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Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping
J. Math. Anal. Appl. 270 (2002) 150–164 www.academicpress.com
Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping Leszek Gasi´nski ∗ and Maciej Smołka Institute of Computer Science, Jagiellonian University, ul. Nawojki 11, 30072 Cracow, Poland Received 31 May 2001 Submitted by H.R. Thieme
Abstract In this paper we prove the existence of solutions for a hyperbolic hemivariational inequality of the form u + Au + Bu + ∂j (u) f, where B is a linear elliptic operator and A is linear and nonnegative (not necessarily coercive). 2002 Elsevier Science (USA). All rights reserved. Keywords: Hemivariational inequalities; Clarke subdifferential; Viscosity damping
1. Introduction The theory of variational inequalities provides us with an appropriate mathematical model to describe many physical problems (cf. Duvaut and Lions [8]). It was started in 60-ties with the pioneer works of G. Fichera, J.L. Lions, and G. Stampacchia. All the inequality problems studied by the use of these methods were related to convex energy functionals and therefore were closely connected with the notion of monotonicity. In the 80-ties, Panagiotopoulos intro* Corresponding author.
E-mail addresses: [email protected] (L. Gasi´nski), [email protected] (M. Smołka). 0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 2 4 7 X ( 0 2 ) 0 0 0 5 7 - 4
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duced the notion of nonconvex superpotential by the use of the general gradient of Clarke [7]. Due to the lack of convexity new types of variational expressions were obtained. These are so called hemivariational inequalities and they are no longer connected with monotonicity. For a comprehensive treatment of the hemivariational inequality problems as well as for many applications, we refer to the monographs of Panagiotopoulos [23, 25], Motreanu and Panagiotopoulos [19], Naniewicz and Panagiotopoulos [20]. In this paper we study the following hyperbolic hemivariational inequality u + Au + Bu + χ = f, u(0) = ψ0 , u (0) = ψ1 in Ω, (1) χ(t, x) ∈ ∂j (u(t, x)) a.e. in (0, T ) × Ω, where A ∈ L(H, V ) is an operator (not necessarily coercive), B ∈ L(V , V ) is a coercive operator, j : R → R is a locally Lipschitz function, ψ0 , ψ1 : Ω → R and f : (0, T ) → V are given functions. The model for our problem is the following second-order nonlinear evolution equation called sine-Gordon equation: 2 ∂ u + α ∂u ∂t − ∆u + γ sin u = f, ∂t 2 ∂u u(0, x) = ψ0 (x), ∂t (0, x) = ψ1 (x), which is of great importance because of its physical applications (cf. Temam [31, Chapter IV.2, p. 188]). Our work allows to introduce nonmonotone multivalued constitutive laws into this model. Moreover, in our framework we can consider damping terms more general than simple multiplication by a positive number (see Section 4). We prove the existence of solutions of (1) using a method similar to the parabolic regularization method from the book of Lions and Magenes [15]; namely we approximate the solution of our problem by a sequence of solutions of some modified problems containing a coercive damping term. For the modified problem we apply the result of Gasi´nski [12].
2. Preliminaries Let X be a Banach space with a norm · X and X its topological dual. By ·, · X ×X we shall denote the duality brackets for the pair (X, X ). If X is in addition a Hilbert space, then by (·, ·)X we shall denote the scalar product in X. In the formulation of our hemivariational inequality the crucial role will be played by the notion of Clarke subdifferential of a locally Lipschitz function. A function j : X → R is said to be locally Lipschitz if for every x ∈ X there exists a neighbourhood U of x and a constant kx > 0 depending on U such that |j (z) − j (y)| kx z − y X
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for all z, y ∈ U . In analogy with the directional derivative of a convex function, we define the generalized directional derivative of a locally Lipschitz function j at x ∈ X in the direction h ∈ X by def
j 0 (x; h) = lim sup x → 0 t 0
j (x + x + th) − j (x + x ) . t
It is easy to check that the function X h → j 0 (x; h) ∈ R is sublinear and continuous and that |j 0 (x; h)| kx h X . Hence by the Hahn–Banach theorem j 0 (x; ·) is the support function of nonempty, convex and w∗ -compact set def ∂j (x) = x ∗ ∈ X : x ∗ , h X ×X j 0 (x; h) for all h ∈ X , known as the Clarke subdifferential of j at x. Note that for every x ∗ ∈ ∂j (x) we have x ∗ X kx . We have also that if j, g : X → R are locally Lipschitz functions, then ∂(j + g)(x) ⊂ ∂j (x) + ∂g(x) and ∂(tj )(x) = t∂j (x) for all t ∈ R. Moreover, if j : X → R is also convex, then the subdifferential of j in the sense of convex analysis coincides with the generalized subdifferential introduced above. Finally, if j is strictly differentiable at x (in particular if j is continuously Gateaux differentiable at x), then ∂j (x) = {j (x)}. Let us introduce the following spaces, needed in the sequel: H = L2 (Ω),
V = H 1 (Ω) = v: v ∈ L2 (Ω), D α v ∈ L2 (Ω) for 0 |α| 1 , V = V (Ω) = H 1 (Ω) . V It is well-known that V ⊂ H ⊂ V form an evolution triple. By cH we will denote “the continuity constant” for the embedding V ⊆ H (so also for the embedding H ⊆ V ). In our evolution case, we will also make use of the following spaces:
H = L2 (0, T ; H ) = L2 ((0, T ) × Ω), V = L2 (0, T ; V ), W = {v: v ∈ V, v ∈ V }.
3. Hyperbolic hemivariational inequality Let T > 0 be any positive real number and let N 1. By Ω ⊂ RN we will denote any open and bounded set. We consider the following hyperbolic hemivariational inequality:
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Find u ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H ) with u ∈ V and χ ∈ H, such that u (t) + Au (t) + Bu(t) + χ(t) = f (t) in V , for a.a. t ∈ (0, T ), (HVI) u(0) = ψ0 , u (0) = ψ1 in Ω, χ(t, x) ∈ ∂j (u(t, x)) for a.a. (t, x) ∈ (0, T ) × Ω, where A ∈ L(H, V ), B ∈ L(V , V ), j : R → R, ψ0 , ψ1 : Ω → R and f : (0, T ) → V are given. For our existence result, we will need the following assumptions: H(j ) j : R → R is a locally Lipschitz function, such that
ξ (i) j (ξ ) = 0 β(s) ds, where β ∈ L∞ loc (R); (ii) for every ξ ∈ R there exist limits limζ →ξ ± β(ζ ); (iii) for every ξ ∈ R, we have |β(ξ )| c0 (1 + |ξ |r ), with some c0 > 0 and 0 r < 1. H(A) A : H → V is a linear operator, such that (i) A is continuous, i.e., there exists αA > 0, such that for all v ∈ H , we have Av V αA v H ; (ii) A|V is nonnegative, i.e., for all v ∈ V , we have Av, v V ×V 0. H(B) B : V → V is a linear operator, such that (i) B is continuous, i.e., there exists αB > 0, such that for all v ∈ V , we have Bv V αB v V ; (ii) B is coercive, i.e., there exists βB > 0, such that for all v ∈ V , we have Bv, v V ×V βB v 2V ; (iii) B is symmetric, i.e., for all v, w ∈ V , we have Bv, w V ×V = Bw, v V ×V . H(f, ψ) f ∈ H, ψ0 ∈ V , ψ1 ∈ H . Now we can state our main result. Theorem 3.1. If hypotheses H(j ), H(A), H(B) and H(f, ψ) hold, then (HVI) admits a solution. First, for any ε > 0 we consider the following regularized hyperbolic hemivariational inequality: Find uε ∈ C([0, T ]; V ) with uε ∈ W and χε ∈ H, such that uε (t) + Auε (t) + εBuε (t) + Buε (t) + χε (t) = f (t), uε (0) = ψ0 , uε (0) = ψ1 , (HVIε ) χε (t, x) ∈ ∂j (uε (t, x)). Lemma 3.2. If hypotheses H(j ), H(A), H(B) and H(f, ψ) hold, then for any ε > 0 there exists at least one solution uε of (HVIε ).
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Proof. This is a consequence of the result of Gasi´nski (see [11] or [12]). To this end note that operator A : (0, T ) × V → V defined by A(t, v) = A|V v + εBv is pseudomonotone (with respect to v-variable), bounded (in a sense that for a.a. t ∈ (0, T ) and all v ∈ V , we have A(t, v) V a 1 (t) + c1 v V , with some a 1 ∈ L2 (0, T ), and c1 > 0) and coercive (namely for a.a. t ∈ (0, T ) and all v ∈ V , we have A(t, v), v V ×V εβB v 2V ). Thus, using Theorem 3.1 and Remark 3.3 of [12], we obtain our lemma. ✷ In the next lemma we show an estimate on selections of ∂j (u). Lemma 3.3. If hypotheses H(j ) hold and u ∈ C([0, T ]; V ) with u ∈ W and η ∈ H are such that η(t, x) ∈ ∂j (u(t, x)) for almost all (t, x) ∈ (0, T ) × Ω, then
η H c(1 + u H ),
(2)
with some constant c = c(Ω, T , c0 ) > 0 not depending on u, η and r. Proof. Using hypothesis H(j )(iii), we obtain T
η H =
T
η(t) 2H
2
dt = 0 Ω
0
T
|η(t, x)|2 dx dt
4c02
2
1 + |u(t, x)| dx dt
T 8c02
0 Ω
|Ω| + u(t) 2H dt
0
8c02 T |Ω| + u 2H , √ √ def so estimate (2) holds with c = c0 2 2 max{ T |Ω|, 1}. ✷ The following lemma gives some estimates on the solutions of (HVIε ). Lemma 3.4. If hypotheses H(j ), H(A), H(B), H(f, ψ) hold and uε is a solution of (HVIε ), then for any ε ∈ (0, 1) we have
√ max uε (t) V + uε (t) H + ε uε V + uε V t ∈[0,T ]
(3) c 1 + ψ0 V + ψ1 H + f H , where c = c(Ω, T , c0 , αA , αB , βB ) > 0 is a constant not depending on ε, ψ0 , ψ1 , A, B, f , j and r. Proof. As uε , uε ∈ V, so in particular uε is an absolutely continuous function and t uε (t) = 0
uε (s) ds + ψ0
for all t ∈ (0, T )
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(see Barbu [3, p. 19, Theorem 2.2]). Thus for any s ∈ (0, T ) we have s
uε (s) 2H
2T
uε (τ ) 2H dτ + 2 ψ0 2H .
(4)
0
From the equality in (HVIε ), taking the duality brackets on uε (s) and integrating over interval (0, t), for any t ∈ (0, T ) we obtain t
uε (s), uε (s) V ×V
t ds +
0
Auε (s), uε (s) V ×V ds
0
t +ε 0 t
+
Buε (s), uε (s) V ×V ds +
t
Buε (s), uε (s) V ×V ds
0
χε (s), uε (s) V ×V ds =
0
t
f (s), uε (s) V ×V ds.
(5)
0
We will estimate separately each term in (5). First, we have t
1 1 uε (s), uε (s) V ×V ds = uε (t) 2H − uε (0) 2H 2 2
0
1 1 = uε (t) 2H − ψ1 2H 2 2 (compare Zeidler [32, pp. 422–423, Proposition 23.23(iv)]). From hypothesis H(A)(ii), we have t
Auε (s), uε (s) V ×V ds 0.
0
Next, hypothesis H(B)(ii) implies t ε
Buε (s), uε (s) V ×V
t ds εβB
0
uε (s) 2V ds.
0
Using the differentiation formula (see Zeidler [32, p. 881, Proof of Theorem 32.E(III)]) and hypotheses H(B)(i) and (ii), we obtain t 0
Buε (s), uε (s) V ×V
1 ds = 2
t 0
d Buε (s), uε (s) V ×V ds ds
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1 1 Buε (t), uε (t) V ×V − Buε (0), uε (0) V ×V 2 2 βB αB
uε (t) 2V −
ψ0 2V . 2 2 =
Next, using hypothesis H(j )(iii), the Young inequality, estimate (4) and the continuity of the embedding V ⊂ H , for all t ∈ (0, T ) we have t
t
χε (s), uε (s) V ×V
ds =
0
χ(s), uε (s)
H
ds
0
t
χ(s) H uε (s) H ds
− 0
1 − 2
t
uε (s) 2H
1 ds − 2
t
uε (s) 2H
t ds
− c02
0
−
1 2
2 c02 1 + |uε (s, x)| dx ds
0 Ω
0
1 − 2
t
|Ω| + uε (s) 2H ds
0
t
uε (s) 2H ds − c02 T |Ω|
0
− c02
t s 2T uε (τ ) 2H dτ + 2 ψ0 2H ds 0
1 − 2
t
0
uε (s) 2H
t s ds
− 2T c02
0
− T c02
V |Ω| + 2cH
ψ0 2V .
uε (τ ) 2H dτ ds
0 0
Finally, from the Young inequality, for all t ∈ (0, T ) we have t
f (s), uε (s) V ×V
0
t ds
f (s), uε (s) H ds
0
t 0
f (s) H uε (s) H ds
L. Gasi´nski, M. Smołka / J. Math. Anal. Appl. 270 (2002) 150–164
1 2
t
uε (s) 2H ds +
0
1 2
t
1 2
157
t
f (s) 2H ds 0
1
uε (s) 2H ds + f 2H . 2
0
Putting all the above estimates into (5), for all t ∈ (0, T ) we obtain 1 βB
uε (t) 2H +
uε (t) 2V + εβB 2 2
t
uε (s) 2V ds
0
1 1 c1 + c2 ψ0 2V + ψ1 2H + f 2H 2 2 t t s + uε (s) 2H ds + 2T c02
uε (τ ) 2H dτ ds, 0
0 0
def
def
V c 2 + α /2. Thus, for all t ∈ (0, T ) we have with c1 = T c02 |Ω| and c2 = 2T cH B 0
t 1 βB 2 2
u (t) H +
uε (t) V + εβB uε (s) 2V 2 ε 2 0
c3 1 + ψ0 2V + ψ1 2H + f 2H t t s 2 + c4 uε (s) H ds + c4
uε (τ ) 2H dτ ds, 0 def
(6)
0 0 def
where c3 = max{1/2, c1, c2 } and c4 = max{1, 2T c02 }. Now, using the generalization of the Gronwall–Bellman inequality (see Pachpatte [21, p. 758, Theorem 1]), for all t ∈ (0, T ) we obtain
(7)
uε (t) 2H c5 1 + ψ0 2V + ψ1 2H + f 2H ,
def where c5 = 2c3 1 + 2T c4 eT (2c4 +1) ; so
uε (t) H c6 1 + ψ0 V + ψ1 H + f H , (8) def √
where c6 =
c5 , and also
uε 2H c7 1 + ψ0 2V + ψ1 2H + f 2H ,
(9)
def
where c7 = T c5 . Applying (7) to (6), we obtain
uε (t) 2V c8 1 + ψ0 2V + ψ1 2H + f 2H
(10)
158
and
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ε uε 2V c8 1 + ψ0 2V + ψ1 2H + f 2H ,
(11)
def
where c8 = (2/βB )(c3 + T c4 c5 (T /2 + 1)). Hence
uε (t) V c9 1 + ψ0 V + ψ1 H + f H and
√ ε uε V c9 1 + ψ0 V + ψ1 H + f H ,
(12)
(13)
def √ where c9 = c8 . Using Lemma 3.3, continuity of the embedding V ⊂ H and estimate (10), for any ε > 0 we have
T
V 2 2 2 2
uε (t) V dt
χε H 2c 1 + uε H 2c 1 + cH 2
2
0
c10 1 + ψ0 2V + ψ1 2H + f 2H , def
V 2 where c10 = 2c2 max{1, T (cH ) c8 }. Finally, using the equation in (HVIε ), hypothesis H(A)(i), continuity of the embedding H ⊂ V , inequalities (9)–(11) and the last inequality, for all ε ∈ (0, 1) we can estimate uε V as
uε 2V
T =
uε (t) 2V dt
0
T 5
A(t, u (t))2 dt ε V
0
T + 5ε
2
B(u (t))2 dt + 5 ε
V
0
T
χε (t) 2V dt + 5
0
T 5αA
B(uε (t))2 dt V
0
T +5
T
f (t) 2V dt 0
uε (t) 2H dt
0
T + 5ε2 αB2 0
uε (t) 2V
T dt
+ 5αB2
uε (t) 2V dt 0
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V 2 + 5 cH
T
χε (t) 2H
V 2 dt + 5 cH
159
T
f (t) 2H dt
0
0
2
uε 2H + 5εαB2 uε 2V + 5αB2 5αA
T
uε (t) 2V dt 0
V 2 V 2 + 5 cH
χε 2H + 5 cH
f 2H
c11 1 + ψ0 2V + ψ1 2H + f 2H ,
def 2 V 2 c7 + αB2 c8 (1 + T ) + (cH ) (c10 + 1) ; so where c11 = 5 αA
uε V c12 1 + ψ0 V + ψ1 H + f H , √ with c12 = c11 . def
Finally, from (8) and (12)–(14), we obtain (3), with c = c6 + 2c9 + c12 .
(14) ✷
Now we are in the position to prove our main result. Proof of Theorem 3.1. From Lemma 3.4, it follows that for any ε ∈ (0, 1), we have
max uε (t) V + uε (t) H + uε V c13 , t ∈[0,T ]
with some constant c13 > 0 not depending on ε ∈ (0, 1). Thus, we can choose a sequence {εn }n1 ⊂ (0, 1), such that εn 0 and uε n → u
weakly∗ in L∞ (0, T ; V ),
uεn uεn
weakly∗
→u
∞
in L (0, T ; H ),
→ u weakly in V .
(15) (16) (17)
But in fact u = u and u = u . It is easy to see that (HVIε ) is equivalent to the following problem: Find uε ∈ C([0, T ]; V ) with uε ∈ W and χε ∈ H, such that ε + εBu ε + Bu ε + χε = f in V , uε + Au (HVIε ) u (0) = ψ0 , uε (0) = ψ1 in Ω, ε χε (t, x) ∈ ∂j (uε (t, x)) for a.a. (t, x) ∈ (0, T ) × Ω, : H → V and B : V → V are the Nemytskii operators corresponding where A to the operators A and B, respectively. Our aim now is to “pass to the limit” in (HVIε ). and B are linear and bounded operators, from (15) and (16) we have As A
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weakly in V , ε → Au Au n εn → Bu weakly in V . Bu
(18)
(19) √ Next, from Lemma 3.4 we see that the sequence { εn uεn }n1 remains bounded in V; hence εn uεn → 0 in V. ε V αB εn uε V ; thus in But using hypothesis H(B)(i), we have that εn Bu n n fact ε → 0 εn Bu n
in V .
(20)
From (15), (16) and the compactness of the embedding W ⊂ H we obtain uε n → u
in H,
and, in particular, possibly passing to a subsequence, uεn (t, x) → u(t, x) for a.a. (t, x) ∈ (0, T ) × Ω.
(21)
Using convergence (15), Lemma 3.3 and extracting a new subsequence if necessary, we obtain χε n → χ
weakly in H,
(22)
with some χ ∈ H; hence also χε n → χ
weakly in L1 ((0, T ) × Ω).
(23)
Now, because of (17)–(20) and (22), we can “pass to the limit” in the equation in (HVIε ) and obtain + Bu +χ =f u + Au
in V .
(24)
Since for all n 1 we have that χεn (t, x) ∈ ∂j (uεn (t, x)) for almost all (t, x) ∈ (0, T ) × Ω, thus, using convergences (21) and (23) and applying Theorem 7.2.2 on p. 273 of Aubin and Frankowska [2] (recall that ∂j is a lower semicontinuous multifunction with convex and closed values), we get χ(t, x) ∈ ∂j (u(t, x))
for a.a. (t, x) ∈ (0, T ) × Ω.
(25) H 1 (0, T ; H ),
Finally, from (15) and (16) we have that uεn → u weakly in hence also weakly in C([0, T ]; H ). Analogously, from (16) and (17) we have that uεn → u weakly in H 1 (0, T ; V ), hence also weakly in C([0, T ]; V ). In particular, we have that uεn (0) → u(0)
weakly in H,
uεn (0) → u (0)
weakly in V .
(26)
To end our proof it remains to show that u ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H ).
(27)
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For this purpose let us recall the definition of the following function space introduced in the book of Lions and Magenes [15]: def Cs ([0, T ]; X) = u ∈ L∞ (0, T ; X): (28) u∗ , u(·) X ×X is continuous ∀u∗ ∈ X . Of course, one has C([0, T ]; X) ⊂ Cs ([0, T ]; X). Moreover, if X and Y are two Banach spaces, X being reflexive, with the dense embedding X ⊂ Y , from [15, p. 297, Lemma 8.1] we know that Cs ([0, T ]; Y ) ∩ L∞ (0, T ; X) = Cs ([0, T ]; X).
(29)
In our case, due to (15)–(17) we have that u ∈ C([0, T ]; H ) ∩ L∞ (0, T ; V ), u ∈ C([0, T ]; V ) ∩ L∞ (0, T ; H ); hence, from (29) we obtain u ∈ Cs ([0, T ]; V ),
(30)
u ∈ Cs ([0, T ]; H ).
(31)
Next, using the same argument as in the proof of Lemma 3.4 (see (5) and the sequel), for any t ∈ [0, T ] we can prove the following energy equality
u (t) 2H + Bu(t), u(t) V ×V = ψ1 2H
+ Bψ0 , ψ0 V ×V
t + 2 f (s) − Au (s) − χ(s), u (s) V ×V ds. 0
This shows that the function
E : [0, T ] t → u (t) 2H + Bu(t), u(t) V ×V ∈ R
is continuous. Take tn , t ∈ [0, T ] such that tn → t and put 2 δn = u (tn ) − u (t)H + Bu(tn ) − Bu(t), u(tn ) − u(t) V ×V
= E(tn ) + E(t) − 2 Bu(t), u(tn ) V ×V − 2 u (tn ), u (t) H . Thanks to (30), (31) and the continuity of E we have that δn → 2E(t) − 2 Bu(t), u(t) V ×V − 2 u (t) 2H = 0, which, together with the inequality 2 2 δn u (tn ) − u (t)H + βB u(tn ) − u(t)V ,
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gives us (27). Now, from (24)–(27) we obtain that u is a solution of the following problem: Find u ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H ) with u ∈ V and χ ∈ H, such that + Bu + χ = f in V , u + Au (HVI ) u(0) = ψ0 , u (0) = ψ1 in Ω, χ(t, x) ∈ ∂j (u(t, x)) for a.a. (t, x) ∈ (0, T ) × Ω, and in particular u is a solution of (HVI). ✷ .
4. Applications and examples As mentioned in the introduction the classical model for our considerations was the sine-Gordon equation u + αu − ∆u + γ sin u = f in Ω × R, (SGE) u(0) = ψ0 , u (0) = ψ1 in Ω. Of course, as | sin u| 1 the assumptions of Theorem 3.1 are satisfied if α 0. This is a “single-valued” case where j (u) = − cos u and ∂j (u) = {sin u}. In the sequel we present some examples of “dampings”, which are admissible in our framework. First let us consider a slight generalization of the one used in the sine-Gordon equation; namely Av = av,
with a ∈ L∞ (Ω), a 0 a.e.
In this case (HVI) has the form ∂u ∂ 2u + a(x) + Bu + ∂j (u) f. 2 ∂t ∂t The above “damping” operators map H into itself. The next two have values in V . This time take a ∈ W 1,∞ (Ω; RN ) and consider A1 v =
N
∂ ai (x)v , ∂xi i=1
A2 v =
N
ai (x)
i=1
It means this time (HVI) has the form N ∂u ∂ 2u ∂ + (x) a + Bu + ∂j (u) f i ∂t 2 ∂xi ∂t i=1
or, respectively, ∂ 2u ∂ 2u + a (x) + Bu + ∂j (u) f. i ∂t 2 ∂t∂xi N
i=1
∂v . ∂xi
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The nonnegativity condition for A1 yields div a 0 in Ω, (a, n) 0 on ∂Ω, where n is the outer normal to the boundary of Ω. In the case V = H01 (Ω) we could drop the second inequality. Similarly, A2 is nonnegative provided that div a = 0 in Ω, (a, n) = 0 on ∂Ω. This time, if we took V = H01 (Ω) we would need only the inequality div a 0 a.e. in Ω.
Acknowledgments This work has been partially supported by the State Committee for Scientific Research of the Republic of Poland (KBN) under research grants No. 7 T07A 047 18 and No. 2 P03A 004 19.
References [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [4] J. Berkovits, V. Mustonen, Monotonicity methods for nonlinear evolution equations, Nonlinear Anal. Theory Methods Appl. 12 (1996) 1397–1405. [5] W. Bian, Existence results for second order nonlinear evolution inclusions, Indian J. Pure Appl. Math. 11 (1998) 1177–1193. [6] K.C. Chang, Variational methods for nondifferentiable functionals and applications to partial differential equations, J. Math. Anal. Appl. 80 (1981) 102–129. [7] F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [8] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. [9] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North Holland, Amsterdam, 1976. [10] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Verlag, Berlin, 1974. [11] L. Gasi´nski, Hyperbolic hemivariational inequalities and their application in the theory of optimal shape design, Ph.D. Thesis, Jagiellonian University, Kraków (2000). [12] L. Gasi´nski, Existence of solutions for hyperbolic hemivariational inequalities, J. Math. Anal. Appl., to appear. [13] S. Hu, N.S. Papageorgiou, Handbook of Multiple Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997. [14] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [15] J.L. Lions, E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris, 1968.
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L. Gasi´nski, M. Smołka / J. Math. Anal. Appl. 270 (2002) 150–164
[16] M. Miettinen, A parabolic hemivariational inequality, Nonlinear Anal. 26 (1996) 725–734. [17] M. Miettinen, M.M. Mäkelä, J. Haslinger, On numerical solutions of hemivariational inequalities by nonsmooth optimization methods, J. Global Optim. 6 (1995) 401–425. [18] S. Migórski, Existence, variational and optimal control problems for nonlinear second order evolution inclusions, Dynamic Systems Appl. 4 (1995) 513–528. [19] D. Motreanu, P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer, Dordrecht, 1998. [20] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Dekker, New York, 1995. [21] B.G. Pachpatte, A note on Gronwall–Bellman inequality, J. Math. Anal. Appl. 44 (1973) 758– 762. [22] P.D. Panagiotopoulos, Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech. 65 (1985) 29–36. [23] P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Basel, 1985. [24] P.D. Panagiotopoulos, Coercive and semicoercive hemivariational inequalities, Nonlinear Anal. 16 (1991) 209–231. [25] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, New York, 1993. [26] P.D. Panagiotopoulos, Hemivariational inequalities and fan-variational inequalities. New applications and results, Atti Sem. Mat. Fis. Univ. Modena 43 (1995) 159–191. [27] P.D. Panagiotopoulos, Modelling of nonconvex nonsmooth energy problems. Dynamic hemivariational inequalities with impact effects, J. Comput. Appl. Math. 63 (1995) 123–138. [28] N.S. Papageorgiou, Existence of solutions for second order evolution inclusions, Bull. Math. Soc. Sci. Math. Roumanie 37 (1993) 93–107. [29] N.S. Papageorgiou, On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities, J. Math. Anal. Appl. 205 (1997) 434–453. [30] N.S. Papageorgiou, F. Papalini, F. Renzacci, Existence of solutions and periodic solutions for nonlinear evolution inclusions, Rend. Circolo Mat. Palermo 48 (1999) 341–364. [31] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [32] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II, Springer-Verlag, New York, 1990.
Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping Leszek Gasi´nski ∗ and Maciej Smołka Institute of Computer Science, Jagiellonian University, ul. Nawojki 11, 30072 Cracow, Poland Received 31 May 2001 Submitted by H.R. Thieme
Abstract In this paper we prove the existence of solutions for a hyperbolic hemivariational inequality of the form u + Au + Bu + ∂j (u) f, where B is a linear elliptic operator and A is linear and nonnegative (not necessarily coercive). 2002 Elsevier Science (USA). All rights reserved. Keywords: Hemivariational inequalities; Clarke subdifferential; Viscosity damping
1. Introduction The theory of variational inequalities provides us with an appropriate mathematical model to describe many physical problems (cf. Duvaut and Lions [8]). It was started in 60-ties with the pioneer works of G. Fichera, J.L. Lions, and G. Stampacchia. All the inequality problems studied by the use of these methods were related to convex energy functionals and therefore were closely connected with the notion of monotonicity. In the 80-ties, Panagiotopoulos intro* Corresponding author.
E-mail addresses: [email protected] (L. Gasi´nski), [email protected] (M. Smołka). 0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 2 4 7 X ( 0 2 ) 0 0 0 5 7 - 4
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duced the notion of nonconvex superpotential by the use of the general gradient of Clarke [7]. Due to the lack of convexity new types of variational expressions were obtained. These are so called hemivariational inequalities and they are no longer connected with monotonicity. For a comprehensive treatment of the hemivariational inequality problems as well as for many applications, we refer to the monographs of Panagiotopoulos [23, 25], Motreanu and Panagiotopoulos [19], Naniewicz and Panagiotopoulos [20]. In this paper we study the following hyperbolic hemivariational inequality u + Au + Bu + χ = f, u(0) = ψ0 , u (0) = ψ1 in Ω, (1) χ(t, x) ∈ ∂j (u(t, x)) a.e. in (0, T ) × Ω, where A ∈ L(H, V ) is an operator (not necessarily coercive), B ∈ L(V , V ) is a coercive operator, j : R → R is a locally Lipschitz function, ψ0 , ψ1 : Ω → R and f : (0, T ) → V are given functions. The model for our problem is the following second-order nonlinear evolution equation called sine-Gordon equation: 2 ∂ u + α ∂u ∂t − ∆u + γ sin u = f, ∂t 2 ∂u u(0, x) = ψ0 (x), ∂t (0, x) = ψ1 (x), which is of great importance because of its physical applications (cf. Temam [31, Chapter IV.2, p. 188]). Our work allows to introduce nonmonotone multivalued constitutive laws into this model. Moreover, in our framework we can consider damping terms more general than simple multiplication by a positive number (see Section 4). We prove the existence of solutions of (1) using a method similar to the parabolic regularization method from the book of Lions and Magenes [15]; namely we approximate the solution of our problem by a sequence of solutions of some modified problems containing a coercive damping term. For the modified problem we apply the result of Gasi´nski [12].
2. Preliminaries Let X be a Banach space with a norm · X and X its topological dual. By ·, · X ×X we shall denote the duality brackets for the pair (X, X ). If X is in addition a Hilbert space, then by (·, ·)X we shall denote the scalar product in X. In the formulation of our hemivariational inequality the crucial role will be played by the notion of Clarke subdifferential of a locally Lipschitz function. A function j : X → R is said to be locally Lipschitz if for every x ∈ X there exists a neighbourhood U of x and a constant kx > 0 depending on U such that |j (z) − j (y)| kx z − y X
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for all z, y ∈ U . In analogy with the directional derivative of a convex function, we define the generalized directional derivative of a locally Lipschitz function j at x ∈ X in the direction h ∈ X by def
j 0 (x; h) = lim sup x → 0 t 0
j (x + x + th) − j (x + x ) . t
It is easy to check that the function X h → j 0 (x; h) ∈ R is sublinear and continuous and that |j 0 (x; h)| kx h X . Hence by the Hahn–Banach theorem j 0 (x; ·) is the support function of nonempty, convex and w∗ -compact set def ∂j (x) = x ∗ ∈ X : x ∗ , h X ×X j 0 (x; h) for all h ∈ X , known as the Clarke subdifferential of j at x. Note that for every x ∗ ∈ ∂j (x) we have x ∗ X kx . We have also that if j, g : X → R are locally Lipschitz functions, then ∂(j + g)(x) ⊂ ∂j (x) + ∂g(x) and ∂(tj )(x) = t∂j (x) for all t ∈ R. Moreover, if j : X → R is also convex, then the subdifferential of j in the sense of convex analysis coincides with the generalized subdifferential introduced above. Finally, if j is strictly differentiable at x (in particular if j is continuously Gateaux differentiable at x), then ∂j (x) = {j (x)}. Let us introduce the following spaces, needed in the sequel: H = L2 (Ω),
V = H 1 (Ω) = v: v ∈ L2 (Ω), D α v ∈ L2 (Ω) for 0 |α| 1 , V = V (Ω) = H 1 (Ω) . V It is well-known that V ⊂ H ⊂ V form an evolution triple. By cH we will denote “the continuity constant” for the embedding V ⊆ H (so also for the embedding H ⊆ V ). In our evolution case, we will also make use of the following spaces:
H = L2 (0, T ; H ) = L2 ((0, T ) × Ω), V = L2 (0, T ; V ), W = {v: v ∈ V, v ∈ V }.
3. Hyperbolic hemivariational inequality Let T > 0 be any positive real number and let N 1. By Ω ⊂ RN we will denote any open and bounded set. We consider the following hyperbolic hemivariational inequality:
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Find u ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H ) with u ∈ V and χ ∈ H, such that u (t) + Au (t) + Bu(t) + χ(t) = f (t) in V , for a.a. t ∈ (0, T ), (HVI) u(0) = ψ0 , u (0) = ψ1 in Ω, χ(t, x) ∈ ∂j (u(t, x)) for a.a. (t, x) ∈ (0, T ) × Ω, where A ∈ L(H, V ), B ∈ L(V , V ), j : R → R, ψ0 , ψ1 : Ω → R and f : (0, T ) → V are given. For our existence result, we will need the following assumptions: H(j ) j : R → R is a locally Lipschitz function, such that
ξ (i) j (ξ ) = 0 β(s) ds, where β ∈ L∞ loc (R); (ii) for every ξ ∈ R there exist limits limζ →ξ ± β(ζ ); (iii) for every ξ ∈ R, we have |β(ξ )| c0 (1 + |ξ |r ), with some c0 > 0 and 0 r < 1. H(A) A : H → V is a linear operator, such that (i) A is continuous, i.e., there exists αA > 0, such that for all v ∈ H , we have Av V αA v H ; (ii) A|V is nonnegative, i.e., for all v ∈ V , we have Av, v V ×V 0. H(B) B : V → V is a linear operator, such that (i) B is continuous, i.e., there exists αB > 0, such that for all v ∈ V , we have Bv V αB v V ; (ii) B is coercive, i.e., there exists βB > 0, such that for all v ∈ V , we have Bv, v V ×V βB v 2V ; (iii) B is symmetric, i.e., for all v, w ∈ V , we have Bv, w V ×V = Bw, v V ×V . H(f, ψ) f ∈ H, ψ0 ∈ V , ψ1 ∈ H . Now we can state our main result. Theorem 3.1. If hypotheses H(j ), H(A), H(B) and H(f, ψ) hold, then (HVI) admits a solution. First, for any ε > 0 we consider the following regularized hyperbolic hemivariational inequality: Find uε ∈ C([0, T ]; V ) with uε ∈ W and χε ∈ H, such that uε (t) + Auε (t) + εBuε (t) + Buε (t) + χε (t) = f (t), uε (0) = ψ0 , uε (0) = ψ1 , (HVIε ) χε (t, x) ∈ ∂j (uε (t, x)). Lemma 3.2. If hypotheses H(j ), H(A), H(B) and H(f, ψ) hold, then for any ε > 0 there exists at least one solution uε of (HVIε ).
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Proof. This is a consequence of the result of Gasi´nski (see [11] or [12]). To this end note that operator A : (0, T ) × V → V defined by A(t, v) = A|V v + εBv is pseudomonotone (with respect to v-variable), bounded (in a sense that for a.a. t ∈ (0, T ) and all v ∈ V , we have A(t, v) V a 1 (t) + c1 v V , with some a 1 ∈ L2 (0, T ), and c1 > 0) and coercive (namely for a.a. t ∈ (0, T ) and all v ∈ V , we have A(t, v), v V ×V εβB v 2V ). Thus, using Theorem 3.1 and Remark 3.3 of [12], we obtain our lemma. ✷ In the next lemma we show an estimate on selections of ∂j (u). Lemma 3.3. If hypotheses H(j ) hold and u ∈ C([0, T ]; V ) with u ∈ W and η ∈ H are such that η(t, x) ∈ ∂j (u(t, x)) for almost all (t, x) ∈ (0, T ) × Ω, then
η H c(1 + u H ),
(2)
with some constant c = c(Ω, T , c0 ) > 0 not depending on u, η and r. Proof. Using hypothesis H(j )(iii), we obtain T
η H =
T
η(t) 2H
2
dt = 0 Ω
0
T
|η(t, x)|2 dx dt
4c02
2
1 + |u(t, x)| dx dt
T 8c02
0 Ω
|Ω| + u(t) 2H dt
0
8c02 T |Ω| + u 2H , √ √ def so estimate (2) holds with c = c0 2 2 max{ T |Ω|, 1}. ✷ The following lemma gives some estimates on the solutions of (HVIε ). Lemma 3.4. If hypotheses H(j ), H(A), H(B), H(f, ψ) hold and uε is a solution of (HVIε ), then for any ε ∈ (0, 1) we have
√ max uε (t) V + uε (t) H + ε uε V + uε V t ∈[0,T ]
(3) c 1 + ψ0 V + ψ1 H + f H , where c = c(Ω, T , c0 , αA , αB , βB ) > 0 is a constant not depending on ε, ψ0 , ψ1 , A, B, f , j and r. Proof. As uε , uε ∈ V, so in particular uε is an absolutely continuous function and t uε (t) = 0
uε (s) ds + ψ0
for all t ∈ (0, T )
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155
(see Barbu [3, p. 19, Theorem 2.2]). Thus for any s ∈ (0, T ) we have s
uε (s) 2H
2T
uε (τ ) 2H dτ + 2 ψ0 2H .
(4)
0
From the equality in (HVIε ), taking the duality brackets on uε (s) and integrating over interval (0, t), for any t ∈ (0, T ) we obtain t
uε (s), uε (s) V ×V
t ds +
0
Auε (s), uε (s) V ×V ds
0
t +ε 0 t
+
Buε (s), uε (s) V ×V ds +
t
Buε (s), uε (s) V ×V ds
0
χε (s), uε (s) V ×V ds =
0
t
f (s), uε (s) V ×V ds.
(5)
0
We will estimate separately each term in (5). First, we have t
1 1 uε (s), uε (s) V ×V ds = uε (t) 2H − uε (0) 2H 2 2
0
1 1 = uε (t) 2H − ψ1 2H 2 2 (compare Zeidler [32, pp. 422–423, Proposition 23.23(iv)]). From hypothesis H(A)(ii), we have t
Auε (s), uε (s) V ×V ds 0.
0
Next, hypothesis H(B)(ii) implies t ε
Buε (s), uε (s) V ×V
t ds εβB
0
uε (s) 2V ds.
0
Using the differentiation formula (see Zeidler [32, p. 881, Proof of Theorem 32.E(III)]) and hypotheses H(B)(i) and (ii), we obtain t 0
Buε (s), uε (s) V ×V
1 ds = 2
t 0
d Buε (s), uε (s) V ×V ds ds
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1 1 Buε (t), uε (t) V ×V − Buε (0), uε (0) V ×V 2 2 βB αB
uε (t) 2V −
ψ0 2V . 2 2 =
Next, using hypothesis H(j )(iii), the Young inequality, estimate (4) and the continuity of the embedding V ⊂ H , for all t ∈ (0, T ) we have t
t
χε (s), uε (s) V ×V
ds =
0
χ(s), uε (s)
H
ds
0
t
χ(s) H uε (s) H ds
− 0
1 − 2
t
uε (s) 2H
1 ds − 2
t
uε (s) 2H
t ds
− c02
0
−
1 2
2 c02 1 + |uε (s, x)| dx ds
0 Ω
0
1 − 2
t
|Ω| + uε (s) 2H ds
0
t
uε (s) 2H ds − c02 T |Ω|
0
− c02
t s 2T uε (τ ) 2H dτ + 2 ψ0 2H ds 0
1 − 2
t
0
uε (s) 2H
t s ds
− 2T c02
0
− T c02
V |Ω| + 2cH
ψ0 2V .
uε (τ ) 2H dτ ds
0 0
Finally, from the Young inequality, for all t ∈ (0, T ) we have t
f (s), uε (s) V ×V
0
t ds
f (s), uε (s) H ds
0
t 0
f (s) H uε (s) H ds
L. Gasi´nski, M. Smołka / J. Math. Anal. Appl. 270 (2002) 150–164
1 2
t
uε (s) 2H ds +
0
1 2
t
1 2
157
t
f (s) 2H ds 0
1
uε (s) 2H ds + f 2H . 2
0
Putting all the above estimates into (5), for all t ∈ (0, T ) we obtain 1 βB
uε (t) 2H +
uε (t) 2V + εβB 2 2
t
uε (s) 2V ds
0
1 1 c1 + c2 ψ0 2V + ψ1 2H + f 2H 2 2 t t s + uε (s) 2H ds + 2T c02
uε (τ ) 2H dτ ds, 0
0 0
def
def
V c 2 + α /2. Thus, for all t ∈ (0, T ) we have with c1 = T c02 |Ω| and c2 = 2T cH B 0
t 1 βB 2 2
u (t) H +
uε (t) V + εβB uε (s) 2V 2 ε 2 0
c3 1 + ψ0 2V + ψ1 2H + f 2H t t s 2 + c4 uε (s) H ds + c4
uε (τ ) 2H dτ ds, 0 def
(6)
0 0 def
where c3 = max{1/2, c1, c2 } and c4 = max{1, 2T c02 }. Now, using the generalization of the Gronwall–Bellman inequality (see Pachpatte [21, p. 758, Theorem 1]), for all t ∈ (0, T ) we obtain
(7)
uε (t) 2H c5 1 + ψ0 2V + ψ1 2H + f 2H ,
def where c5 = 2c3 1 + 2T c4 eT (2c4 +1) ; so
uε (t) H c6 1 + ψ0 V + ψ1 H + f H , (8) def √
where c6 =
c5 , and also
uε 2H c7 1 + ψ0 2V + ψ1 2H + f 2H ,
(9)
def
where c7 = T c5 . Applying (7) to (6), we obtain
uε (t) 2V c8 1 + ψ0 2V + ψ1 2H + f 2H
(10)
158
and
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ε uε 2V c8 1 + ψ0 2V + ψ1 2H + f 2H ,
(11)
def
where c8 = (2/βB )(c3 + T c4 c5 (T /2 + 1)). Hence
uε (t) V c9 1 + ψ0 V + ψ1 H + f H and
√ ε uε V c9 1 + ψ0 V + ψ1 H + f H ,
(12)
(13)
def √ where c9 = c8 . Using Lemma 3.3, continuity of the embedding V ⊂ H and estimate (10), for any ε > 0 we have
T
V 2 2 2 2
uε (t) V dt
χε H 2c 1 + uε H 2c 1 + cH 2
2
0
c10 1 + ψ0 2V + ψ1 2H + f 2H , def
V 2 where c10 = 2c2 max{1, T (cH ) c8 }. Finally, using the equation in (HVIε ), hypothesis H(A)(i), continuity of the embedding H ⊂ V , inequalities (9)–(11) and the last inequality, for all ε ∈ (0, 1) we can estimate uε V as
uε 2V
T =
uε (t) 2V dt
0
T 5
A(t, u (t))2 dt ε V
0
T + 5ε
2
B(u (t))2 dt + 5 ε
V
0
T
χε (t) 2V dt + 5
0
T 5αA
B(uε (t))2 dt V
0
T +5
T
f (t) 2V dt 0
uε (t) 2H dt
0
T + 5ε2 αB2 0
uε (t) 2V
T dt
+ 5αB2
uε (t) 2V dt 0
L. Gasi´nski, M. Smołka / J. Math. Anal. Appl. 270 (2002) 150–164
V 2 + 5 cH
T
χε (t) 2H
V 2 dt + 5 cH
159
T
f (t) 2H dt
0
0
2
uε 2H + 5εαB2 uε 2V + 5αB2 5αA
T
uε (t) 2V dt 0
V 2 V 2 + 5 cH
χε 2H + 5 cH
f 2H
c11 1 + ψ0 2V + ψ1 2H + f 2H ,
def 2 V 2 c7 + αB2 c8 (1 + T ) + (cH ) (c10 + 1) ; so where c11 = 5 αA
uε V c12 1 + ψ0 V + ψ1 H + f H , √ with c12 = c11 . def
Finally, from (8) and (12)–(14), we obtain (3), with c = c6 + 2c9 + c12 .
(14) ✷
Now we are in the position to prove our main result. Proof of Theorem 3.1. From Lemma 3.4, it follows that for any ε ∈ (0, 1), we have
max uε (t) V + uε (t) H + uε V c13 , t ∈[0,T ]
with some constant c13 > 0 not depending on ε ∈ (0, 1). Thus, we can choose a sequence {εn }n1 ⊂ (0, 1), such that εn 0 and uε n → u
weakly∗ in L∞ (0, T ; V ),
uεn uεn
weakly∗
→u
∞
in L (0, T ; H ),
→ u weakly in V .
(15) (16) (17)
But in fact u = u and u = u . It is easy to see that (HVIε ) is equivalent to the following problem: Find uε ∈ C([0, T ]; V ) with uε ∈ W and χε ∈ H, such that ε + εBu ε + Bu ε + χε = f in V , uε + Au (HVIε ) u (0) = ψ0 , uε (0) = ψ1 in Ω, ε χε (t, x) ∈ ∂j (uε (t, x)) for a.a. (t, x) ∈ (0, T ) × Ω, : H → V and B : V → V are the Nemytskii operators corresponding where A to the operators A and B, respectively. Our aim now is to “pass to the limit” in (HVIε ). and B are linear and bounded operators, from (15) and (16) we have As A
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weakly in V , ε → Au Au n εn → Bu weakly in V . Bu
(18)
(19) √ Next, from Lemma 3.4 we see that the sequence { εn uεn }n1 remains bounded in V; hence εn uεn → 0 in V. ε V αB εn uε V ; thus in But using hypothesis H(B)(i), we have that εn Bu n n fact ε → 0 εn Bu n
in V .
(20)
From (15), (16) and the compactness of the embedding W ⊂ H we obtain uε n → u
in H,
and, in particular, possibly passing to a subsequence, uεn (t, x) → u(t, x) for a.a. (t, x) ∈ (0, T ) × Ω.
(21)
Using convergence (15), Lemma 3.3 and extracting a new subsequence if necessary, we obtain χε n → χ
weakly in H,
(22)
with some χ ∈ H; hence also χε n → χ
weakly in L1 ((0, T ) × Ω).
(23)
Now, because of (17)–(20) and (22), we can “pass to the limit” in the equation in (HVIε ) and obtain + Bu +χ =f u + Au
in V .
(24)
Since for all n 1 we have that χεn (t, x) ∈ ∂j (uεn (t, x)) for almost all (t, x) ∈ (0, T ) × Ω, thus, using convergences (21) and (23) and applying Theorem 7.2.2 on p. 273 of Aubin and Frankowska [2] (recall that ∂j is a lower semicontinuous multifunction with convex and closed values), we get χ(t, x) ∈ ∂j (u(t, x))
for a.a. (t, x) ∈ (0, T ) × Ω.
(25) H 1 (0, T ; H ),
Finally, from (15) and (16) we have that uεn → u weakly in hence also weakly in C([0, T ]; H ). Analogously, from (16) and (17) we have that uεn → u weakly in H 1 (0, T ; V ), hence also weakly in C([0, T ]; V ). In particular, we have that uεn (0) → u(0)
weakly in H,
uεn (0) → u (0)
weakly in V .
(26)
To end our proof it remains to show that u ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H ).
(27)
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For this purpose let us recall the definition of the following function space introduced in the book of Lions and Magenes [15]: def Cs ([0, T ]; X) = u ∈ L∞ (0, T ; X): (28) u∗ , u(·) X ×X is continuous ∀u∗ ∈ X . Of course, one has C([0, T ]; X) ⊂ Cs ([0, T ]; X). Moreover, if X and Y are two Banach spaces, X being reflexive, with the dense embedding X ⊂ Y , from [15, p. 297, Lemma 8.1] we know that Cs ([0, T ]; Y ) ∩ L∞ (0, T ; X) = Cs ([0, T ]; X).
(29)
In our case, due to (15)–(17) we have that u ∈ C([0, T ]; H ) ∩ L∞ (0, T ; V ), u ∈ C([0, T ]; V ) ∩ L∞ (0, T ; H ); hence, from (29) we obtain u ∈ Cs ([0, T ]; V ),
(30)
u ∈ Cs ([0, T ]; H ).
(31)
Next, using the same argument as in the proof of Lemma 3.4 (see (5) and the sequel), for any t ∈ [0, T ] we can prove the following energy equality
u (t) 2H + Bu(t), u(t) V ×V = ψ1 2H
+ Bψ0 , ψ0 V ×V
t + 2 f (s) − Au (s) − χ(s), u (s) V ×V ds. 0
This shows that the function
E : [0, T ] t → u (t) 2H + Bu(t), u(t) V ×V ∈ R
is continuous. Take tn , t ∈ [0, T ] such that tn → t and put 2 δn = u (tn ) − u (t)H + Bu(tn ) − Bu(t), u(tn ) − u(t) V ×V
= E(tn ) + E(t) − 2 Bu(t), u(tn ) V ×V − 2 u (tn ), u (t) H . Thanks to (30), (31) and the continuity of E we have that δn → 2E(t) − 2 Bu(t), u(t) V ×V − 2 u (t) 2H = 0, which, together with the inequality 2 2 δn u (tn ) − u (t)H + βB u(tn ) − u(t)V ,
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gives us (27). Now, from (24)–(27) we obtain that u is a solution of the following problem: Find u ∈ C([0, T ]; V ) ∩ C 1 ([0, T ]; H ) with u ∈ V and χ ∈ H, such that + Bu + χ = f in V , u + Au (HVI ) u(0) = ψ0 , u (0) = ψ1 in Ω, χ(t, x) ∈ ∂j (u(t, x)) for a.a. (t, x) ∈ (0, T ) × Ω, and in particular u is a solution of (HVI). ✷ .
4. Applications and examples As mentioned in the introduction the classical model for our considerations was the sine-Gordon equation u + αu − ∆u + γ sin u = f in Ω × R, (SGE) u(0) = ψ0 , u (0) = ψ1 in Ω. Of course, as | sin u| 1 the assumptions of Theorem 3.1 are satisfied if α 0. This is a “single-valued” case where j (u) = − cos u and ∂j (u) = {sin u}. In the sequel we present some examples of “dampings”, which are admissible in our framework. First let us consider a slight generalization of the one used in the sine-Gordon equation; namely Av = av,
with a ∈ L∞ (Ω), a 0 a.e.
In this case (HVI) has the form ∂u ∂ 2u + a(x) + Bu + ∂j (u) f. 2 ∂t ∂t The above “damping” operators map H into itself. The next two have values in V . This time take a ∈ W 1,∞ (Ω; RN ) and consider A1 v =
N
∂ ai (x)v , ∂xi i=1
A2 v =
N
ai (x)
i=1
It means this time (HVI) has the form N ∂u ∂ 2u ∂ + (x) a + Bu + ∂j (u) f i ∂t 2 ∂xi ∂t i=1
or, respectively, ∂ 2u ∂ 2u + a (x) + Bu + ∂j (u) f. i ∂t 2 ∂t∂xi N
i=1
∂v . ∂xi
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The nonnegativity condition for A1 yields div a 0 in Ω, (a, n) 0 on ∂Ω, where n is the outer normal to the boundary of Ω. In the case V = H01 (Ω) we could drop the second inequality. Similarly, A2 is nonnegative provided that div a = 0 in Ω, (a, n) = 0 on ∂Ω. This time, if we took V = H01 (Ω) we would need only the inequality div a 0 a.e. in Ω.
Acknowledgments This work has been partially supported by the State Committee for Scientific Research of the Republic of Poland (KBN) under research grants No. 7 T07A 047 18 and No. 2 P03A 004 19.
References [1] R. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkhäuser, Boston, 1990. [3] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [4] J. Berkovits, V. Mustonen, Monotonicity methods for nonlinear evolution equations, Nonlinear Anal. Theory Methods Appl. 12 (1996) 1397–1405. [5] W. Bian, Existence results for second order nonlinear evolution inclusions, Indian J. Pure Appl. Math. 11 (1998) 1177–1193. [6] K.C. Chang, Variational methods for nondifferentiable functionals and applications to partial differential equations, J. Math. Anal. Appl. 80 (1981) 102–129. [7] F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York, 1983. [8] G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976. [9] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North Holland, Amsterdam, 1976. [10] H. Gajewski, K. Gröger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Academie-Verlag, Berlin, 1974. [11] L. Gasi´nski, Hyperbolic hemivariational inequalities and their application in the theory of optimal shape design, Ph.D. Thesis, Jagiellonian University, Kraków (2000). [12] L. Gasi´nski, Existence of solutions for hyperbolic hemivariational inequalities, J. Math. Anal. Appl., to appear. [13] S. Hu, N.S. Papageorgiou, Handbook of Multiple Analysis, Volume I: Theory, Kluwer, Dordrecht, 1997. [14] J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. [15] J.L. Lions, E. Magenes, Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris, 1968.
164
L. Gasi´nski, M. Smołka / J. Math. Anal. Appl. 270 (2002) 150–164
[16] M. Miettinen, A parabolic hemivariational inequality, Nonlinear Anal. 26 (1996) 725–734. [17] M. Miettinen, M.M. Mäkelä, J. Haslinger, On numerical solutions of hemivariational inequalities by nonsmooth optimization methods, J. Global Optim. 6 (1995) 401–425. [18] S. Migórski, Existence, variational and optimal control problems for nonlinear second order evolution inclusions, Dynamic Systems Appl. 4 (1995) 513–528. [19] D. Motreanu, P.D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Kluwer, Dordrecht, 1998. [20] Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Dekker, New York, 1995. [21] B.G. Pachpatte, A note on Gronwall–Bellman inequality, J. Math. Anal. Appl. 44 (1973) 758– 762. [22] P.D. Panagiotopoulos, Nonconvex problems of semipermeable media and related topics, Z. Angew. Math. Mech. 65 (1985) 29–36. [23] P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser, Basel, 1985. [24] P.D. Panagiotopoulos, Coercive and semicoercive hemivariational inequalities, Nonlinear Anal. 16 (1991) 209–231. [25] P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer-Verlag, New York, 1993. [26] P.D. Panagiotopoulos, Hemivariational inequalities and fan-variational inequalities. New applications and results, Atti Sem. Mat. Fis. Univ. Modena 43 (1995) 159–191. [27] P.D. Panagiotopoulos, Modelling of nonconvex nonsmooth energy problems. Dynamic hemivariational inequalities with impact effects, J. Comput. Appl. Math. 63 (1995) 123–138. [28] N.S. Papageorgiou, Existence of solutions for second order evolution inclusions, Bull. Math. Soc. Sci. Math. Roumanie 37 (1993) 93–107. [29] N.S. Papageorgiou, On the existence of solutions for nonlinear parabolic problems with nonmonotone discontinuities, J. Math. Anal. Appl. 205 (1997) 434–453. [30] N.S. Papageorgiou, F. Papalini, F. Renzacci, Existence of solutions and periodic solutions for nonlinear evolution inclusions, Rend. Circolo Mat. Palermo 48 (1999) 341–364. [31] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1997. [32] E. Zeidler, Nonlinear Functional Analysis and Its Applications, II, Springer-Verlag, New York, 1990.