u ’ u0 ¤

L∞ ([0,T ];L2 („¦))

T

f ’ u0 + ∆u0 L1 (]0,T [;L2 („¦)) ,

+ (53)

and the solution is unique up to a constant.

Proof Let us search for solutions u = u0 + v where

‚v ‚u0

(t, x) ’ ∆v(t, x) = f (t, x) ’ (t, x) ∈ R — „¦,

(t, x) + ∆u0 (t, x), (54)

‚t ‚t

and

‚v ‚u0

(t, x) = g(t, x) ’ (t, x) ∈ R — ‚„¦.

(t, x) = 0, (55)

‚n ‚n

32 Periodic solutions for evolution equations

Consider the operator AN : D(AN ) ‚ L2 („¦) ’ L2 („¦) given as

AN v = ’∆v

with domain

‚v

D(AN ) = v ∈ H 2 („¦) : (x) = 0, ∀ x ∈ ‚„¦ .

‚n

The operator AN is linear monotone:

=’

(AN v, v) ∆v(x)v(x)dx

„¦

‚v

v(x) 2 dx

=’ (x)v(x)dσ +

‚n

‚„¦ „¦

v(x) 2 dx ≥ 0, ∀ v ∈ D(AN ).

= (56)

„¦

Since the equation »v ’ ∆v = f has unique solution in D(AN ) for every f ∈

L2 („¦), » > 0 it follows that AN is maximal (see [6]). Moreover, it is symmetric

v1 (x) · ∀ v1 , v2 ∈ D(AN ).

(AN v1 , v2 ) = v2 (x)dx = (v1 , AN v2 ),

„¦

Note that by the hypothesis the second member in (54) f ’ u0 + ∆u0 belongs to

C 1 (R; L2 („¦)). Therefore the Theorem 3.8 applies and hence the problem (54),

(55) has periodic solutions if and only if there is w ∈ D(AN ) such that

T

1 du0

’∆w = {f (t) ’ (t) + ∆u0 (t)}dt.

T dt

0

T du0 T

1

Since u0 is T -periodic we have (t)dt = 0 and thus w + u0 (t)dt is

0 dt T 0

solution for the elliptic problem

T T

1 1

’∆ w + u0 (t)dt = f (t)dt = F,

T T

0 0

with the boundary condition

T T

‚ 1 ‚w 1 ‚u0

w+ u0 (t)dt = + (t)dt

‚n T ‚n T ‚n

0 0

T

1

= g(t)dt = G.

T 0

As known from the general theory of partial di¬erential equations (see [6]) this

problem has solution if and only if ‚„¦ G(x)dσ + „¦ F (x)dx = 0 or

T T

g(t, x)dtdσ + f (t, x)dtdx = 0.

‚„¦ 0 „¦ 0

The estimate (53) follows from Theorem 3.8.

For the heat equation with Dirichlet boundary condition we have the follow-

ing existence result.

Mihai Bostan 33

Theorem 3.13 Assume that f ∈ C 1 (R; L2 („¦)) is T -periodic and g(t, x) =

u0 (t, x), (t, x) ∈ R — ‚„¦ where u0 ∈ C 1 (R; H 2 („¦)) © C 2 (R; L2 („¦)) is T -periodic.

Then the heat problem (50), (51) has an unique T -periodic solution u in

C(R; H 2 („¦)) © C 1 (R; L2 („¦)) and there is a constant C(„¦) such that

u ’ u0 ¤ C(„¦) f + ∆u0 L∞ ([0,T ];L2 („¦))

L∞ ([0,T ];L2 („¦))

√

T

f ’ u0 + ∆u0 L2 (]0,T [;L2 („¦))

+

2

T

+ f ’ u0 + ∆u0 L1 (]0,T [;L2 („¦)) , (57)

2

and

1

√ f ’ u0 + ∆u0 L2 (]0,T [;L2 („¦))

u ’ u0 ¤

L∞ ([0,T ];L2 („¦))

T

+ f ’ u0 + ∆u0 L1 (]0,T [;L2 („¦)) . (58)

Proof This time we consider the operator AD : D(AD ) ‚ L2 („¦) ’ L2 („¦)

given as

AD v = ’∆v

with domain

D(AD ) = v ∈ H 2 („¦) : v(x) = 0, ∀x ∈ ‚„¦ ,

As before AD is linear, monotone and symmetric and thus our problem reduces

to the existence for an elliptic equation:

T

1

’∆w = {f (t) + ∆u0 (t)}dt,

T 0

with homogenous Dirichlet boundary condition w = 0 on ‚„¦. Since the previous

problem has a unique solution verifying

1T

¤ C(„¦) {f (t) + ∆u0 (t)}dt

w L2 („¦) L2 („¦)

T0

¤ C(„¦) f + ∆u0 L∞ ([0,T ];L2 („¦)) , (59)

we prove the existence for (50), (51). Here we denote by C(„¦) the Poincar´™s

e

constant,

1/2 1/2

|w(x)|2 dx w(x) 2 dx 1

¤ C(„¦) ∀w ∈ H0 („¦).

,

„¦ „¦

Moreover in this case the operator AD is strictly monotone. Indeed, by using

the Poincar´™s inequality, for each v ∈ D(AD ), we have have

e

1/2 1/2

|v(x)|2 dx v(x) 2 dx = C(„¦)(AD v, v)1/2 .

¤ C(„¦)

„¦ „¦

Hence if (AD v, v) = 0 we deduce that v = 0. Therefore, by Proposition 3.2 we

deduce the uniqueness of the periodic solution for (50), (51). The estimates of

the solution follow immediately from (59) and Theorem 3.8.

34 Periodic solutions for evolution equations

3.4 Non-linear case

Throughout this section we will consider evolution equations associated to sub-

di¬erential operators. Let • : H ’]’∞, +∞] be a lower-semicontinuous proper

convex function on a real Hilbert space H. Denote by ‚• ‚ H — H the sub-

di¬erential of •,

‚•(x) = y ∈ H; •(x) ’ •(u) ¤ (y, x ’ u), ∀u ∈ H , (60)

and denote by D(•) the e¬ective domain of •:

D(•) = x ∈ H; •(x) < +∞ .

Under the previous assumptions on • we recall that A = ‚• is maximal mono-

tone in H — H and D(A) = D(•). Consider the equation

x (t) + ‚•x(t) f (t), 0 < t < T. (61)

We say that x is solution for (61) if x ∈ C([0, T ]; H), x is absolutely continuous

on every compact of ]0, T [ (and therefore a.e. di¬erentiable on ]0, T [) and sat-

is¬es x(t) ∈ D(‚•) a.e. on ]0, T [ and x (t) + ‚•x(t) f (t) a.e. on ]0, T [. We

have the following main result [1]

Theorem 3.14 Let f be given in L2 (]0, T [; H) and x0 ∈ D(‚•). Then the

Cauchy problem (61) with the initial condition x(0) = x0 has a unique solution

x ∈ C([0, T ]; H) which satis¬es:

√

x ∈ W 1,2 (]δ, T [; H) t · x ∈ L2 (]0, T [; H), • —¦ x ∈ L1 (0, T ).

∀ 0 < δ < T,

Moreover, if x0 ∈ D(•) then

• —¦ x ∈ L∞ (0, T ).

x ∈ L2 (]0, T [; H),

We are interested in ¬nding su¬cient conditions on A = ‚• and f such that

equation (61) has unique T -periodic solution, i.e. x(0) = x(T ). Obviously, if

such a solution exists, by periodicity we deduce that it is absolutely continuous

on [0, T ] and belongs to W 1,2 (]0, T [; H). It is well known that if • is strictly

convex then ‚• is strictly monotone and therefore the uniqueness holds

Proposition 3.15 Assume that • : H ’] ’ ∞, +∞] is a lower-semicontinuous

proper, strictly convex function. Then equation (61) has at most one periodic

solution.

Proof By using Proposition 3.2 it is su¬cient to prove that ‚• is strictly

monotone. Suppose that there are u1 , u2 ∈ D(‚•), u1 = u2 such that

(‚•(u1 ) ’ ‚•(u2 ), u1 ’ u2 ) = 0.

Mihai Bostan 35

We have

•(u2 ) ’ •(u1 ) ≥ (‚•(u1 ), u2 ’ u1 )

= ’(‚•(u2 ), u1 ’ u2 )

≥ •(u2 ) ’ •(u1 ),

and hence

•(u2 ) ’ •(u1 ) = (‚•(u1 ), u2 ’ u1 ).

We can also write for » ∈]0, 1[

•((1 ’ »)u1 + »u2 ) •(u1 + »(u2 ’ u1 ))

=

≥ •(u1 ) + (‚•(u1 ), »(u2 ’ u1 ))

•(u1 ) + »(‚•(u1 ), u2 ’ u1 )

=

•(u1 ) + »(•(u2 ) ’ •(u1 ))

=

(1 ’ »)•(u1 ) + »•(u2 ).

=

Since • is strictly convex we have also

•((1 ’ »)u1 + »u2 ) < (1 ’ »)•(u1 ) + »•(u2 ),

which is in contradiction with the previous inequality. Thus u1 = u2 and hence

‚• is strictly monotone. We state now the result concerning the existence of

periodic solutions.

Theorem 3.16 Suppose that • : H ’] ’ ∞, +∞] is a lower-semicontinuous

proper convex function and f ∈ L2 (]0, T [; H) such that

lim {•(x) ’ (x, f )} = +∞, (62)

x ’∞

and every level subset {x ∈ H; •(x) + x 2 ¤ M } is compact. Then equation

(61) has T -periodic solutions x ∈ C([0, T ]; H) © W 1,2 (]0, T [; H) which satisfy

• —¦ x ∈ L∞ (0, T ).

¤f x(t) ∈ D(•) ∀ t ∈ [0, T ],

x L2 (]0,T [;H) ,

L2 (]0,T [;H)

Before showing this result, notice that the condition (62) implies that the

lower-semicontinuous proper convex function ψ : H ’] ’ ∞, +∞] given by

ψ(x) = •(x) ’ (x, f ) has a minimum point x0 ∈ H and therefore f ∈

Range(‚•) since 0 = ‚ψ(x0 ) = ‚•(x0 ) ’ f .

Proof As previous for every ± > 0 we consider the unique periodic solution

x± for

±x± (t) + x± (t) + ‚•x± (t) = f (t), 0 < t < T. (63)

(In order to prove the existence and uniqueness of the periodic solution for (63)

consider the application S± : D(‚•) ’ D(‚•) de¬ned by S± (x0 ) = x(T ; 0, x0 ),

36 Periodic solutions for evolution equations

where x(·; 0, x0 ) denote the unique solution of (63) with the initial condition x0

and apply the Banach™s ¬xed point theorem. By the previous theorem it follows

that the periodic solution x± is absolutely continuous on [0, T ] and belongs to

C([0, T ]; H)©W 1,2 (]0, T [; H)). First of all we will show that (x± )±>0 is uniformly

bounded in L2 (]0, T [; H). Indeed, after multiplication by x± (t) we obtain

T T T

2