Recent results on the Cahn - Hilliard Equation with dynamic Boundary conditions Текст научной статьи по специальности «Математика»

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ

MSC 35K55, 35K50, 82C26 DOI: 10.14529/mmpl70101

RECENT RESULTS ON THE CAHN - HILLIARD EQUATION WITH DYNAMIC BOUNDARY CONDITIONS

P. Colli, Dipartimento di Matematica "F. Casorati", Universitä di Pavia and Research Associate at the IMATI - C.N.E. PAVIA, Pavia, Italy, pierluigi.colli@unipv. it,

G. Gilardi, Dipartimento di Matematica "F. Casorati", Universitä di Pavia and Research Associate at the IMATI - C.N.R. PAVIA, Pavia, Italy, gianni.gilardi@unipv.it,

J. Sprakels, Weierstrass Institute for Applied Analysis and Stochastics; Department of Mathematics, Humboldt-Universität zu Berlin, Berlin, Germany, sprekels@wias-berlin.de

The pure or viscous Cahn - Hilliard equation with possibly singular potentials and dynamic boundary conditions is considered and the well-posedness of the related initial value problem is discussed. Then, a boundary control problem for the viscous Cahn -Hilliard system is studied and first order necessary conditions for optimality are shown. Moreover, the same boundary control problem is addressed for the pure Cahn - Hilliard system, by investigating it and passing to the limit in the analogous results for the viscous Cahn - Hilliard system as the viscosity coefficient tends to zero.

Keywords: Cahn - Hilliard equation; dynamic boundary conditions; phase separation; well-posedness; boundary control problem; optimality conditions.

Dedicated, to our friend Angelo Favini on the occasion of his 70th birthday with best wishes.

Introduction

The classical Cahn - Hilliard equation and the so-called viscous Cahn - Hilliard equation can be written as

dty - Aw = 0 Mid w = rdty - Ay + ß(y) + n(y) - g in Q x (0,T), (0.1)

according to the case t = 0 or t > 0, respectively. Here, Q 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), y denotes the order parameter and w represents the chemical potential. Moreover, / and n are the derivatives of the convex part / and of the concave perturbation 3 of a double-well potential f := / + 3, and g is a source term. Important examples of f are the everywhere defined regular potential freg and the logarithmic double-well potential fiog given by

freg (r) = 4(r2 - 1)2 , r £ R, (0.2)

flog(r) = (1 + r)ln(1 + r) + (1 - r)ln(1 - r) - or2 , r £ (-1,1), (0.3)

where o > 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 ^ specified by

fdobs(r) = /[-1,1] (r) - or2, r £ R (0.4)

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

/[-1,1](r) = 0 if r £ [-1,1] and /[-1,1](r) = otherwise. (0.5)

In this case, / is no longer a derivative, but it represents the subdifferential d/[-1;1] of the indicator function of the interval [-1,1], that is,

< 0 if r = -1,

s £ d/[-1;1](r) if and only if s { = 0 if - 1 <r< 1, (0.6)

> 0 if r =1.

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

dny + dtyr - Aryr + 3r(yr) + nr(yr) = gr (0.8)

on £ := r x (0,T), where

• yr denotes the trace y s on the boundary £;

• - Ar stands for ^te - Beltrami operator on r;

• /r and nr are nonlinearities playing the same role as ^d n but now acting on the boundary value of the order parameter;

• finally, gr is a boundary source torn with no relation with g acting on the bulk. We aim to point out that the corresponding initial-boundary value problem

dty - Aw = 0 in Q :=n x (0,T), (0.9)

g Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 1, pp. 5-21

w = Tdty - Ay + f'(y) - g in Q, (0.10)

dnw = 0 on E, (0.11)

yr = y ^d dny + dtyr - Aryr + fr(yr) = 0r on E, (0.12)

y(0)= yo m П, (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 fr = /Зг + Зг. 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, n and вг, пг

f

In this note, we follow a strategy developed in [18] to investigate the Allen - Cahn

equation with dynamic boundary conditions, which consists in letting fr be the leading

f

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);

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 t > 0. Moreover, if t > 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 t > 0 0 = 0 and gr = ur, the control being then the source term ur that appears in the dynamic boundary condition (cf. (0.8) and (0.12))

dny + dtyr - Aryr + er(yr) + nr (yr) = ur on E. (0.14)

Namely, the cost functional

a(y,yr,ur) := Y \\y - zq\\2l2{q) + у llyr - zs||L2(S) + b-0 |Ы|?,2(Е) (0.15)

is considered, for some given functions zq, z^ and nonnegative constants bQ,b^,b0. The control problem then consists in minimizing J(y,yr,ur) subject to the state system and to the constraint ur E Uad, where the control box Uad is specified by

Uad := {ur E H 1(0,T; Hr) П L~(E) :

Ur min < Ur < Ur max Oil E, ||5tMr|L2(E) < MQ). (0.16)

Here, the functions ur>min , ur>max G L^ (E) and the positive constant M0 are prescribed in order that the control box Uad be nonempty: this is guaranteed if, for instance, at least one of ur>min or ur>max actually belongs to Uad. 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].

These results are then used in [23], where the optimal control problem is discussed for the same state system, but when t = 0. The technique adopted in [23] essentially consists in starting from the known results for t > 0 and then letting the parameter t tend to zero. In doing that, some of the ideas of [20] and [24] are used: indeed, these papers [20,24] deal with the Allen - Cahn and the viscous Cahn - Hilliard equations, respectively, and address similar control problems related to the nondifferentiable double-obstacle potential fdobs defined by (0.4).

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 t > 0 g = 0 and gr = ur, 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, r, dn, Vr and Ar 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 E := r 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

: R ^ [0, are convex, proper, and 1.s.c. and /3(0) = /3r(0) = 0, (1.1) n,nr : R ^ R are Lipschitz continuous with n(0) = nr(0) = 0. (1.2)

We introduce the primitives 3 and 7rr of n and nr that vanish at the origin and define the

potentials / and /г and the graphs в and вг in R x R as follows

7r(r) := /0 n(s) ds and Дг(г) := /0 nr(s) ds for r E R, (1.3)

/ := Д + Д and /г : Дг + Дг, (1-4)

в := дД and вг := дДг- (1.5)

Notice that both [ and [r are maximal monotone with some eflective domains D([) and D([3r). Due to (1.1), we have 3(0) 3 0 and 3r(0) 3 0. Clearly, all the basic examples of the Introduction fit the previuos assumptions. For the graphs [ and [r we assume the following compatibility condition

D([r) C D([) and 3°(r)|< nl3^(r)| + C

for some n, C > 0 and every r G D([r), (1.6)

where [°(r^d ¡r(r) are the dements of [(r^d [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 equation.

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

V := H*(Q), H := L2(Q), Hr := L2(r^d Vr := H 1(r), (1.7)

V := {(v,vr) G V x Vr : vr = v\r} and H := H x Hr, (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 way, i.e., such that (u,v) = fQ uvdx u G H v G V

At this point, we can describe the state problem. For the data, we assume that

g E L2(0,T; H) and дг E L2(0,T; Яг), (1.9)

g E H 1(0,T; H) if т = 0, (1.10)

y0 E V, у0|г E VT , ДЫ E L1(Q) and Дг(У0г) E L1(r), (1.11)

m0 := (y0)n lies in the interior of D(вг). (1.12)

Our problem consists in looking for a quintuplet (у,уг,1^,£,£г) such that

у E H 1(0,T; V*) П L™(0,T; V) П L2(0,T; H2(Q)^d т ду E L2(0,T; H), (1.13)

уг E H 1(0,T; Hг) П L™(0,T; Уг) П L2(0,T; H2(Г)), (1.14)

УгСО = y(t)\г for a.a. t E (0,T), (1.15)

w E L2(0,T; У), (1.16)

С E L2(0, T; H) and С E в(у) a.e. in Q, (1.17)

Сг E L2(0,T; Hг) and Сг E вг(уг) a.e. on E, (1.18)

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

(дty(t), v) + I Vw(t) •Vv = 0, (1.19)

Jo,

/ w(t)v = Tdty(t) v + dtyr(t) v + Vy(t) •Vv + Vpyp(t) • Vpv

JQ JQ JV JQ JV

+ / (e(t) + n(y(t)) — g(t)) v + / (£r(t) + nr(yr(t)) - gr(t)) v (1.20)

JQ JV

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

y(0) = yo • (1.21)

The light notation Tdty stands for dt(Ty). In particular, it means zero if t = 0. Clearly, equations (1.19) - (1.20) are the variational formulation of the boundary value problem

dty — Aw = 0 and w E t dty — Ay + 3(y) + n(y) — g in Q, (1.22)

dnw = 0, yr = y s and dny + dtyr — Apyp + ¡p(yp) + nr(yr) 9 gr on E. (1.23)

We notice that the duality pairing that appears in (1.19) can be replaced by a usual integral if t > 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 dty.

Remark 1. It is worth to note a fact that is typical for Cahn - Hilliard equations. To this end, if u E V* and u E L1(0,T; V*), we define their generalized mean values uQ E R and uQ E L1(0,T) by setting

uQ := (u, 1) Mid uQ(t) := (u(t))Q for a.a. t E (0,T). (1.24)

Clearly, the relations in (1.24) give the usual mean values when applied to elements of H or L1(0,T; H). By testing (1.19) by the constant 1/|H|, we obtain

(dty(t))Q = 0 for a.a. t E (0,T) and y(t)Q = m0 for every t E [0,T] (1.25)

y

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, Thm. 2.2]):

Theorem 1. Assume (1.1) - (1.5) and let (gi, gr,i,y0,i), i = 1,2, be two sets of data satisfying (1.9) and such that y0)1,y0)2 belong to V and have the same mean value. Then, if (yi>yr,i,wi,£i,£r,i) are any two corresponding solutions to problem (1.13) - (1-21), the inequality

Iy1 — y2|||~(0,T;V *) + T ly1 — y2||L~(0,T;ff) + 11^,1 — yr,2 || L~(0,T;ffr) + ||V(y1 — y2)||2(0,T ;H) + ||Vr(yr,1

— yr,2)||2(0,T ;Hr)

< ^^0,1 — y0,2f* + t^0,1 — mAli + ||y0,1r — y0,2r|Hr

+ Hgt— g2^2L2(0T ;H) + ygr,1 — gr^^T ;Hr)} (i-26)

| () Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming

& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 1, pp. 5-21

holds true with a constant c that depends only on Q, T, and the Lipschitz constants of n and nr. In particular, any two solutions to problem (1.13) - (1.21) have the same components y yr and Moreover, even the components w and £ of such solutions are the same if [ is single-valued.

The above theorem is proved in [21] and is quite similar to the results stated in [11, Thm. 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, Thm. 2.3].

Theorem 2. Assume (1.1) - (1.6) and (1.9) - (1.12). Then, there exists a quintuplet (y, yr, 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 ^d yr of the solution to problem (1.19) - (1.21) given by the above theorems also satisfy

y e W 1'~(0,T; V*) n H 1(0,T; V) n L™(0,T; H2(Q)^d rdty e L~(0,T; H), (1.27)

yr G W 1'^(0,T; Hr) n H 1(0,T; Vr) n L™(0,T; H2(r)), (1.28) whence also

y G L~(Q) and yr G L~(E). (1.29) To this aim, we make further assumptions on the data. Namely

g G H1(0,T; H) and gr G H1(0,T; Hr), (1.30)

yo G H2(Q) and yo|r G H2(r), (1.31)

there exists £0 G H such that £0 G [(y0) a.e. in Q, (1.32)

there exists £r>0 G Hr such that £r>0 G [r(y0|r) a.e. on E, (1.33)

and, if t = 0, we reinforce (1.32) by requiring that

the family { —Ay0 — [£(y0) — g(0) : e G (0, e0)} is bounded in V (1-34)

for some e0 > 0 In (1.34), the symbol [£ stands for the Yosida regularization of [ at fevel 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 [£(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 th at r- <r_ < y0 < r+ < r+ a.e. in Q, (r'_, r+) C D([) and ^te restricti on of [to (r-, r+) is a single-valued Lipschitz continuous function.

Here is our first regularity result (see [21, Thm. 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 t > 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), £r G L~(0,T; Hr) . (1.35)

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

T>0

g E L~(Q), gr E L~(E) and 3°(y0) E L~(Q). (1.36)

Then, there exists a solution to problem (1.19) - (1.21) that also satisfies (1-27) - (1-29), (1.35) and

w E L(X(0, T; H2(H)) c L~(Q) and £ E L~(Q). (1.37)

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

D(3) and D(3r) are the same open interval I. (1.38)

This condition is fulfilled if / and fr 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 ¡and 3r- We observe th at, if I is not the who le of R and r0 is an end-point of it, then 3° has an infinite limit at r0 since the interval I is open. Hence, the second property in (1.37) yields that y(x, t) remains bounded away from r0. Moreover, if I is unbounded, one can account for (1.29). As D(3r) = D(3) properties of this type for £ and y imply similar properties for £r- 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.

T > 0

structure and (1.36) on the data. Then, there exists a solution (y,yr,w,£,£r) to problem (1.13) - (1.21) that also satisfies (1.27) - (1.29), (1.35), (1.37) and

y(x,t) E K for a.a. (x,t) E Q and some compact subset K c I, £r E L~(E).

Moreover, if 3 and 3r are single-valued, the unique solution also satisfies

3'(y) E L~(Q), 3r(y) E L~(E)

as well as, if / and fr are C2 functions in addition,

f''(y) E Lrx(0, T; V) and /£(y) E L~(0,T; Vr) .

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 3 and 3r satisfy (1.38) and are single-valued smooth function on their common domain. Here are the precise assumptions we add to (1.1) - (1.6):

D(3) = D(3r) = (r_, r+) with — to < r- < 0 < r+ < (2.1)

f, fr are С3 functions on (r—,r+), (2.2)

\f'(r)| < n \fr(r)| + С for some n, С > 0 and every r E (r—,r+), (2.3)

lim f'(r) = lim f(r) = — го and lim f'(r) = lim f (r) = (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 = freg and f = fiog 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 W T; V*) П H1 (0, T; V) П L~(0, T; H2(П)), (2.5)

rdty E L™(0,T; H), (2.6)

yr E W 1'^(0,T; Нг) П H 1(0,T; Vr) П L™(0,T; Н2(Г)), (2.7)

r- < infess y < sup ess y < r+, (2.8)

Q Q

w E L™(0,T; H2(Q)). (2.9)

In particular, all the components y, yr and w are bounded, as well as f /г^У ) f°r

i = 1, 2, 3. We notice that the assumptions on yo included in (1.31) and (1.36) mean that

y0 E H 2(П), yor E H 2(Г) and r— < y0(x) < r+ for eve ry x E П (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 gr, which we term ur now. We rewrite the full system for clarity:

I dty(t) v + J Vw(t) •Vv = 0, (2.11)

Jo, Jq

/ w(t) v = T dty(t) v + dtyr (t) vr + Vy(t) •Vv + Vr yr (t) • Vr vr Jo Jo Jr Jo Jr

+ i f'(y(t)) v + i (ff (yr(t)) — ur(t)) vr, (2.12)

Jo Jr

y(0) = yo, (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,vr) E V, respectively. We call (y, yr) the state corresponding to the control ur, and this is the most important part of the solution. Indeed, the other components are completely determined by it. The control box Uad is given by

Uad := {ur E H 1(0,T; Hr) П L~(E) :

ur,min < ur < ur,ma^ оn E, \\dturЦ2 < Mo} (2.14)

where the constant M0 and the functions ur,min and ur>max satisfy

Mo > 0, u

,min, ur,max E L (E) and Uad is ^^nempty. (2.15)

Finally, given the functions and the constants

Zq E L2(Q) , zs E L2(E^d bQ, bs, b0 E [0, (2.16)

we set

J(y,yr,ur) := y № — QLhq) + y y^r— z^Li^) + 70 ||ur!L2(s) (2-17)

for, say, y E C0([0,T]; H), yr E C0([0,T]; Hr) and up E L2(E). At this point, the control problem consists in minimizing the cost functional (2.17) subject to the constraint ur E Uad and to the state system (2.11) - (2.13). The following result holds true (see [22, Thm. 2.3] for the viscous case and [23, Thm. 2.5] for the pure one):

Theorem 5. Assume (2.10). Then, there exists ur E Uad such that

J(y,yv,uv) < J(y,yp,up) for every ur E Uad, (2.18)

where y, yr, y and yr are the components of the solutions (y, yr,w) and (y,yr,w) to the state system (1.13) - (1.21) corresponding to the controls ur and ur, respectively.

Once such an existence result is established, one looks for necessary conditions for a given ur to be an optimum control. The natural strategy is the introduction of suitable Banach spaces X and Y with the following properties: i) the control box Uad is a closed subset of X; ii) for eve ry ur in some neighbour hood U of Uad, the state system has a unique solution and the corresponding pair (y, yr) belongs to Y; iii) the map S that associates such a pair (y, yr) to the arbitrary ur E U is Frechet differentiate.

This project is difficult to realize in the general case, due to the low regularity of the

L2(0, T; V*)

is different in the viscous case due to (2.6).

T > 0

results corresponding to the above program are proved in [22] with the following choice of the spaces:

X := H 1(0,T; Hr) n L~(E) and Y := H 1(0,T; H) n L™(0,T; V). (2.19)

U Uad

Then, since the functional to be minimized is Uad 9 ur M J(ur) := 3(S(«r),Mr^d Uad is convex, the natural necessary condition is the following: (D3(ur), vr — ur) > 0 for every vr E Uad, where D3(ur) E X* is the Frechet derivative of 0 at ur- However, because of the chain rule, this contains the value at hr := vr — ur of ^te ^^^^tet derivative DS(ur), which turns out to be the solution to the problem obtained by linearizing (1.13) - (1.21) around ur and taking hr 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

Vq = bQ(y — zQ) and = bs(yr — zs), (2.20)

where (y,yr) is the state associated to the optimal control ur under attention. Then, the adjoint problem is the following: find a triplet (p, q, qr) that fulfills the regularity requirements

p E H1 (0, T; H2(Q)) n L2(0, T; H4(Q)), (2.21)

q E H 1(0,T; H) П L2(0,T; H2(H)), (2.22)

qr E H 1(0,T; Hr) П L2(0,T; H2(Г)), (2.23)

qr(t) = q(t)r fora.a. t E (0,T), (2.24)

and solves the variational equations

qv = Vp • Vv a.e. in (0,T) and for all v E V, (2.25)

'o Jq

— dt (p + Tq)v + Vq •Vv + f'(y) qv

' o

— dtqr vr + Vrqr • Vrvr + f'(yr) qrvr = Фq v + vr Jr Jr Jr Jo Jr

a.e. in (0,T) and every (v,vr) E V (2.26)

and the final condition

(P + Tq)(T) v + qr (T) vr = 0 for every (v, vr) E V. (2.27)

Jo Jr

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

Theorem 6. Assume (2.10) and t > 0, and let ur and (y,yr) = S(ur) be an optimal control and the corresponding state. Then the adjoint problem (2.25) - (2.27) has a unique solution (pT,qT,qT) 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, Thm. 2.6])

Theorem 7. Assume (2.10) and t > 0, and let ur be an optimal control. Moreover, let (y,yr) = S(ur) and (pT,qT,qT) be the associate state and the unique solution to the adjoint problem (2.25) - (2.27) given by Theorem 6. Then we have

J (ql + bour)(v r — ur) > 0 for every vr E Uad. (2.28)

Remark 2. In particular, if b0 > 0, (2.28) says that

ur is the orthogonal projection of —ql/b0 on Uad (2.29)

with respect to the standard scalar product in L2(E).

T=0

this is done in [23]. The idea is to take the limit as t n 0 in the above results.

Even though the adjoint problem (2.25) - (2.27) involves a triplet (pT,qT,ql) as

q T

qT q T

mention the result proved in [22] that deals with the pair (qT,qT). 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

dom N := [v* E V* : v? = 0} Mid N : dom N ^ [v E V : vn = 0} (2.30)

by setting, for v* E dom N,

Nv* E V, (Nv*)n = 0, and / VNv* • Vz = (v*,z) for every z E V. (2.31)

Jn

Thus, Nv* is the solution v to the generalized Neumann problem for — A with datum v* that satisfies vn = 0. Indeed, if v* E H, the above variational equation means that — ANv* = v* and dnNv* = 0. As Q is bounded, smooth, and connected, it turns out that (2.31) yields a well-defined isomorphism. Furthermore, we introduce the spaces Hn and Vn by setting

Hn := {(v,vr) E H : vn = 0} and Vn := Hn n 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 t > 0. Then, with the notation (2.20), there exists a unique pair (qT ,qT) satisfying the regularity conditions

qT E H 1(0,T; H) n L2(0,T; H2(Q)) and q^ E H 1(0,T; Hr) n L2(0,T; H2(r)) (2.33)

and solving the following problem:

(qT ,qT )(t) E Vn for eve ry t E [0, T ], (2.34)

— i dt(N(qT) + TqT) v + i VqT •Vv + / f'(y T) qT v Jn Jn Jn

— J dtqr vr + J Vrqr • Vrvr + J fp (yf) qr vr = pqv + tpY>vr a.e. in (0,T) and for every (v,vr) E Vn, (2.35)

Jn Jr

i (NqT + TqT) (T) v + i qr(T) vr = 0 for every (v, vr) E Vn. (2.36)

Jn Jr

Moreover, the pair (qT, qr) is the same as the couple of components of the unique solution (pT,qT, qr) to the 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 (Vn, Hn, Vn) (see [22, formula (5.25)]). Thus, one proves that it can be solved (see [22, pp. 21-22]). On the other hand, if (q, qr) is its unique solution, one shows that on can reconstruct p in order that the triplet (p,q,qr) solves problem (2.25) - (2.27) (see [22, Thm. 5.4], in particular formulas [22, (5.10) - (5.11)]).

At this point, we let t tend to zero in (2.34) - (2.36) rather than in (2.25) - (2.27). By

pT

We introduce the spaces

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

Proposition 1. A functional F : W0 ^ R belongs to W0 if and only if there exist A and Ar satisfying

Л G (H 1(0, T ; V *) П L2(0, T ; V)) * and Ap G (Я1(0,Т; Vr*) П L2(0,T; Vr)) *, (2.39)

where the duality products (•, ^)q and (•, • )s are related to the spaces X* and X with X = H 1(0, T; V*) n L2(0, T; V) and X = H 1(0,T; Vr*) n L2(0, T; Vr), respectively.

However, this representation is not unique, since different pairs (A, A r) 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 t = 0 and a first-order necessary condition for optimality similar to (2.28) (see [23, Thm. 2.7]).

Theorem 9. Assume (1.1) - (1.6) and (1.9) - (1-12), and let 3 and Uad be defined by (2.17) and (2.14) under the assumptions (2.15). Moreover, let ur be any optimal control

related to the state system with t = 0. Then, there exist A and Ar satisfying (2.39), and (q, q )

W := L2(0,T; Vn) П (H 1(0,T; V*) x H 1(0,T; Vr*)), Wo := {(v,vr) G W : (v,vr)(0) = (0, 0)}

(2.37)

(2.38)

((F, (v, vr))) = (Л, v)q + (Лг, vr)s for every (v, vr) G Wo ,

(2.40)

q G L(X(0, T ; V*) П L2(0,T ; V), qr G Lrx(0, T; Hr) П L2(0,T; Vr), (q,qr )(t) G Vn fora. e. t G [0,T],

(2.41)

(2.42)

(2.43)

as well as

í (dtv, Nq) + í (dtvr,qr )r W Vq •Vv + / Vr qr • Vr vr

( v, v ) G Wo ,

such that

for every vr E Uad. Remark 4. In particular, if b0 > 0, (2.45) says that

(2.45)

ur is the orthogonal projection of -qr/b0 on Uad

(2.46)

with respect to the standard scalar product in L2(E).

One recognizes in (2.44) a problem that is analogous to (2.35) - (2.36). Indeed, if A, A r and the solution (q, qr) were regular functions, then its strong form should contain both a generalized backward parabolic equation like (2.35) and a final condition for (Nq, qr) of type (2.36), since the definition of Wo allows its elements to be free at t = T. However, the terms f''(yT )qT and ff (yf)qf are just replaced by the fu nctionals A and Ar 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 PAnalisi Matematica, la Probability e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica).

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Received November 30, 2016

УДК 517.9 DOI: 10.14529/mmpl70101

ПОСЛЕДНИЕ РЕЗУЛЬТАТЫ ДЛЯ УРАВНЕНИЯ КАНА - ХИЛЛИАВДА С ДИНАМИЧЕСКОЙ ГРАНИЦЕЙ

17. Колли, Дою. Доюиларди, Ю. Спрекелс

В статье рассматривается уравнение Кана - Хиллиарда «чистое» или с вязкостью с возможно сингулярными потенциалами и динамическими граничными условиями. Обсуждается корректность соответствующей начальной задачи. Изучается задача граничного управления для системы Кана - Хиллиарда с вязкостью и находятся необходимые условия оптимальности первого порядка. Кроме того, ставится аналогичная задача граничного управления для «чистой» системы Кана - Хиллиарда, результаты получаются помощью предельного перехода в случае системы Кана - Хиллиарда с вязкостью, когда коэффициент вязкости стремится к нулю.

Ключевые слова: уравнение Кана - Хиллиарда; динамические граничные условия; разделение фаз; корректность; оптимальное граничное управление; условия оптимальности.

Пьерлуиджи Колли, кафедра математики имени Ф. Касорати, университет Па-вии; Институт прикладной математики и информационных технологий имени Энрико Мадженеса (г. Павия, Италия), pierluigi.colli@unipv.it.

Джанни Джиларди, кафедра математики имени Ф. Касорати, университет Павии; Институт прикладной математики и информационных технологий имени Энрико Мадженеса (г. Павия, Италия), gianni.gilardi@unipv.it.

Юрген Спрекелс, Институт прикладного анализа и стохастики имени Вей-ерштрасса; кафедра математики, Берлинский университет имени Гумбольдта (г. Берглин, Германия), sprekels@wias-berlin.de.

Поступила в редакцию 30 ноября 2016 г.