Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential

This paper is concerned with a distributed optimal control problem for a nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version of a model for two-species phase segregation on an atomic lattice under the presence of diffusion. The local model has been investigated in a series of papers by P. Podio-Guidugli and the present authors; the nonlocal model studied here consists of a highly nonlinear parabolic equation coupled to an ordinary differential inclusion of subdifferential type. The inclusion originates from a free energy containing the indicator function of the interval in which the order parameter of the phase segregation attains its values. It also contains a nonlocal term modeling long-range interactions. Due to the strong nonlinear couplings between the state variables (which even involve products with time derivatives), the analysis of the state system is difficult. In addition, the presence of the differential inclusion is the reason that standard arguments of optimal control theory cannot be applied to guarantee the existence of Lagrange multipliers. In this paper, we employ recent results proved for smooth logarithmic potentials and perform a so-called `deep quench' approximation to establish existence and first-order necessary optimality conditions for the nonsmooth case of the double obstacle potential.

The state system (1.2)-(1.6)constitutes a nonlocal version of a phase field model of Cahn-Hilliard type describing phase segregation of two species (atoms and vacancies, say) on a lattice, which was recently studied in [21].In the (simpler) original local model, which was introduced in [30] (see also [10] and [15]), the nonlocal term B[ρ] is a replaced by the diffusive term −∆ρ .The local model has been intensively discussed in the past years (cf.[7-11, 13-15, 18]).In particular, in [12] the analogue of the control problem (P 0 ) for the local case was investigated for g(ρ) = ρ ; for this special case, also the optimal boundary control problem was studied (see [17]).
The state variables of the model are the order parameter ρ , interpreted as a volumetric density, and the chemical potential µ ; for physical reasons, we must have 0 ≤ ρ ≤ 1 and µ > 0 almost everywhere in Q .The control function u on the right-hand side of (1.2) plays the role of a microenergy source.We remark at this place that the requirement encoded in the definition of U ad , namely that u be nonnegative, is indispensable for the forthcoming analysis; it is needed to guarantee the nonnegativity of the chemical potential µ .
The nonlinearity F is assumed smooth, while I [0,1] is the indicator function of the interval [0, 1] , so that the specific local free energy F loc := I [0,1] + F is typically a (nonsmooth) double obstacle potential.In this connection, the subdifferential ∂I [0,1] of I [0,1] is defined by .
The presence of the nonlocal term B[ρ] in (1.3) constitutes the main difference to the local model.Simple examples are given by integral operators of the form k(t, s, x, y) ρ(y, s) ds dy (1.8) and purely spatial convolutions like with sufficiently regular kernels.
The mathematical literature on control problems for phase field systems involving equations of viscous or nonviscous Cahn-Hilliard type is still scarce and quite recent.We refer in this connection to the works [4,5,19,20,26,33].Control problems for convective Cahn-Hilliard systems were studied in [31,34,35], and a few analytical contributions were made to the coupled Cahn-Hilliard/Navier-Stokes system (cf.[23][24][25]27,28]).The recent contribution [16] deals with the optimal control of a Cahn-Hilliard type system arising in the modeling of solid tumor growth.On the other hand, let us quote [2] for the analysis of the optimal control problem for a phase field system that couples an energy balance equation with an ordinary differential inclusion characterized by the presence of the graph ∂I [0,1] acting on the phase variable.[21] for the case u = 0 (no control), and in [22] we have investigated the control problem (P 0 ) for the case of smooth (but singular) nonlinearities.
In this paper, we aim to employ the results established in [22] to treat the nondifferentiable double obstacle case when ξ satisfies the inclusions (1.4).Recall that it is well known that in this case all of the classical constraint qualifications fail, so that the existence of suitable Lagrange multipliers cannot be guaranteed using standard methods of optimal control.Instead, our approach is guided by a strategy employed in [4] for viscous Cahn-Hilliard systems (see also [6] for for the simpler case of the Allen-Cahn equation): in [4], necessary optimality conditions for the double obstacle case could be established by performing a so-called 'deep quench limit' in a family of optimal control problems with differentiable nonlinearities of a form that had been previously treated in [20].The general idea is briefly explained as follows: we replace the inclusion (1.4) by where h is defined as the logarithmic potential and ϕ ∈ C 0 ((0, 1]) is a positive function satisfying lim αց0 ϕ(α) = 0. (1.12) We remark that we could simply choose ϕ(α) = α p for some p > 0 ; however, there might be situations (e. g., in the numerical approximation) in which it is advantageous to let ϕ have a different behavior as α ց 0 .
. Hence, in particular, we have that Hence, we may regard the graphs of the functions ϕ(α) h ′ as approximations to the graph of the subdifferential ∂I [0,1] .
We then consider, for any α ∈ (0, 1] , the optimal control problem (later to be denoted by (P α ) ), which results if in (P 0 ) the relation (1.4) is replaced by (1.10).For this type of problem, in [20] the existence of optimal controls u α ∈ U ad as well as first-order necessary optimality conditions have been derived.Proving a priori estimates (uniform in α ∈ (0, 1] ), and employing compactness and monotonicity arguments, we will be able to show the following existence and approximation result: whenever {u αn } ⊂ U ad is a sequence of optimal controls for (P αn ) , where α n ց 0 as n → ∞ , then there exist a subsequence of {α n } , which is again indexed by n , and an optimal control u ∈ U ad of (P 0 ) such that where, here and in the following, will always denote the control space.In other words, optimal controls for (P α ) are for small α > 0 likely to be 'close' to optimal controls for (P 0 ) .It is natural to ask if the reverse holds, i. e., whether every optimal control for (P 0 ) can be approximated by a sequence {u αn } of optimal controls for (P αn ) , for some sequence α n ց 0 .
Unfortunately, we are not able to prove such a 'global' result that applies to all optimal controls of (P 0 ) .However, a 'local' result can be established.To this end, let u ∈ U ad be any optimal control for (P 0 ) .We introduce the 'adapted' cost functional and consider for every α ∈ (0, 1] the adapted control problem of minimizing J subject to u ∈ U ad and to the constraint that (ρ, µ) solves the approximating system (1.2), (1.3), (1.10), (1.5), (1.6).It will then turn out that the following is true: (i) There are some sequence α n ց 0 and minimizers u αn ∈ U ad of the adapted control problem associated with α n , n ∈ IN , such that (1.17) (ii) It is possible to pass to the limit as α ց 0 in the first-order necessary optimality conditions corresponding to the adapted control problems associated with α ∈ (0, 1] in order to derive first-order necessary optimality conditions for problem (P 0 ) .
The paper is organized as follows: in Section 2, we give a precise statement of the problem under investigation, and we derive some results concerning the state system (1.2)-(1.6)and its α -approximation which is obtained if in (P 0 ) the relation (1.4) is replaced by the relation (1.10).In Section 3, we then prove the existence of optimal controls and the approximation result formulated above in (i).The final Section 4 is devoted to the derivation of the first-order necessary optimality conditions, where the strategy outlined in (ii) is employed.
Throughout this paper, we will use the following notation: for a (real) Banach space X, we denote by • X its norm and the norm of X × X × X , by X ′ its dual space, and by •, • X the dual pairing between X ′ and X .If X is an inner product space, then the inner product is denoted by (•, •) X .The only exception from this convention is given by the L p spaces, 1 ≤ p ≤ ∞ , for which we use the abbreviating notation • p for the norms in L p (Ω) .Furthermore, we put We have the dense and continuous embeddings , where u, v V = (u, v) H and u, w W = (u, w) H for all u ∈ H , v ∈ V , and w ∈ W .We will make repeated use of the Young inequalities which are valid for all a, b ∈ IR , γ > 0 , and p, q ∈ (1, +∞) with 1 p + 1 q = 1 , as well as of the fact that for three dimensions of space and smooth domains the embeddings V ⊂ L p (Ω) , 1 ≤ p ≤ 6 , and H 2 (Ω) ⊂ C 0 (Ω) are continuous and (in the first case only for 1 ≤ p < 6 ) compact.In particular, there are positive constants K i , i = 1, 2, 3 , which depend only on the domain Ω , such that For convenience, we also put, for t ∈ [0, T ] , Please note the difference in the position of t .About time derivatives of a time-dependent function v , we warn the reader that we will use both the notations ∂ t v, ∂ 2 t v and the shorter ones v t , v tt .
(iii) For every v, w ∈ L 1 (0, T ; H) and t ∈ (0, T ] , it holds that (iv) It holds, for every v ∈ L 2 (0, T ; V ) and t ∈ (0, T ] , that ) of B at v has for every v ∈ L 2 (Q) and t ∈ (0, T ] the following properties: In the above formulas, C B,p and C B denote given positive structural constants.We also notice that (2.7) implicitly requires that DB[v](w)| Qt depends only on w| Qt ; this is, however, a consequence of (2.2).
The statements related to the control problem (P 0 ) depend on the assumptions made in the introduction.We recall them here: (A4) J and U ad are defined by (1.1) and (1.7), respectively, where (2.10) Remark 1: In [22] it was shown that the integral operator (1.9) satisfies all of the conditions (2.2)-(2.8)provided that the integral kernel satisfies k ∈ C 1 (0, +∞) and fulfills, with suitable constants Notice that these growth conditions are met by, e. g., the three-dimensional Newtonian potential, where k(r) = c r with some c = 0 .
The following well-posedness result for the state system is a direct consequence of [21, Thms.2.1 and 2.2].
Theorem 2.3.Suppose that (A1)-(A4) are satisfied.Then there is some constant K * 2 > 0 , which depends only on the given data, such that for every u ∈ U ad and every α ∈ (0, 1] the corresponding solution (µ α , ρ α ) to (2.15)-(2.18)satisfies (2.24) Proof: Let u ∈ U ad be fixed and (µ α , ρ α ) = S α (u) for α ∈ (0, 1] .We establish the validity of (2.24) in a series of steps.In what follows, we denote by C i , i ∈ IN , positive constants that may depend on the data of our problem but not on α ∈ (0, 1] .For the sake of a better readability, we will omit the arguments of functions if there is no danger of confusion, as well as the superscript α ∈ (0, 1] , which will only be written at the end of each step of estimation.
Step 1: Hence, multiplication of (2.15) by µ and integration over Q t , for 0 < t ≤ T , yields the identity u µ dx ds , whence, using (A1), Young's inequality, and Gronwall's lemma, we easily deduce that Moreover, since the embedding (L ∞ (0, T ; H) ∩ L 2 (0, T ; V )) ⊂ L 10/3 (Q) is continuous, we have (by possibly choosing a larger constant C 1 ) that Step 2: Next, we multiply (2.16) by ρ t and integrate over Q t , where 0 < t ≤ T .It follows Thus, employing (2.1), (2.23), and (2.25), we readily obtain from Young's inequality that Step 3: Now observe that we have the identity Taking the euclidean scalar product of this identity with ∇ρ , and integrating the result over Q t , where 0 < t ≤ T , we find that where, since h ′′ (ρ) ≥ 0 , the integral on the left-hand side is nonnegative, while, thanks to the fact that g ′′ (ρ) ≤ 0 and µ ≥ 0 , the second integral on the right-hand side is nonpositive.Moreover, by using (2.5), (2.23), (2.25), and Young's inequality, we easily find that the first integral on the right-hand side is bounded by an expression of the form ) .Hence, we can infer from Gronwall's lemma and (2.27) that In particular, this yields the desired bound for the last term on the right-hand side of (2.24).

.36)
We may therefore account for the conditions µ 0 ∈ W and u L ∞ (Q) ≤ R in order to repeat the argument developed in the proof of [10, Thm.2.3], which is based on the above summability of ρ α t .We should remark that the quoted proof is performed for g(ρ) = ρ and u ≡ 0 , but only minor changes are needed to arrive at the same conclusion in the present situation (see also the proof of the analogous [15,Thm. 3.7] in an even more complicated case).We thus can infer the bound Finally, we use the previously shown estimates to conclude that the expression independently of α ∈ (0, T ] , which then also holds true for ∆µ α , by comparison in (2.15).We thus have This concludes the proof of the assertion.

Existence and approximation of optimal controls
Our first aim is to show the existence result stated below.For its proof we do not use the direct method since this would force us to find bounds for the states corresponding to the chosen minimizing sequence of controls.On the contrary, Theorem 2.3 already provides a number of estimates.For that reason, we pass through the related approximating control problem.
To this end, we recall that h is convex and bounded in [0, 1] and that ϕ is nonnegative.We thus have, for every n ∈ IN , Now, we notice that the first term and the last one of this chain tend to zero as n → ∞ by (1.12).Hence, by ignoring the middle line of (3.11) and invoking (3.5) and (3.6), passage to the limit as n → ∞ yields This entails that ξ is an element of the subdifferential of the extension It remains to show that ((µ, ρ, ξ), u) is in fact optimal for (P 0 ) .To this end, let v ∈ U ad be arbitrary.In view of the convergence properties (3.1)-(3.7),and using the weak sequential lower semicontinuity properties of the cost functional, we obtain J((µ, ρ), u) = J(S 0 (u), u) ≤ lim inf n→∞ J(S αn (u αn ), u αn ) where for the last equality the continuity in L 2 (Q) × L 2 (Q) of the cost functional with respect to (µ, ρ) was used (see however the next statement).With this, the assertion is completely proved.

.13)
Proof: By the same arguments as in the first part of the proof of Theorem 3.1, we can conclude that (3.2)-(3.10)hold true at least for some subsequence.But, as we have just seen, the limit is given by the unique solution triple to the state system (1.2)-(1.6).Hence, the limit is the same for all convergent subsequences, and thus (3.2)-(3.10)are true for the entire sequence, as claimed.Now, let v ∈ U ad be arbitrary.Then, owing to (3.6), (3.7), S αn (v) converges strongly to The validity of (3.13) is then a consequence of the fact that J is continuous in L 2 (Q) × L 2 (Q) with respect to (µ, ρ) .
Theorem 3.1 does not yield any information on whether every solution to the optimal control problem (P 0 ) can be approximated by a sequence of solutions to the problems (P α ) .As already announced in the introduction, we are not able to prove such a general 'global' result.Instead, we give a 'local' answer for every individual optimizer of (P 0 ) .For this purpose, we employ a trick due to Barbu [1].Now let ū ∈ U ad be an arbitrary optimal control for (P 0 ) , and let (μ, ρ, ξ) be the associated solution triple to the state system (1.2)-(1.6) in the sense of Theorem 2.1.In particular, (μ, ρ) = S 0 (ū) .We associate with this optimal control the 'adapted cost functional' and a corresponding 'adapted optimal control problem' ( P α ) Minimize J((µ α , ρ α ), u) subject to u ∈ U ad and (2.15)- (2.18).
With a standard direct argument that needs no repetition here, we can show the following result.
We are now in the position to give a partial answer to the question raised above.We have the following result.
We now aim to prove that u = ū .Once this is shown, the uniqueness result of Theorem 2.1 yields that also (µ, ρ, ξ) = (μ, ρ, ξ) , which then implies that (3.16) holds true.Indeed, we have, owing to the weak sequential lower semicontinuity of J , and in view of the optimality property of ((μ, ρ), ū) for problem (P 0 ) , On the other hand, the optimality property of ((µ αn k , ρ αn k ), u αn k ) for problem ( P αn k ) yields that for any k ∈ IN we have whence, taking the limit superior as k → ∞ on both sides and invoking (3.13) in Corol- Optimal control of a phase field system with double obstacle Combining (3.19) with (3.21), we have thus shown that 1 2 u − ū 2 L 2 (Q) = 0 , so that in fact u = ū and thus also (µ, ρ, ξ) = (μ, ρ, ξ) .Moreover, (3.19) and (3.21) also imply that which proves (3.17) and, at the same time, also (3.15).The assertion is thus completely proved.

The optimality system
In this section, we aim to establish first-order necessary optimality conditions for the optimal control problem (P 0 ) .This will be achieved by a passage to the limit as α ց 0 in the first-order necessary optimality conditions for the adapted optimal control problems ( P α ) that can be derived by arguing as in [22] mith minor changes.This procedure will yield certain generalized first-order necessary conditions of optimality in the limit.In this entire section, we assume that ū ∈ U ad is a fixed optimal control of problem (P 0 ) and that (μ, ρ, ξ) is the associated solution to the state system (1.2)-(1.6)established in Theorem 2.1, that is, we have (μ, ρ) = S 0 (ū) and ξ ∈ ∂I [0,1] (ρ) almost everywhere in Q .We begin our analysis by formulating for arbitrary α ∈ (0, 1] the adjoint state system for the adapted control problem ( P α ) corresponding to ū .We assume that ūα ∈ U ad is an arbitrary optimal control for ( P α ) and that (μ α , ρα ) = S α (ū α ) is the corresponding solution to the associated state system (2. ) that the corresponding adjoint system has the form where denotes the adjoint operator associated with the operator , which is defined by the identity According to [22,Thm. 4.2], the system (4.1)-(4.3) has for every α ∈ (0, 1] a unique solution pair (p α , q α ) such that Moreover, as in [22,Cor. 4.3] it follows that the following variational inequality is satisfied: We now prove an a priori estimate that will be fundamental for the derivation of the optimality conditions for (P 0 ) .To this end, we set, for all α ∈ (0, 1] , and we introduce the function space which is a Hilbert space when equipped with the standard inner product of We have the following result.
Suppose that the conditions (A1)-(A4) are fulfilled.Then there is a constant K * 3 > 0 , which depends only on the data of problem (P 0 ) , such that, for every α ∈ (0, 1] , Proof: In the following, C i , i ∈ IN , denote positive constants, which are independent of α ∈ (0, 1] .For the sake of a better readability, we will in the following steps omit the superscript α , writing it only at the final estimate of each step.We also will make repeated use of the global bounds (2.21), (2.23), (2.24) without further reference.
Step 1: We first add p on both sides of (4.1), multiply the result by −p t , and integrate over Q t (recall (1.22)), where 0 ≤ t < T .Using the fact that g is nonnegative, we then obtain that where the quatities I j , 1 ≤ j ≤ 4 , will be specified and estimated below.At first, we obtain from Young's inequality the estimates |p| 2 dx ds , (4.13) Moreover, owing to Hölder's and Young's inequality, and using the continuity of the embedding V ⊂ L 6 (Ω) , Step 2: We now multipy (4.2) by q and integrate over Q t , where 0 ≤ t < T .We obtain that where, since ϕ(α)h ′′ (ρ) ≥ 0 and µ g ′′ (ρ) ≤ 0 , the integral on the left-hand side is nonnegative and the quantities J j , 1 ≤ j ≤ 5 , will be defined and estimated below.We have as well as, using Young's inequality, where in the last estimate we have used (4.5) and (2.7).Finally, we take advantage of the continuity of the embedding H 2 (Ω) ⊂ L ∞ (Ω) and of Hölder's and Young's inequality to conclude that, for every γ > 0 (to be chosen later), where the constant C Ω > 0 depends only on the domain Ω .Combining all the estimates (4.18)-(4.24),we have therefore shown that Step 3: We now use (4.1) to estimate ∆p α directly.Indeed, we have Now, owing to (1.20), it holds that Optimal control of a phase field system with double obstacle In conclusion, we have shown that are bounded in L 1 (0, T ) independently of α ∈ (0, 1] , we conclude from Gronwall's lemma (taken backward in time) the estimate Now assume that v ∈ Y is arbitrary.As q α (T ) = v(0) = 0 , integration by parts with respect to time (which is permitted since q α , v ∈ H 1 (0, T ; H) ) yields and it follows from (4.29) that It remains to show the bound for λ α (see (4.8)).To this end, notice that (4.2) yields where it easily follows from the estimates (2.23), (2.24) and (4.29) that the last five summands on the right-hand side are bounded in L 2 (Q) (and thus in Y ′ ) independently of α ∈ (0, 1] .To estimate the remaining term, let v ∈ Y be arbitrary and observe that (2.19), (4.6), the continuous embedding V ⊂ L 6 (Ω) and Hölder's inequality imply whence also g ′ (ρ α ) μα t p α ∈ L 2 (Q) .Therefore, using (4.10), (2.24) and (4.29), as well as the continuity of the embeddings H 2 (Ω) ⊂ L ∞ (Ω) and H 1 (0, T ; H) ⊂ C 0 ([0, T ]; H) , we have In consequence, we have that which concludes the proof of the assertion.
We draw some consequences from the previously established results.Assume that {α n } ⊂ (0, 1] satisfies α n ց 0 as n → ∞ .Then, thanks to Theorem 3.4, there is a subsequence, without loss of generality {α n } itself, such that, for any n ∈ IN , we can find an optimal control ūαn ∈ U ad for ( P αn ) and an associated state (μ αn , ραn ) such that the convergence given by (3.15) and (3.17) (where one reads α n in place of α n k ) holds true.As in the proof of Theorem 3.1, we may without loss of generality assume that, for 1 ≤ q < 6 and Φ ∈ {F ′′ , g, g ′ , g ′′ } , Also, by virtue of Lemma 4.1, we may without loss of generality assume that the corresponding adjoint state variables (p αn , q αn ) satisfy q αn → q weakly-star in L ∞ (0, T ; H), for a suitable triple (p, q, λ) .Therefore, passing to the limit as n → ∞ in the variational inequality (4.7), written for α n , n ∈ IN , we obtain that p satisfies Next, we will show that in the limit as n → ∞ a limiting adjoint system for (P 0 ) is satisfied.To this end, we note that it is not difficult to check that g(ρ αn ) p αn t → g(ρ) p t , g ′ (ρ αn ) ραn t p αn → g ′ (ρ) ρt p, g ′ (ρ αn ) q αn → g ′ (ρ) q, all weakly in L 1 (Q).

.40)
We also have that (see (4.35)) ∂ n p = 0 a. e. on Σ , p(T ) = 0 a. e. in Ω. (4.41) In order to derive an equation resembling (4.2), we multiply (4.2) (written for α n ) by an arbitrary element v belonging to the space which is a dense subset of Y .Integrating over Q and by parts with respect to t , we then obtain the equation By virtue of the previously established convergence properties of the involved sequences, it is not difficult to show that we may pass to the limit as n → ∞ in all of the summands occurring in (4.42) to the penultimate one.We may leave the details of these straightforward calculations to the reader.For the penultimate summand, we have (q αn − q) DB[ρ](v) dx dt .
While the second summand tends to zero as n → ∞ , nothing can be said about the first one: we need an additional assumption.The following condition is obviously sufficient to guarantee that also the first summand approaches zero as n → ∞ : In conclusion, if (A5) is valid, then the passage to the limit as n → ∞ results in the following identity: We claim that the variational equation (4.43) holds in fact true for all v ∈ Y .To see this, we employ a standard density argument.Indeed, if v ∈ Y is given, then there is some sequence {v n } ⊂ Y 0 such that v n → v in the norm of H 1 (0, T ; H) .Now observe that (4.43) is valid for v = v n .It is now easily checked that we may pass to the limit as n → ∞ in each term occurring in (4.43).As an example, we give the details for the most difficult term, which is the last one on the left-hand side.We have, denoting by C i , i ∈ IN , constants that do not depend on n : ≤ C 3 v − v n H 1 (0,T ;H) → 0 as n → ∞.
Next, we show that the limit λ satisfies some sort of a complementarity slackness condition.Indeed, we have lim inf Moreover, there is some indication that the limit λ should somehow be concentrated on the set where ρ = 0 or ρ = 1 (which, however, we cannot prove rigorously).To this end, we test λ αn by the function ραn (1 − ραn ) φ , where φ is any smooth test function satisfying φ(0) = 0 .By recalling that h ′′ (r) = We now collect the results established above.We have the following statement.
Theorem 4.2: Let the assumptions (A1)-(A5) be satisfied, and let ū ∈ U ad be an optimal control for (P 0 ) with the associated solution (μ, ρ, ξ) to the state system (1.2)-(1.6) in the sense of Theorem 2.1.Moreover, let {α n } ⊂ (0, 1] with α n ց 0 as n → ∞ be such that there are optimal pairs ((μ αn , ραn ), ūαn ) for the adapted problem ( P αn ) satisfying (3.15)-(3.17)(such sequences exist by virtue of Theorem 3.4) and having the associated adjoint variables {(p αn , q αn )} .Then, for any subsequence {n k } k∈IN of IN , there are a subsequence {n k ℓ } ℓ∈IN and some triple (p, q, λ) such that • p ∈ H 1 (0, T ; H) ∩ L ∞ (0, T ; V ) ∩ L 2 (0, T ; W ) , q ∈ L ∞ (0, T ; H) , and λ ∈ Y ′ , • the relations Remark: We are unable to show that the limit triple (p, q, λ) solving the adjoint problem associated with the optimal pair ((μ, ρ), ū) is uniquely determined.Therefore, it may well happen that the limiting pairs differ for different subsequences.However, it follows from the variational inequality (4.38) that, for any such limit, it holds with the orthogonal projection IP U ad onto U ad with respect to the standard inner product in L 2 (Q) that in the case β 3 > 0 we have ū = IP U ad −β −1 3 p .