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470

Proof.

The

square

of the

norm

obviously less than

constant multiplied

by the

right hand side

(61).

Let us

prove

the

opposite. Using Poincare's inequality

Hi(0,7r)

and integrating

in y2, we see

that

the

square

of the

norm

in V is

more than

constant

multiplied

by the

first term

of the

right hand side

(61).

only remains

prove

analogous inequality

for the

second term

in the

right hand side

(61).

But it

follows

from

the

previous considerations that

ll-llv>C||.||

L2(Rv2;L2(

o,.

)vi)

(62)

and obviously

>C\\Dlv\\

LHRs2]L2(0ir)yiy

(63)

Moreover,

in H

(R) the

norms

IMI^(R)

and

IMI^(R)

||£>

HlL(R)

are equivalent (this

is a

classical result, which holds

for

bounded

well

as for

unbounded

intervals,

see

[7], Theorem 2.3,

p. 19).

Then, adding

(62) and (63) the

conclusion follows.

•

In addition, classical methods

translation

and

convolution with

smooth function

([7],

p. 14)

show that

Lemma

3.2 We

have

the

identity between spaces

V = L

(H^JHJIO,^)

f]L

((0,7r)

;fl

(H,

)) . (64)

Remark

3.3 We

note that

v e V

implies that

the

traces

of v on yi = 0 and yi = n

vanish,

but the

trace

of D\V

does

not

vanish

general (compare with

H^B^)).

The

process

completion

construct

implies

loss

principal boundary conditions

0 and

y\ = IT.

Lemma

3.4

The functional

($,w)

defined

in (31) (see

also

(50)) is

continuous

on V.

Proof.

Immediate, using Lemma 3.1

and the

trace theorem

in H

(Rj,

•

consequence,

the

following limit problem

is a

Lax-Milgram problem, enjoying

existence

and

uniqueness

Find

DSV

such that

, .

VweV: a

(f,w)

($,w).

*• '

We then have

the

convergence theorem

Theorem

3.5 Let v

and

the

solutions

of (57) and (65)

respectively. Then

—>

strongly

in V- (66)

471

The proof of this theorem is exactly analogous to that of Theorem 2.4 and even simpler,

as equivalence classes are not involved. It will not be given here.

Let us consider now the limit problem Vo (65). It is not hard to solve it by separation

of variables, as it involves functions (and no longer equivalence classes, as in section 2).

The equation associated with the variational formulation (65) is clearly

(-D\ + B}) v = F(

)6'(y

) on B

(0,

TT)

X R. (67)

We add the boundary conditions

w(0,lft) = «(jr,ift) = 0, (68)

v (2/1,3/2) tends to 0 as \y

—>

00. (69)

Obviously, conditions (68) are contained in the definition of the energy space V (30).

As for (69), they will be understood in the sense of the exponential decreasing for the

Fourier components involved in the separation of variables process (see later on) ; they

are consistent with the definition of V and allow to construct the unique solution of (65).

Let A

and w

) be the eigenvalues and eigenvectors (normalized in L

(0,7r)) of

—D\

with the boundary conditions

w„(0) = w

(n) = 0. (70)

Let

(&)

(y^'

(

)

n=l

F = Y, FnVn (Vl) , (72)

n=l

be the expansions of the unknown v and the datum F respectively. Equation (65) amounts

A»«» (») + D\v

(jft) = FJ (a,). (73)

The distribution 5' at the right hand side of (73) may be described in terms of the

jumps. Denoting [a] = a(0+)

—

a(0—) :

K] = 0,

] = 0,

[Z?K] = F,

] = 0.

The function v

) are odd and satisfy :

(0+) = 0,

(0+) = F/2

472

and we have explicitly

' u

(2/2) is odd,

< u

) = -^— (e"*» - e^») for y

> 0, (74)

, where a± = 2^/2

i/4

(

1±i)

We then have the explicit expression of the solution, which is exponentially decreasing

for I2/2I -* 00. It should be mentioned that, as the exponents are not real, the graph of

(2/2) cuts the axis of abcissas infinitely many times.

Remark 3.6 The same problem may be considered with a singularity of lower order in

the right hand side, namely

f{x

) = F(x

)6(x

)

which does not belong to the dual of V

defined in (52). The developments are analogous

to the previous ones, but (56) is replaced by

(x) = -v^(y) (75)

and the right hand side of (57) is presently

($,«,) = / F{y

)w(y

,0)dy

Obviously, in the limit problem (67), 6' must be replaced by 6. Its explicit solution is

obtained as before and is now an even function in y

4 Layers along a boundary with a Neumann bound-

ary condition

In this section, we consider the previous problems in the domain

n = (0,jr)x(0,l), (76)

instead of (13). The boundary conditions will be of Neumann type on x

= 0, whereas

they remain unchanged on the rest of the boundary.

We first consider (14), (15) on the space

V = {v | v e

(Q),

v = d

v = 0 on x

= 0, x

TT,

= l} (77)

and the functional of the right hand side is still given by (24). This amounts to

(-A +

)

= 0 infi, (78)

473

with

the

Neumann boundary conditions

(-<9

)

= 0

oni

= 0, (79)

= -F

oni

(80)

and

course

(18) on the

rest

of the

boundary. After

the

change,

the

problem

analogous

to (29) but in the

space

| v

), v = d

v = 0 on y

= 0, y

TT,

= 1/e)

(0,7r)x(0,l/e).

Moreover,

(32) is

replaced

((M)

;

(0,l/e)/R),

(81)

e<l

where

(0,1/e)

denotes

the

space

of the

functions

of H

(0,1/e) extended

for

1/e

with

constant value equal

to the

trace

at y

= 1/e and

considered

up to an

additive

constant.

Finally

the

solution

of the

limit problem

is,

instead

of (44) :

v(yi,y2)

)(l-e-y>).

(82)

The problem

section

admits

analogous treatment

in the

domain

(76). In

particular, Lemmas

3.1 and 3.2

hold, after replacing

R^ by the

half-axis

> 0. The

convergence Theorem

3.5

holds true,

and the

solution

the limit problem

merely twice

the expression

(74) for y

> 0.

References

[1] Eckhaus

W.,

Asymptotic analysis

singular perturbations. North-Holland, Amster-

dam,

1979.

[2] Egorov

Yu. V. and

Shubin

M.A.,

Linear partial differential equations. Foundations

of classical theory.

In :

Encyclopaedia

Mathematical Sciences (Part.

diff. Eq. I),

Springer,

1992, 30, p.

345-375.

[3] Gerard

P.,

Solutions conormales analytiques d'equations hyperboliques

non

lineaires.

Commun. Part.

Diff. Eq., 13

(1988), 345-375.

[4] Gerard

P. and

Sanchez-Palencia

E.,

Sensitivity phenomena

for

certain thin elastic

shells with edges. Math. Meth. Appl.

Sci., 23

(2000), 379-399.

[5] IPin

A. M.,

Matching

asymptotic expansions

solutions

boundary value prob-

lems. Amer. Math.

Soc, 1991.

474

[6] J.L. Lions, Perturbations singulieres dans les problemes aux limites et en controle

optimal. Springer, Berlin, 1973.

[7] Lions J.L. and Magenes E., Problemes aux limites non homogenes et applications.

Vol. 1, Dunod, Paris, 1968.

[8] Sanchez-Palencia E., On a singular perturbation going out of the energy space. J.

Math. Pures Appl., 79 (2000), 591-602.

[9] Sanchez-Palencia E., Propagation of singularities along a characteristic boundary for

a model problem of shell theory and relation with the boundary layer. Compt. Rend.

Ac.

Sci., ser. lib, 329 (2001), 249-254.

[10] Vishik M.I. and Lusternik L., Regular degenerescence and boundary layer for linear

differential equations with small parameter. Usp. Mat. Nauk, 12(5) (1957),

1-122.

Regularity and uniqueness results for a phase change

problem in binary alloys*

Jean-Frangois Scheid

, Giulio Schimperna

1 '

Institut de Mathematiques Elie Cartan, Universite de Nancy 1

F-54506 Vandoeuvre-les-Nancy Cedex, Prance

Dipartimento di Matematica, Universita di Pavia

Via Ferrata, 1,

1-27100

Pavia, Italy

Email : scheid@iecn.u-nancy.fr ; giulio@dimat.unipv.it

Abstract

An isothermal model describing the separation of the components of a binary

metallic alloy is considered. A phase transition process is also assumed to occur in

the solder; hence, the state of the material is described by two order parameters,

i.e., the concentration c of the first component and the phase Held ip. Existence

of a solution to the related initial and boundary value problem has been proved

in a former paper, where, anyway, uniqueness was obtained only in a very special

case.

Here some further regularity and uniqueness results are shown in a more

general setting using an a priori estimates - compactness argument. A key point

of the proofs is the analysis of the fine continuity properties of the inverse map of

the solution-dependent elliptic operator characterizing one of the equations of the

system.

1 Introduction and mathematical preliminaries

In this paper, we aim at presenting some regularity and uniqueness results for the system

dttp-Atp = F

{ip) + c +

{^),

—

div (ji

(tp,

c) Vw) = 0, (1)

w e -Ac +

describing the diffusive separation of components in a binary metallic alloy possibly un-

dergoing a phase transition phenomenon. In the above relations (1), the unknown c

represents the relative concentration of either of the components, while tp is the phase

parameter, with

= 0 denoting the solid, tp = 1 the liquid, and 0 <

< 1 a mixture.

Moreover, F

, 7 and g are smooth coupling terms whose meaning is outlined in the

'This work was partially supported by the Istituto di Analisi Numerica del CNR-Pavia, Italy

475

476

paper [4]. Finally, /i is the mobility coefficient, possibly depending on both the unknowns,

but assumed to be nondegenerate, and

is a maximal monotone graph guaranteeing the

"physical" constraint 0 < c < 1.

Let us point out that the system (1) has to be complemented with the Cauchy con-

ditions for ip and c and with homogeneous Neumann boundary conditions for

<p,

w and

c. Under such assumptions, problem (1) has been studied in [4], where existence and

regularity results were proved for a variational formulation of that system. However, the

question of uniqueness was solved just in the very particular case when fi is a constant.

Hence, in this paper we deal with two more general settings, where the mobility

fj,

may

depend either on the sole unknown

or on both the unknowns

(</?,

c), and we are able to

show two distinct uniqueness and continuous dependence results. Indeed, both situations

seem to be significant on the physical viewpoint [4, Introduction]. In order to prove the

second result, we have to assume very strong smoothness properties of the solutions. In

this direction, we actually prove a new regularity theorem for the solution of (1) that fits

with the uniqueness setting at least for suitably small final times. The main analytical

instrument on which we rely consists in a fine analysis of the continuity properties of

the solution-dependent elliptic operators resulting from the nonconstant mobility fi. This

follows the lines of the work [1], where similar arguments were used for the study of the

Cahn-Hilliard system with nonconstant mobility.

Let us now start briefly presenting some mathematical notations and tools that will

be used throughout the rest of the paper. Let fi be a smooth, bounded and connected

domain in M

(the situation in the two-dimensional case is analogous and is even easier in

the one-dimensional one) and let T > 0 be a given final time. Set T := <3fi, E := T x (0, T),

:= fi x (0,t), for t e (0,T] and Q := Q

. Set also H := L

(fi) and V := H\n) and

endow the latter space with its usual scalar product.

{{v,w)) := / vwdx+ Vu -Vwdx. (2)

J n J n

We identify H and its dual, in order that the compact inclusion H C V holds and

(V, H, V) form a Hilbert triplet. Denote by (•, •) the scalar product of both H and H

(fi),

by |-1 the associated norms and by ||

•

the norm of the generic Banach space X. Finally,

indicate by (.,.) the duality pairing between V and V and by ((., .))„ the associated scalar

product on V.

Given any £ € V, let us now set

(

:=±(C,l); V^.= {CeV:Cn = 0},

:=VnVi.

(3)

Let also 0 < a < fi

be assigned constants and let

fi 6 Lip

ioc

), with a< fi<n

a.e. in K

. (4)

If v, z : Q

—>

R are measurable functions, then we can define the operators

VlZ

) : V -> V, (B

u, y) := /

(v, z)

•

Vydx, for u, y € V, (5)

J n

477

B:V^V,

{Bu,y) := / Vu

•

Vycte, for

u,yeV.

(6)

J n

Clearly, B and f$(„,

) map V onto VJJ and their restrictions to

turn out to be

isomorphisms of V

onto

VQ.

Then, we can denote by Af the inverse of B and by Af(

,z)

the inverse of -B(„

). Just by applying the definition (5), one can readily check that for

any u € V and (e VJ there holds

u,N(?,z)0 = (

(v,z)N

t,u) = (C,u> . (7)

Let now p £

[l,oo],

u, Z be measurable functions on Q, and u, £ be measurable

functions of time with values in V and

VQ,

respectively. Then, for a.e. t € (0,T), we can

put (B

u)(t) := B

(v(t)iZ(t

)){u{t)) and (A/"

(

„

iZ)

C)(£) := A/(„(i),*(t))(C(<))- This provides a

natural extension of the above operators to a time-dependent setting.

Prom this point on, k will stand for a positive constant (possibly not always the same)

depending only on a,

fiQ,

O. A positive constant depending on one, or more additional

parameters (say 6), will be noted as kg, instead. We have (see [4, Sec. 2] or [1, Sec. 2] for

the

proof,

which is essentially based on the Poincare-Wirtinger inequality):

Lemma 1.1 For all ( £ V

' and all measurable functions v, z

—>

M, we have

||JW||

IKHv

<C,JV(»,.)C) > k

\\C\\

(8)

Moreover, ifv, z are measurable functions of Q into K, p £

[l,oo],

then

: 77(0, T; V) -

V^T'X)

and M

: i7(0,T; V

') - L"(0,T; V

introduced as noted above, are well

defined.

In addition, the S(^,

) are (surjective) iso-

morphisms of L

(0,T;V

) onto 1^(0,T;

VQ).

Finally, we have

(

)\\c(LP(0,T;V),LP(0,T;V^) —

k and

II^W) \\c(LP(p,T;V£),W(0,T;V

)) —

> ^>

i.e., their norms do not depend on

(v,z).

A consequence of (8) and of the compact embedding V CC H is that, if 6 > 0,

£ Vo,

—

B(„

ju for a suitable u £ V, then

< «|VC|

+ fcs Ut, < <5|VC|

+ fc,|V«|

. (10)

Now, let us analyze some deeper continuity properties of the operator Af(

,z)- We

report the proofs, which follow the lines of

[1,

pp. 493-494], for the sake of completeness:

Lemma 1.2 Let v, z £

(fl),

f 6 ii" n VJ,', and u £ V be such that £ :=

V<Z

)U.

Let

also \x e W

1,0

°(R

). T/ien, i/iere ezisfe fc > 0 such that

IMI^(n) < fc (iCI +

ll(^)ll(tf

|Vu|) . (11)

478

Proof.

Since

£ £ H, our

hypotheses

on v, z,

(J,

guarantee that

u is in

(£l)

and

satisfies

—div(/i(v,

z)Vu) = £, in H.

We deduce that

—n(v,

z)Au =

(Vfi(v,

z))

•

Vu + ( in H and

consequently

/ |-Aw|

dz<fc|C|

+ fc / (\Vv\

\Vz\

)\Vu\

dx.

Now, using elementary interpolation, Sobolev's embedding theorem,

and

Young's

in-

equality,

obtain

IMI

(«)

k(\u\

\-Au\

)

k\C\

k(\\vf

)

k\C\

+ k(\H

HHn)

\\H'(CI)J

IMIjfCn))

|V«|

\H\

{n)

|V«|

+ \

||«||^

(n)

Lemma

1.3 Lei u, w, z, (

functions from

into

wifft

£ =

^)U,

a.e. in

(0,T).

Then,

the

following formulas hold whenever they make sense

i /»

((B

,u)

-—{B

u,u)+ n(v,z)

|Vu|

dx,

(12)

({B(

VlZ

)u),u

)

-—(B(

ViZ)

u,u)- fj,(v,z)

\Vu\

dx.

(13)

Proof.

Let us

assume that

all the

functions

are

sufficiently regular

give sense

to the

integrations

parts below. Then

have

((•B(„,

)u)

u) = — {B

u, u) - (B

u, d

{ii{v,z)\Vu\

)dx-\

[

n(v,z)d

\Vu\

<9t(M^)|Vu|

)<fa

+ - /

^,z)

|Vu|

dz,

•/It

J it

so that

(12) is

proved

in the

regular case

and in

general

we can

conclude using

density

argument.

The

proof

of (13) is

analogous.

•

2 Main results

Let

give

the

main assumptions

on the

data

of the

problem.

Let K > 0 and

\Fi\

•

l-Pal.

M ,

Ifll,

\F{\, \K\, W\,

\g'\

< K a.e. in R,

>Po

e H,

C 1 x R is a

maximal monotone graph such that

0 £

/? (0),

^ '

intD

(/?),

where

•

479

Here

t/>

: K—•

[0,

+00] is a convex and lower semicontinuous function such that 0 = dip

and

(0) = 0 and the domain of the graph

is required to fulfill

£(/3):={reR:/?(r)^0}c[O,l]. (15)

Then, the following existence-regularity result [4, Thms. 3.1, 3.3, 3.4] holds :

Theorem 2.1 Let assumptions (4), (14)-(15)

hold.

Then there exists a quadruple

Up,

w, f)

such that

(0,

T; V) n L°°

(0,

T; V)

l~l

(0,

T; if

(fi)),

«J6L

(0,T;F),

£eL

(0,T;H).

The quadruple (tp,c,w,£) satisfies

(16)

ip +

B<p

= Fi (p) + cF

{ip)

in V, a.e. in

(0,T),

c +

B(p

= 0 in V, a.e. in (0,T).

w = Be + £ + 7 (c) + 5

(</>)

inV, a.e. in (0,T).

£ € /3(c) o.e. in Q,

¥>(.,0) = </3

(.), c(.,0) = co(.) o.e. infi,

c(«)

= c

/oraW*e[0,T].

(17)

If in addition

¥>o

e V, (18)

i/ien

</>

satisfies

<p£H

(0,T;H)n C°

([0,T];

V) n L

(0,T; tf

(H)) (19)

and i/

£H

{Q,),

5„v?

= 0 a.e. onT, (20)

iften

ftas i/ie additional regularity

'™

(0,

T; H) n ff

(0,

T; V) n L°°

(0,

T; ff

(«)) fll

(0, T; #

(fi)) . (21)

Finally, just under the assumptions (4),

(14)>

(fi

) *

constant, then the solution

(ip,c,w,^) is unique.

Under more restrictive hypotheses on the data, we are able to prove additional regu-

larity properties of the solution(s). We will possibly assume

fi = \i

Up)

only depends on

<p,

, ,

new

'™,

||MlU-<ff-

{

Theorem 2.2 Let (4), (14)-(15), (20) and (22) hold for a couple of initial data

ipo

Co,i

and

</?o,2)

co,2 and assume that

(cb,i)

= (co,

)„

•

(23)