Recent results on the Cahn - Hilliard equation with dynamic boundary conditions

The classical Cahn - Hilliard ecpiation and the so-called viscous Cahn - Hilliard equation can be written as dty — Aw = 0 aiid w = Tdty — Ay + в(У) + п(y) — g in D x (0, T),      (0.1)

according to the case т = 0 о г т >  0 , respectively. Here, D C R³ stands for the bounded smooth domain where the evolution takes place and T denotes some final time.

The set of Cahn - Hilliard equations (0.1) provide a description of the evolution phenomena related to solid-solid phase separations. We refer to, in chronological order, [15] for some pioneering contributions on these models and problems. In general, an evolution process goes on diffusively. However, the process of the solid-solid phase separation does not seem to comply with this structure: more precisely, each phase concentrates and the so-called spinodal decomposition occurs. A comparative discussion on the modelling approach for phase separation, spinodal decomposition and mobility of atoms between cells can be found in [6-10]).

About the variables appearing in (0.1), у denotes the order parameter and w represents the chemical potential. Moreover, в and п are the derivatives of the convex part в and of the concave perturbation b of a double-well potential f := в + b, ^and g is a source term. Important examples of f are the everywhere defined regular potential f reg and the logarithmic double-well potential f_log given by

. . .1

freg (r) = 4(r 2 - 1)2 , r E R,(0.2)

flog (Г) = (1 + Г)ln(1+ Г) + (1 - Г )ln(1 - Г) - СГ2 , Г E (-1, 1),(0.3)

where c > 0 in (0.3) is large enough in order that flog be nonconvex. Another important example refers to the so-called double-obstacle problem and corresponds to the nonsmooth potential fdobs : R ^ (то, + to] specified by fdobs (r) = I [ -1,1] (r) - cr2, r E R(0.4)

with c >  0 and where the indicator function of the interval [ - 1 , 1] fulfills

I [ - 1 , 1] ( r ) = 0 if r E [ - 1 , 1] and I [ _- 1 , 1] ( r ) = + to otherwise. (0.5)

In this case, в is n° longer a derivative, but it represents the subdifferential dI[- 1,1] of the indicator function of the interval [-1, 1], that is, s E dI[-1,1](r)

{ < 0

=0

> 0

If r = - 1 ,

If - 1  1 ,

If r = 1 .

(0.6)

We are interested in the coupling of (0.1) with the usual no-flux condition for the chemical potential dnw = 0                                  (0.7)

and with the dynamic boundary condition дпУ + dtyг - Аг У г + в г( У г) + пг( У г) = g г                   (0.8)

on X := Г х (0 ,T ). where

• •    у г denotes the trace у _s on the boundary X:

• •    - А_г stands for the Laplace - Beltrami operator on Г:

• •    в г arк! п _г are nonlinearities pla. ylng the same role as в and п but now acting on the boundary value of the order parameter;

• •    fimilly, g _г Is a boundary source term with no relation with g acting on the bulk.

We aim to point out that the corresponding initial-boundary value problem dty - Аw = 0 1 ii Q :=Q х (0,T),

(0.9)

w = т d t y - A y + f‘ ( y ) - g  in Q,                    (0.Ю)

d n w = 0 on X ,                              (0.11)

y г = y _S arid   d n y + d t y г - Ar y г + fГ ( y г) = g г  (m X ,          (0.12)

y (0) = y о inO ,                                  (0.13)

has been first addressed in [11]. Actually, the Cahn - Hilliard system (0.9) - (0.13), or better some variation of it including dynamic boundary conditions, has drawn much attention in recent years: let us quote [12-16] among other contributions. In particular, the existence and uniqueness of solutions as well as the behavior of the solutions as time goes to infinity have been studied for regular potentials f and f г = в г + тг_г. Moreover, a wide class of potentials, including especially singular potentials like (0.3) and (0.4), has been considered in [11, 17]: in these two papers the authors were able to overcome the difficulties due to singularities and to show well-posedness results along with the long-time behavior of solutions. The approach of [11,17] is based on a set of assumptions for в, п and в г , п г that gives the role of the dominating potential to f and entails some technical difficulties.

In this note, we follow a strategy developed in [18] to investigate the Allen - Cahn equation with dynamic boundary conditions, which consists in letting f _г be the leading potential with respect to f: it turns out that this approach simplifies the analysis. Moreover, we discuss the optimal boundary control problem for the viscous and pure Cahn - Hilliard equation with dynamic boundary conditions, in analogy with the corresponding contributions for the Allen - Cahn equation (see [19] and [20]). In particular, we review the results proved in the three research papers

• •    [21] (well-posedness and regularity);

• •    [22] (optimal control problem for the viscous Cahn - Hilliard equation);

• •    [23] (optimal control problem for the pure Cahn - Hilliard equation).

The paper [21] contains a number of results on the state system (0.9) - (0.13). More precisely, existence, uniqueness and regularity results are proved in [21] for general potentials that include (0.2) - (0.3), and are valid for both the viscous and pure cases, i.e., by assuming just т >  0. Moreover, if т >  0, further regularity and properties of the solution are ensured.

On the other hand, the paper [22] deals with a control problem for the state system (0.9) - (0.13) when т >  0, g = 0 and g _г = и _г, the control being then the source term u г that appears in the dynamic boundary condition (cf. (0.8) and (0.12))

d n y + d t y г - А г y г + в г ( y г ) + п г ( y г ) = и г с rn X.              (0.14)

Namely, the cost functional

^{J( y} , ^y г , ^u г ) ^:= "Q || ^{y —} z Q ^ L ² ( Q ) + "Q ^ ^y г ^— z "  I I L ² (S) + "Q l l u г I I L ² (S)            (0.15)

is considered, for some given functions z q ,z " and nonnegative constants b Q ,b _S , b 0. The control problem then consists in minimizing J( y,y _г , u _г) subject to the state system and to the constraint и г G U_ad. where the control box U_ad is specified by

U ad := { и г G H 1 (0 ,T ; H г ) П I (X) :

u Г , min < u Г < u Г , max a.e. Oil S , ||d_tU г || _L 2 ( _S ) < M 0} . (0.16)

Here, the functions u г , min , u г , _max G L” (S) and the positive constant M0 are prescribed in order that the control box U_ad be nonempty: this is guaranteed if, for instance, at least one of u г , min оr u г , _max actually belongs to U_ad. The existence of an optimal control and first-order necessary conditions for optimality are proved and expressed in terms of the solution of a proper adjoint problem in [22].

Now, we think it is important to recall some related contributions. The paper [25] deals with the well-posedness of the system (0.9) - (0.13) in which also an additional mass constraint on the boundary is imposed. The case of a dynamic boundary condition also of Cahn - Hilliard type, i.e. admitting a chemical potential on the boundary too, has been studied in [26]. Recently, Cahn - Hilliard systems have been rather investigated from the viewpoint of optimal control. In this connection, we refer to [27-29] and point out the contributions [30,31] dealing with the convective Cahn - Hilliard equation; the case with a nonlocal potential is studied in [32]. The paper [33] investigates the second-order optimality conditions for the state system (0.9) - (0.13) when т >  0, g = 0 and g г = u p, starting from the results of [22]. There also exist articles addressing some discretized versions of general Cahn - Hilliard systems, cf. [34,35].

The present paper is organized as follows. In the next section, we list our assumptions, state the problem in a precise form and present our well-posedness and regularity results. In the last section we deal with boundary control problems both for the viscous and the pure case.

1.    Well-Posedness and Regularity

In this section, we describe the problem more carefully and present some basic results. As in the Introduction, Q is the body where the evolution takes place. We assume Q C R³ to be open, bounded, connected, and smooth, and we write | Q | for its Lebesgue measure. Moreover, Г, d n, V _r and А_г stand for the boundary of Q, the outward normal derivative, the surface gradient and the Laplace - Beltrami operator, respectively. Finally, T is a given finite final time and we use the notation

Q := Q x (0 , T ) and S := Г x (0 , T ) .

Now, we specify the assumptions on the structure of our system. In order to include both regular and singular potentials, like the examples (0.2), (0.3) and (0.4) of the Introduction, every potential is split into a convex part and a perturbation, with mild assumptions on the former and regularity assumptions on the latter. So, we assume that

• в, в_Г : R ^ [0 , + to ] are convex, prop er. and l.s.c. and в (0) = в г(0) = 0 , (1.1) п,п г : R ^ R are Lipschitz continuous with п (0) = п _г(0) = 0 . (1.2)

We introduce the primitives b and b_r оf п and п _r that vanish at the origin and define the potentials f and f г and the graphs в and в г in R x R ^as follows

b(r) := Jr п(s) ds and f : = в + b в := db br(r) := JJ пr(s) ds for r G R, and fг : br + br, and вг := дв г •

(1.3)

(1.4)

(1.5)

Notice that both в and в г are maximal monotone with some effective domains D ( в ) and D ( в г)• Due to (1.1), we have в (0) 9 0 and в r(0) Э 0. Clearly, all the basic examples of the Introduction fit the previuos assumptions. For the graphs в and в г ^we assume the following compatibility condition

D (вr) C D (в) arid 1в° (r) |< п1вГ (r) | + C for some n, C > 0 and every r G D(вг),                 (1-6)

where в ° ( r ) and в Г ( r ) are the elements of в ( r ) and в r( r ), respectively, having minimum modulus. Roughly speaking, condition (1.6) is opposite to the one postulated in [11]. On the contrary, it is the same as the one introduced in the paper [18], which however deals with the Allen - Cahn ecpiation.

The above assumptions are sufficient for satisfactory well-posedness results. In order to present them with a. simplified notation, we set

V := H 1(0) , H := L 2(0) , H г := L 2(Г) arк! V г := H 1(Г) ,        (1.7)

V := { ( v,v r) g V x V r : v r = v _r } arid H : = H x H г ,              (1.8)

and endow these spaces with their natural norms. Furthermore, the symbol (■, ■) stands for the duality pairing between V* , the dual space of V , and V itself. In the following, it is understood that H is embedded in V* in the usual war". i.e.. such that (u, v) = J_Q uv dx for every u G H and v G V.

At this point, we can describe the state problem. For the data, we assume that g G L2(0,T; H) arid gr G L2(0,T; Hг), (1.9) g G H 1(0,T; H) if т = 0, (1.10) yо G V, yo|r G Vr , в(yo) G L 1(0) aiid /5r(yo|r) G L 1(Г), U-i-1) mо := (У0)q lies in tire interior of D(вг)• (1-12)

Our problem consists in looking for a quintuplet (y,yr,w,£,£r) such that y G H 1(0 ,T; V *) П L“ (0 ,T; V) C L 2(0 ,T; H 2(0)) arid т dty G L2 (0 ,T; H),   (1.13)

y r G H 1(0 , T ; H r) П L“ (0 , T ; V г ) C L 2(0 , T ; H ²(Г)) ,                              (1.14)

y r( t ) = y ( t )|_r for a.a. t G (0 ,T ) ,                                                          (1.15)

w G L 2(0 ,T ; V ) ,                                                                    (1.16)

^ G L 2(0 , T ; H ) ar id ^ G в ( y ) ^a-^e • in Q,                                         (1-17)

^ _r G L ²(0 ,T ; H г) arid ^ _r G в г( y г ) а.е. on D ,                                 (1-18)

and satisfying for a.a. t G (0 , T ) the variational equations

^{( d} t y ⁽ t ) ,v) +

/

Ω

Vw ( t ) ■ Vv = 0 ,

(1.19)

I w ( t ) v = I Td t y ( t ) v + I d t y r ( t ) v + I Vy ( t ) • Vv + J V г y г ( t ) • V p v + 1 ( £ ⁽ t ^{) + n (} у ⁽ t )) - g ⁽ t )) v + 1 ( £ r ⁽ t ) + ⁿ r ⁽ y r ⁽ t )) - g r ⁽ t )) v

(1.20)

(1.21)

for every v E V and every v E V, respectively, and the Cauchy condition

У (o) = y о .

The light notation Tdty stands for dt(ту). In particular. 11 means zero If т = 0. Clearly, ecpiations (1.19) - (1.20) are the variational formulation of the boundary value problem dty — Aw = 0 arid w E т dty — Ay + в(У)+ п(y) — g in Q, (1.22 ) dnW = 0, y г = y E arid dny + dty г — Ary г + вг( Уг) + п г( у г) 9 g г < mH.  (1.23)

We notice that the duality pairing that appears in (1.19) can be replaced by a. usual integral if т >  0 thanks to the last (1.13), while it has to be kept as it is in the opposite case due to the low level of regularity of d t y.

Remark 1. It is worth to note a. fact that is typical for Cahn - Hilliard ecpiations. To this end. if u E V * ar id u E L 1(0, T; V *). we define their generalized mean values uQ E R and uQ E L 1(0, T) by setting uQ "= |Q ^u’ 1 ^  Ш^  uQ(t) "= (^(t^  ^°Г a'a't E (0’T)'         (1-21)

Clearly, the relations in (1.24) give the usual mean values when applied to elements of H or L 1(0 ,T ; H ). By testing (1.19) by the constant 1 /| Q | . we obtain

( d t y ( t )) q = 0 for a.a. t E (0 ,T ) ar id y ( t ) q = m ₀ for every t E [0 ,T ]      (1.25)

with the notations (1.24) and (1.12). Thus, the mean value of y is conserved during the evolution. For that reason, this model has to be included in the class of the so-called conserved models for two phase systems.

Now, we present a. number of results proved in [21]. As far as uniqueness and continuous dependence are concerned, we have (see [21, Thin. 2.2]):

Theorem 1. Assume (1.1) - (1.5) and let ( gi,g_г^,у ₀ ,i ), i = 1 , 2, be two sets of data satisfying (1.9) and such that y ₀ , 1 ,y ₀ ,2 belong to V and have the same mean value. Then, if ( yi,y_г,_i’W_i’$_li’$_{l г} ,_i ) are any two corresponding solutions to problem (1.13) - (1.21), the inequality

^ ^y 1     ^y 2 || L ^ (0 ,T ; V * ) ^{+ T} ll y 1     y 2 I I L ” (0 ,T ; H ) ⁺ ll y г, 1     ^y г, 2 || L ^ (0 ,T ; H _r )

+ ll ^{V (} y 1 ^— y 2 ) ¹ L ² (0 ,T ; H ) + ^V l' ^{( y} г, 1 ^{— y} г , 2) ¹ L 2 ² (0 ,T ; H _r )

A c | ^{l y} 0 , 1 ^{— y} 0 , 2 ¹ * + ^{T l y} 0 , 1 ^{— y} 0 , 2 ¹ H + ^{l y} 0 , 1|_г ^{— y} 0 , 2|_г ¹ H _r

+ ^{l g} 1 ^{— g} 2 ¹ L ² (0 ,T ; H ) + ^{l g} г, 1 ^— g г, 2 ¹ L ² (0 ,T ; H r) }                      (1-26)

holds true with a constant c that depen ds only on O, T, and the Lipschitz constants of n and п p. In particular, any two solutions to problem, (1.13) - (1.21) have the same components y. y г anul £ r■ Moreover, even the. components w and £ of such solutions are the same if в is single-valued.

The above theorem is proved in [21] and is quite similar to the results stated in [11, Thin. 1 and Rem. 9]. In the latter paper (see [11, Rem. 4 and Rem. 8]), it is also shown that partial uniqueness and conditionally full uniqueness as in the above statement are the best one can prove. As for existence, here is our general result [21, Thin. 2.3].

Theorem 2. Assume (1.1) - (1.6) and (1.9) - (1.12). Then, there exists a quintuplet ( y,y г , w,£,£ r) satisfying (1.13) - (1.18) and solving problem (1.19) - (1.21).

Next goal is regularity. First, we want to prove that the components y and yг of the solution to problem (1.19) - (1.21) given by the above theorems also satisfy y G W1 'x(0,T; V*) П H 1(0,T; V) П Lx(0,T; H2(0))  arid  mly G L“(0,T; H), (1.27)

y г G W ¹ ’^ (0 ,T ; H г) П H 1(0 ,T ; V Д П L“ (0 ,T ; H ²(Г)) ,                          (1.28)

whence also y G L^ (Q) aiid yг G L” (S).                      (1.29)

To this aim, we make further assumptions on the data. Namely g G H 1(0,T; H) arid gг G H 1(0,T; Hr),                       (1.30)

y о G H 2(0) arid y 0|_r G H ²(Г) ,                                  (1.31)

there exists £ 0 G H such that £ 0 G в ( y 0) ^a-^e• in Q,                (1-32)

there exists £ p0 G H _r such that £ p0 G в r( y о|_г) ^a'^e' °ⁿ S ,         (1.33)

and, if т = 0, we reinforce (1.32) by requiring that the family {—Ay0 — ве(yо) — g(0) : E G (0, E0)} is Ironnded in V (1-34)

for some e ₀ >  0. In (1.34). the symbol в_е stands for the Yosiв at level e (see, e.g., [36, p. 28]). Clearly, in order to ensure (1.34), one can assume that Ay₀+g(0) G V and that в_е(y0) remains bounded in V for e small enough. A sufficient condition for the latter is the following: there exist r±,r± G R such that r— ₀< r₊< r + a.e. in 9, (r-,r +) C D(в) and the restriction of в tо (r-,r +) is a single-valued Lipschitz continuous function.

Here is our first regularity result (see [21, Thin. 2.4]). It regards general potentials and both the viscous and pure cases.

Theorem 3. Assume (1.1) - (1.6) on the structure and suppose that the data satisfy (1.30) - (1.33) and (1.12). Moreover, assume either т > 0 or (1.34). Then, there exists a solution to problem (1.19) - (1.21) that also satisfies (1.27) - (1.29) as well as w G L” (0 ,T; V), £ G L” (0 ,T; H), £ г G L” (0 ,T; Hr).         (1.35)

The next result regards the viscous case, only, but it still allows general potentials (see [21, Thin. 2.6]).

Theorem 4. In addition to the assumptions of Theorem 3, suppose that т > 0 and that g G L" (Q), g г G L" (E) ш id в ° (У о) G L" (Q) ■             (T36)

Then, there exists a solution to problem (1.19) - (1.21) that also satisfies (1.27) - (1.29), (1.35) and w G L"(0,T; H2(fi)) C L"(Q) auid £ G L"(Q)■             (1.37)

It is worth noting an interesting consequence that holds in the following case:

D ( в ) arid D ( в г) are the same open interval I.                (1.38)

This condition is fulfilled if f and f г are, for instance, the same everywhere defined smooth potential (0.2) or the same logarithmic potential (0.3). On the contrary, potentials whose convex part is an indicator function like (0.4) are excluded. However, (1.38) still allows multi-valued operators в and в г- We observe that, if I is not the whole of R and r 0 is an end-point of it, then в° has an infinite limit at r 0 since the interval I is open. Hence, the second property in (1.37) yields that y ( x,t ) remains bouncled away from r0. Moreover, if I is unbounded, one can account for (1.29). As D ( в г) = D ( в ) properties of this type for f and y imply similar properties for f г- Therefore, if (1.38) holds, the next statement (see [21, Cor. 2.7]) easily follows from the results already presented. Let us recall (1.4)-(1.5) before stating it.

Corollary 1. In addition to the hypotheses of Theorem 3, assume т >  0 and (1.38) on the structure and (1.36) on the data. Then, there exists a solution ( y,y г , w,^, ^ г) to problem (1.13) - (1.21) that also satisfies (1.27) - (1.29), (1.35), (1.37) and

y ( x,t ) G K for a. a. ( x, t ) G Q and some, compact subset K C I,

( г G L" (E) ■

Moreover, if в and вг йге single-valued, the unique solution also satisfies в ‘(y) G L" (Q),   в г (У) G L" (E)

as well as. if f and f _г a re C ² functions in addition.

f" ( y ) G L" (0 ,T ; V ) ш id #( y ) G L" (0 ,T ; V г) ■

2.    Control Problems

In dealing with control problems, it might be easy to prove the existence of an optimal control, while, in general, it is more difficult to establish first-order necessary conditions for optimality. To this aim, one often needs that the state corresponding to the optimal control under attention is very smooth. For that reason, we reinforce our assumptions on the structure. In particular, we also assume that в and в г satisfy (1.38) and are singlevalued smooth function on their common domain. Here are the precise assumptions we add to (1.1) - (1.6):

D ( в ) = D ( в г) = ( r—,r +) with — to < r- <  0 < r ₊ <  + to ,        (2.1)

f, f г ai'e C ³ functicms on ( r-,r + ) ,

|f‘ ( r ) | < n Ifr ( r ) | + C for some n, C >  0 and every r E ( r-,r ₊) ,

lim f‘ ( r ) = lim f Г( r ) = —to and

r ^ r.

r ^ r.

lim f‘ ( r ) = lim f‘ ( r ) = + to . r ^ r +           r / r +

(2.2)

(2.3)

(2.4)

Clearly, (2.3) and (2.4) follow from (1.1) - (1.6) if both r- and r ₊ are finite. Notice that, once more, the choices f = f_reg and f = f log corresponding to (0.2) and (0.3) are allowed. On the contrary, the double-obstacle potential (0.4) is excluded. It is understood that all the assumptions (1.1) - (1.6) and (2.1) - (2.4) on the structure are in force throughout the whole section.

If the data, satisfy (1.30) - (1.33) and (1.12), then the solution is unique and enjoys the following regularity y E W1 ”(0, T; V*) П H1 (0, T; V) О L”(0, T; H2(Q)),(2.5)

Tdty E L”(0,T; H),(2.6)

yг E W1 ’”(0,T; Hг) П H 1(0,T; V) О L”(0,T; H2(Г)),(2.7)

r- < infess y < sup ess y < r +,(2.8)

QQ w E L”(0,T; H2(Q)).(2.9)

In particular, all the components y, yг and w are bounded, as well as f(i)(y) and f^i)(yг) for i = 1,2,3. We notice that the assumptions on yо included in (1.31) and (1.36) mean that yо E H2(Q), yо|г E H2(Г) and r- < yo(x) < r + for every x E Q (2.10)

in the present case.

At this point, we can address the corresponding control problem. The state system is (1.13) - (1.21) with g = 0 and the (control is g p. which лve term u г now. We rewrite the full system for clarity:

I d t y ( t ) v + I Vw ( t ) • Vv = 0 ,                                           (2.11)

I w ( t ) v = т I d t y ( t ) v + I d t y r( t ) v г + j Vy ( t ) •Vv + j V r y r( t ) • V r v г

+ ^ f ‘ ( y ( t )) v + ^( f‘ ( y r( t ))

y ⁽⁰⁾ = y о ,

-

u r( t )) v г ,

(2.12)

(2.13)

where (2.11) and (2.12) hold for a.a. t E (0 , T ) and for every v E V and every ( v,v r) E V, respectively. We call ( y,y r) the state corresponding to the control u r, and this is the most important part of the solution. Indeed, the other components are completely determined by it. The control box U_ad is given by

Uad := { u г E H 1(0 ,T ; H г) О L” (S) :

u Γ , min < u г < u г ’ max me. ОП S , ^d t u г У 2 < M э}          (2.14)

where the constant M о and the functions u г , _min and u г , _max satisfy

M о >  0 , u г , min , u г , max E L” (S) arid Uad is поп empty .          (2.15)

Finally, given the functions and the constants zQ E L2(Q),   zs E L2(S)  aiid  bQ, bs, bо E [0, + to),            (2.16)

we set

^{J( y}, У г , ^u г) ^: = y Q ll ^{y — z}Q^ L ²( Q ) + ll y Г ^— z s I I L ² (s) + "2" l l u г I I L ²(s)           (--A )

for, say, y E C "([0 , T ]; H ), y г E C "([0 , T ]; H г) and u г E L ²(S). At this point, the control problem consists in minimizing the cost functional (2.17) subject to the constraint u _г E U_ad and to the state system (2.11) - (2.13). The following result holds true (see [22, Thin. 2.3] for the viscous case and [23, Thin. 2.5] for the pure one):

Theorem 5. Assume (2.10). Then, there exists u _г E U_ad such that

J( y,y г ,u г) <  J( y,y г , u г) for every u г E Uad ,                (2.18)

where y. y _г. y and y г are. the. componen ts о J the. solutions ( y, y _г ,w ) and ( y,y г ,w ) to the. state system (1.13) - (1.21) corresponding to the controls u г a nd u г, respectively.

Once such an existence result is established, one looks for necessary conditions for a given u г to be an optimum control. The natural strategy is the introduction of suitable Banach spaces X and Y with the follow dug properties: i ) the control box U_ad is a closed subset of X; ii ) for every u г in some neighbourhood U of U_ad, the state system has a unique solution and the corresponding pair ( y,y г) belongs to Y; iii ) the map S that associates such a pair ( y,y г) to the arbitrary u г E U is Frechet differentiable.

This project is difficult to realize in the general case, due to the low regularity of the time derivative of the state, which only belongs to L 2(0 ,T ; V* ) (see (2.5)). The situation is different in the viscous case due to (2.6).

So, we split our discussion in two parts, and we first assume that т >  0. Then, the results corresponding to the above program are proved in [22] with the following choice of the spaces:

X := H 1(0 ,T ; H г) F l L“ (S) arid Y := H 1(0 ,T ; H) F l L“ (0 ,T ; V) .        (2.19)

Moreover, U is an arbitrary open neighbourhood of U_ad (see [22, Prop. 2.4 and Thin. 4.2]). Then, since the functional to be minimized is U_ad Э u _г ^ J( u г) := J(S( u г) , u г) and U_ad is convex, the natural necessary condition is the following: (D J( u г) , v _г — u _г ) >  0 for every v г E U_ad. where D J( u г) E X * is the Frechet derivative of J at u _г. However, because of the chain rule, this contains the value at h г := v г — u г of the Freehet derivative D S( u г), which turns out to be the solution to the problem obtained by linearizing (1.13) - (1.21) around u г and taking h _г in the linear term that corresponds to the position of the control in the nonlinear problem (see [22, Prop. 6.1 and formula (2.42)]). This can be eliminated by introducing a proper adjoint problem. We set for brevity

ФQ = bQ ( y — ZQ ) arid ф s = b s( y г — z s) ,                 (2.20)

where (y, yг) is the state associated to the optimal control uг under attention. Then, the adjoint problem is the following: find a triplet (p, q, qг) that fulfills the regularity requirements p E H1 (0,T; H2(fi)) П L2(0,T; H4(fi)),                   (2.21)

q Е H 1(0 ,T ; H ) П L 2(0 ,T ; H 2(0)) , (2.22) q г Е H 1(0 , T ; H г) П L2(0, T ; H ² (Г)) , (2.23) q r( t ) = q ( t )| _г for a.a. t Е (0 ,T ) , (2.24)

and solves the variational equations j qv = j Vp • Vv a.e. in (0, T) and for all v Е V,

• -    I dt ( p + Tq ) v + I Vq •Vv + J f‘‘ ( y ) qv

• -    dtq г v г + V r q г • V г v г + fГ ( У г) q г v г = ФQ v +

Γ    Γ     Γ     ΩΓ a.e. in (0, T) arid every (v,vг) Е V

(2.25)

φ Σ v Γ

(2.26)

and the final condition

У ( p + Tq )( T ) v + У q p( T ) v _г = 0 for every ( v,v _г ) Е V .

(2.27)

We have the following result (see [22, Thin. 2.5]):

Theorem 6. Assume (2.10) and t >  0, and let u _г a nd ( y, y _г ) = S( u _г ) be an optimal control and the corresponding state. Then the adjoint problem (2.25) - (2.27) has a unique solution ( p^T,q^T,qг ) satisfying the regularity conditions (2.21) - (2.24).

Finally, the necessary condition involving the linearized problem takes a particularly simple form if the solution of the adjoint problem is used. Namely, we have (see [22, Thin. 2.6])

Theorem 7. Assume (2.10) and t >  0, and let и _г be an optimal control. Moreover, let ( У, У г) = S( u г) м^ ( P^T, q^T,q г) be the associate state and the unique solution to the adjoint problem (2.25) - (2.27) given by Theorem 6. Then we have

( qг + Ъ 0 u г)( v г - u г) >  0 for every v г Е Uad .

(2.28)

Σ

Remark 2. In particular, if Ъ0 > 0, (2.28) says that иг is the orthog'onal projection of —qг/Ъ0 оn Uad              (2.29)

with respect to the standard scalar product in L 2(E).

The next step is to treat the pure Cahn - Hilliard system, i.e., the case t = 0, and this is done in [23]. The idea is to take the limit as t ^ 0 in the above results.

Even though the adjoint problem (2.25) - (2.27) involves a triplet (pT,qT,qг) as an adjoint state, only the third component qг enters the necessary condition (2.28) for optimality. On the other hand, qT a nd q г are strictly related to each other. Hence, we mention the result proved in [22] that deals with the pair (qT,qг). To this end, we recall a tool, the generalized Neumann problem solver N, that is often used in connection with the Cahn - Hilliard equations. With the notation for the mean value introduced in (1.24), we define domN := {v* G V* : vQ = 0} arid N : domN > {v G V : vQ = 0}       (2.30)

by setting, for v* G domN.

N v ∗ ∈ V,

(N v* )^Q = 0 , and

/

Ω

V N v* ■ Vz = (v*, z)

for every z G V.

(2.31)

Thus. N v* is the solution v to the generalized Xeumaini problem for — A with (latum v* that satisfies v ^Q = 0. hideed. If v* G H. the above variational equation means that — AN v* = v* and dn N v* = 0. As Q is bounded, smooth, and connected, it turns out that (2.31) yields a well-defined isomorphism. Furthermore, we introduce the spaces H_Q and V_Q by setting

Hq := { ( v, v r) G H : v ^Q = 0 } aiid Vq := Hq П V ,

(2.32)

and endow them with their natural topologies as subspaces of H and V, respectively. We have the following result.

Theorem 8. Assume т > 0. Then, with the notation (2.20), there exists a unique pair (qT,qГ) satisfying the regularity conditions qT G H 1(0 ,T; H) П L 2(0 ,T; H 2(Q)) ал id qГ G H1 (0 ,T; Hг) П L 2(0 ,T; H 2(Г)) (2.33)

and solving the following problem:

(2.34)

(2.35)

(2.36)

⁽ q^T ,q r⁾⁽ t ) ^{G v}q f^{or ev(}M/ t^G [° ,T ^] ,

— J dt (N( q^T ) + Tq^T ) v + J Vq^T ■Vv + J f ‘‘ ( y ^T ) q^T v

^— J ^d tqГ ^v г + J ^VrqГ ■ ^V r v г + J^f Г ⁽ yГ ) qГ ^v г

= J фQV + J ф s v _Г a. i■. in (0 , T ) and for every ( v, v r) G V_Q.

J (N q^T + Tq^T ) ( T ) v + ^ q r( T ) v _Г = 0 for every ( v,v r) G V_Q .

Moreover, the pair ( q^T,q Г) is the same as the couple of components of the unique solution ( P^T,q^T,q Г) ^° ^^e adjoint problem (2.25) - (2.27) given by Theorem 6.

Remark 3. It is worth to notice that our presentation does not follow [22] in the detail. Indeed, [22] uses this problem to solve the adjoint problem (2.25) - (2.27) as follows. From one hand, the system (2.34) - (2.36) can be seen as a backward Cauchy problem in the framework of the Hilbert triplet ( V_q , H_q , VQ) (see [22, formula (5.25)]). Thus, one proves that it can be solved (see [22, pp. 21-22]). On the other hand, if ( q, q _Г) is its unique solution, one shows that on can reconstruct p in order that the triplet ( p,q,q _Г) solves problem (2.25) - (2.27) (see [22, Thin. 5.4], in particular formulas [22, (5.10) - (5.11)]).

W := L 2(0 ,T ; V_Q) П (H 1(0 ,T ; V* ) x H 1(0 ,T ; V* )) ,            (2.37)

Wo := { ( v,v r) G W : ( v,v r)(0) = (0 , 0) }                       (2.38)

and endow them with their natural topologies. Moreover, we denote by ((•, - ^ the duality product between W0 and W_o. We have the following representation result for the elements of the dual space W * (see [23, Prop. 2.6]):

Proposition 1. A functional F : W_o ^ R belongs to W * if and only if there exist Л and Л_г satisfying

Л G (H 1(0 ,T ; V* ) П L 2(0 ,T ; V )) *  aiid  Лг G (H 1(0 ,T ; V* ) П L 2(0 ,T ; V Д) *,  (2.39)

«F, ( v, v r) )) = ( Л , v ) q + ( Лг , v г ) s for every ( v, v r) G Wo ,

(2.40)

where the duality products (•, -) q a nd (•, •) _s are related t о the spaces X * a nd X with X = H 1(0 ,T ; V* ) П L 2(0 ,T ; V ) aa zd X = H 1(0 ,T ; V* ) П L 2(0 ,T ; V r). respectively.

However, this representation is not unique, since different pairs (Л , Л_г) satisfying (2.39) could generate the same functional F through formula (2.40).

At this point, we can state our last result. The following theorem gives both a generalized solution to a proper adjoint problem with т = 0 and a first-order necessary condition for optimality similar to (2.28) (see [23, Thin. 2.7]).

Theorem 9. Assume (1.1) - (1.6) and (1.9) - (1.12), and let J and Uad be defined by (2.17) and (2.14) under the assumptions (2.15). Moreover, let Uг be any optimal control related to the state system with т = 0. Then, there exist Л and Лг satisfying (2.39), and a pair (q, qr) satisfying q G L^(0,T; V*) П L2(0,T; V), qг G L '(0,T; Hг) П L2(0,T; VД, (q,qr)(t) G VQ for a.e. t G [0,T],

(2.41)

(2.42)

(2.43)

as well as

T(dyv, Nq) + ^ T(dyv г , q г ) г +    Vq • Vv +    V r q г • V r v г

+ ( Л ,v ) q + ( Лг , v Г ) s = I ФQ v + I ф s v г     foi• every ( v,v r) G Wo ,      (2.44)

such that

I ( q г + b o u r)( v г - u г) >  0 for every v г G Uad • Remark 4. In particular, if b 0 >  0, (2.45) says that

U г is the ortLogon al projection of —q _r /b 0 о n U_ad with respect to the standard scalar product in L 2(B).

(2.45)

(2.46)

One recognizes in (2.44) a problem that is analogous to (2.35) - (2.36). Indeed, if Л, Л_г and the solution ( q, q _r) were regular functions, then its strong form should contain both a generalized backward parabolic equation like (2.35) and a final condition for (N q, q _r) of type (2.36), since the definition of W₀ allows its elements to be free at t = T . However, the terms f‘‘ ( y ^T ) q ^T aiid ff ( yГ ) q Г are just replaced by the functionals Л and Л_Г and cannot be identified as products, unfortunately.

Acknowledgements. PC and GG gratefully acknowledge some financial support from the MIUR-PRIN Grant 2015PA5MP7 "Calculus of Variations" and the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probability e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).