Problem (P). More precisely, we have the following Then, the solution to Problem (P ) given by Theorem 2.1 and by Theorem 3.1, respectively, is unique.
In two spatial dimensions, we can prove a stronger continuous dependence result. To this aim we need the following conditions.
Theorem 3.5. Assume that d = 2 and that (A1)-(A4), (C1)-(C4), (D1)-(D2) are satisfied. Suppose in addition that be the corresponding strong solutions given by Theorem 3.2. Then, there exists a positive constant K 5 which depends only on Ω, T , J, and on the data of the system such that

Existence of strong solutions
Let us first recall two useful lemmas.
We refer the reader to [17, Chap. IX, Sec. 18] for the proof and just recall that the space H 1/2 (Γ) is endowed with the following seminorm where dΓ stands for the surface measure on the boundary Γ.
Another advantage of considering admissible kernels in the sense of Definition 3.1 is the validity of the following result, whose proof can be found in [3,Lem. 2].
Lemma 3.7. Assume that the kernel J is admissible in the sense of Definition 3.1. Then, for every p ∈ (1, ∞), there exists a positive constant C p such that (3.10)

Proof of Theorem 3.1
We apply the argument outlined in [29,Sec. 4] to the system given by (1.6a), (1.7) only, where σ is taken as the weak solution given by Theorem 2.1. For fixed ε > 0, we introduce the regular potential F ε , and the functions m ε , P ε given by . Then, owing to (A1), it is clear that the function λ ε := m ε F ′′ ε satisfies the bounds Moreover, we claim that there exist two constants k 1 , k 2 > 0, independent of ε, such that the following bound holds we first approximate (1.6a), (1.7) with the following system where Λ ε (s) := A s 0 λ ε (r)dr for every s ∈ R. We then prove that, for every ε > 0, system (3.14)-(3.15) admits a solution ϕ ε in the class (3.1).
Before we derive uniform discrete estimates, let us collect a useful elementary identity and several useful inequalities established in [29], more precisely (3.19), (3.20) and (3.21) below can be derived from equations (4.16), (4.25) and (4.27) of [29], respectively. In the following δ denotes positive constants whose values are yet to be determined, while C denotes positive constants independent of N, τ and ε. For n ≤ N − 1, it holds that and that Now, integrating (3.16) yields on account of the boundedness of P ε F ′ ε and σ k ≤ σ L ∞ (0,T ;H) ≤ C. In particular, Hence, by the Poincaré inequality Then, testing (3.16) with ϕ k+1 , summing over k from k = 0 to k = n ≤ N − 1, employing the identity (3.18), estimates (3.19), (3.22) and (3.11), and choosing δ = α 0 A/4 yields An immediate consequence of (3.23) is , summing from k = 0 to k = n and using (3.18) for ∇Λ ε (ϕ k ) yields As in [29, (4.19)-(4.20)], the first and third terms on the right-hand side are bounded above by Together with the fact that ∇Λ ε (ϕ 0 ) ≤ Aλ ∞ ∇ϕ 0 , we find that By taking τ small enough so that Cτ < 1 4 , and by applying the discrete Gronwall lemma, we deduce that while by (3.21) and (3.25) it holds that Employing elliptic regularity, (3.20) and the above estimates, we infer that and consequently a similar argument to that used in [ We mention that in the case d = 3, a Moser-Alikakos iteration is used in [29, Proof of Thm. 6.1, p. 718-719] to first establish ϕ k+1 L ∞ (Ω) ≤ C( ϕ 0 L ∞ (Ω) ) for all k = 0, . . . , n with n < N − 1, which is then used to show (3.28). In our setting we have the additional source term S k : (3.16), and at present we cannot directly replicate the Moser-Alikakos argument as σ k is currently not bounded in L ∞ (0, T ; L ∞ (Ω)).
However, let us claim that the control given by (3.28) can be achieved also for d = 3 assuming n = 1 with similar arguments provided we consider the nutrient variable σ k to possess the stronger regularity pointed out by Theorem 3.2 which in turn would give us σ k ∈ L ∞ (0, T ; L ∞ (Ω)) (cf. Remark 3.1). In fact, it turns out that the stronger regularity for the nutrient, in the case n = 1, just requires further assumption on σ 0 and that we still have ϕ ∈ H 1 (0, T ; H) ∩ L ∞ (0, T ; V ) ∩ L 2 (0, T ; W ) also for the case d = 3 without the assumption n = 1 (see (3.32)-(3.37) below).
We multiply (1.6b) by −∆σ (which is a valid test function in the Galerkin approximation) and integrate over Ω and by parts to obtain that where the terms on the right-hand side are denoted by I 1 and I 2 . By means of (A4) along with the Young inequality and the previous estimate, we obtain that for a positive δ yet to be determined. As for the first term, if d = 2, we use the boundedness of n ′ , the Gagliardo-Nirenberg inequality (2.2) and the elliptic estimate to find that It is worth noting that in the last term, accounting for the above estimates, we have that ∇ϕ ∈ L ∞ (0, T ), ϕ 2 W ∈ L 1 (0, T ), and σ ∈ L ∞ (0, T ) due to (2.5). Therefore, we insert An estimate for ∂ t σ can also be deduced, by means of a comparison argument in (1.6b). Indeed, we can write (1.6b) in the form ∂ t σ = n(ϕ)∆σ + n ′ (ϕ)∇ϕ · ∇σ − P (ϕ)(σ − AF ′ (ϕ) + BJ ⋆ ϕ), (3.43) and estimate the second term on the right-hand side (present only in the case d = 2) as which, on account of the L 2 (0, T ; W )-regularity for ϕ and for σ, entails that Hence, from this estimate, (3.42), (A2), (C1) and (3.43) we have for d = 2 the estimate In the case d = 3, since we are assuming n = 1, the second term on the right-hand side of (3.43) is not present, and the L 2 (0, T ; H)-regularity for ∂ t σ proceeds with a similar argument. Therefore, (3.2) is proven.
It then remains to establish the improved regularity (3.3) for σ for the case d ∈ {2, 3} and n = 1. To this aim, we formally differentiate (1.6b) in time and test with ∂ t σ, which can be made rigorous by returning to a Galerkin approximation of (1.6b) for σ treating ϕ possessing the above improved regularity H 1 (0, T ; H) as given data, and differentiating the Galerkin approximation in time. Since n = 1 and σ 0 ∈ W , the latter implies ∂ t σ 0 := ∆σ 0 − P (ϕ 0 )(σ 0 − AF ′ (ϕ 0 ) + BJ ⋆ ϕ 0 ) ∈ H, and we obtain, after integration over Ω, so that by Gronwall's lemma we have σ ∈ W 1,∞ (0, T ; H) ∩ H 1 (0, T ; V ). Then, by a comparison of terms in (1.6b) with n = 1, we see that ∆σ ∈ L ∞ (0, T ; H) and so by elliptic regularity it holds that The proof of Theorem 3.1 is complete.
Remark 3.1. We point out that estimate (3.28), which yields the control of the discretized solutions ϕ + N in L 2 (0, T ; W ) can be recovered also for the case d = 3, provided the improved regularity for σ given by (3.3) is assumed. Indeed, this allows to reproduce the Moser-Alikakos type argument of [29, Proof of Thm 6.1] which establishes the crucial bound ϕ k+1 L ∞ (Ω) ≤ C( ϕ 0 L ∞ (Ω) ) for all k = 0, . . . , n with n < N − 1. Let us just sketch the main points of this argument. We return to the time-discrete problem (3.16)-(3.17) taking the above improved regularity for σ into account, which implies that the source term . Appealing now to the Moser-Alikakos computation in [29, p. 718-719], which involves testing (3.16) with ϕ p j −1 k+1 where p j := 2 j , integrating over Ω and summing the resulting identity over k, for k = 0, . . . , n with 0 ≤ n ≤ N − 1, we obtain

45)
where p ′ j = p j p j −1 is the conjugate of p j . The new element in the analysis is the last term on the right-hand side which, owing to the uniform boundedness of S k and Young's inequality, can be handled as with a constant C independent of δ, the index j and N. Choosing δ = p j and noting that p −p j j → 0 as j → ∞ we infer that with a constant C independent of index j and N. The convolution term in (3.45) can be handled as in [29, (6.14)] so that, after multiplying (3.45) by p j and estimating the convolution term and the source term, we arrive at the following inequality which is exactly [29, (6.15)]. Then, we may argue as in [29] to deduce that (3.28) also holds in the case d = 3 with a modified constant Q( ϕ 0 V , σ 0 W ) on the right-hand side.

.71)] and (3.25), it holds that
and on account of (3.25), of estimate ∇ϕ k = ∇Λ ε (ϕ k )/λ ε (ϕ k ) , and of (3.12), we obtain (3.52) A similar argument as in [29] shows that By the Cauchy-Schwarz inequality, the fundamental theorem of calculus, and recalling estimate (3.44), we have that Hence, by employing this last estimate in (3.52) the discrete Gronwall lemma entails that This implies that the interpolation functionsφ N introduced in the proof of Theorem 3.1 now also satisfy . Arguing as in [29], and using the estimate Hence, after passing to the limit as N → ∞ and then as ε → 0, we deduce (3.5).
We now turn to the regularity (3.3) for σ in the case d = 2. By formally differentiating (1.6b) in time and testing with ∂ t σ we obtain where the right-hand side is bounded above by Then, by the Gagliardo-Nirenberg inequality and the fact that ∂ t ϕ ∈ L ∞ (0, T ; H) ∩ L 2 (0, T ; V ) and σ ∈ L ∞ (0, T ; V ) ∩ L 2 (0, T ; W ), Hence, choosing δ sufficiently small, and using (A2), (C1) and (3.44), we infer by Gronwall's lemma that By comparison of terms in (1.6b), it is easy to see that ∆σ ∈ L ∞ (0, T ; H), and by a classical elliptic regularity argument, we also deduce σ ∈ L ∞ (0, T ; W ).

Proof of Theorem 3.3
Let us denote for convenience . First let us consider the case d = 2. We take the difference of (1.6b) tested by σ, which yields, after integration over Ω and for some δ > 0 to be fixed later, on account of the fact that σ 1 ∈ L ∞ (0, T ; V )∩L 2 (0, T ; W ) and of the Gagliardo-Nirenberg inequality. Moreover, we have used (C1) and the bound |ϕ i | ≤ 1, for i = 1, 2. Next, testing the difference of (1.6a) with ϕ yields Thanks to the formulae (3.60) and from (3.58) and (A1) we get the Gagliardo-Nirenberg inequality. Moreover, in the first inequality we have used (C1), (C2) and (C4). Estimating the term Sσ in (3.57) in a similar fashion using the Lipschitz continuity of P and P F ′ , and adding the result to the above inequality yields d dt on account of (A2), and after choosing δ sufficiently small. Then, Gronwall's lemma applied to (3.62) gives the L ∞ (0, T ; H) ∩ L 2 (0, T ; V )-estimate of (3.7) for the case d = 2.
For the H 1 (0, T ; V * )-estimate we obtain from the difference of (1.6a) and (1.6b) for any v, w ∈ V . In light of the above estimates, as well as the calculations in (3.57), we readily infer where the constants C depend on the L 4 (0, T ; L 4 (Ω))-norms of ∇ϕ 1 and ∇σ 1 . Applying the L ∞ (0, T ; H) ∩ L 2 (0, T ; V )-estimate of (3.7) then finishes the proof.
For d = 3, we note that the term (n 1 − n 2 )∇σ 1 · ∇σ in (3.57), and the term (λ(ϕ 1 ) − λ(ϕ 2 ))∇ϕ 1 · ∇ϕ in (3.58) both vanish, since, in this case, n and λ are assumed to be constant. Hence, it is immediate to check that we can infer an analogous differential inequality to (3.62) (the term ϕ 1 2 W will no longer appear on the right hand side). This allows us, by means of Gronwall's lemma, to recover the assertion (3.7).

Application to the inverse identification problem
In this section we study the constrained minimisation problem (1.10). The standard procedure to deriving first-order optimality conditions is to first establish the differentiability of the solution operator S : ϕ 0 → ϕ, establish well-posedness results for the corresponding linearised system and adjoint system, and then use them to derive the optimality condition.
We restrict our analysis to the two dimensional case. In preliminary calculations not shown here, it appears that the Fréchet differentiability of the solution operator would require a continuous dependence estimate for ϕ and σ in L 4 (0, T ; V ) ∩ L 2 (0, T ; W ). While this is guaranteed for strong solutions given by Theorem 3.2, there is a requirement on the initial condition ϕ 0 ∈ W to fulfil the compatibility condition (c.f. Theorem 3.2) [∇Λ(ϕ 0 ) − Bm(ϕ 0 )(∇J ⋆ ϕ 0 )] · n = 0 on Γ. (4.1) Hence, as a first attempt we can take the space of admissible controls as U = u ∈ W : |u| ≤ 1 a.e. in Ω and (4.1) holds . However, this set is not convex due to the nonlinear constraint (4.1), and the resulting optimality condition would involve Lagrange multipliers, which further complicates the numerical implementation. Moreover, instead of using the norm · H 1 (Ω) in (1.10), one would employ a norm · U which need to match with the expected regularity of the control. In this case we can choose for example u U := u 2 + ∇u 2 + ∆u 2 1/2 , but we can expect the strong form of the optimality condition involves a fourth order differential operator. Hence, in light of both analytical and numerical complications arising from working with solutions of the highest level of regularity, we consider regularities obtained from Theorem 3.1. It turns out that they are enough to establish Gâteaux differentiability of the solution operator, and we can consider, for κ > 0 fixed, the convex set
For the adjoint system, again we derive formal estimates and mention that the argument can be made rigorous with a Faedo-Galerkin approximation. Testing (4.9a) with p On the other hand, taking w = ξ in (4.9a), z = η in (4.9b), and integrating by parts on (0, T ) the resulting identities, we get where we have used η(0) = 0 and q(T ) = 0. Moreover, the following identity has been employed Then, (4.12) is a consequence of comparing the sum of (4.13a)-(4.13b) and the sum of (4.14a)-(4.14b).

Gâteaux differentiability of the solution mapping
The main result of this section is the following Gâteaux differentiability of the solution mapping.  to (h, ϕ, σ).
In light of the fact that [ξ, η] is independent of the subsequence (τ k ) k∈N chosen for the weak/strong convergences, we conclude the assertion is valid for any null sequence.

Analysis of the constrained minimisation problem
The main result concerning (1.10) is formulated as follows. (i) For any α > 0, there exists at least one solution ϕ α 0 ∈ U to the minimisation problem (1.10).