Convergence of a Time Discretization for Nonlinear Second Order Inclusion
Krzysztof Bartosz, Leszek Gasiński, Zhenhai Liu, Paweł Szafraniec
aa r X i v : . [ m a t h . A P ] J a n Convergence of a Time Discretization forNonlinear Second Order Inclusion ∗ Krzysztof Bartosz , Leszek Gasi´nski † , Zhenhai Liu and Pawe l Szafraniec , Jagiellonian University, Faculty of Mathematics and Computer Scienceul. Lojasiewicza 6, 30348 Krakow, Poland College of Sciences, Guangxi University for NationalitiesNanning, Guangxi 530006, Peoples Republic of China
January 24, 2019
Abstract.
We study an abstract second order inclusion involving two nonlin-ear single-valued operators and a nonlinear multivalued term. Our goal is toestablish the existence of solutions to the problem by applying numerical schemebased on time discretization. We show that the sequence of approximate solu-tion converges weakly to a solution of the exact problem. We apply our abstractresult to a dynamic, second order in time differential inclusion involving Clarkesubdifferential of a locally Lipschitz, possibly nonconvex and nonsmooth poten-tial. In two presented examples the Clarke subdifferential appears either in asource term or in a boundary term.
Keywords: time discretization; differential inclusions; nonconvex potential;weak solution; Rothe method ∗ The research was supported by the Marie Curie International Research Staff ExchangeScheme Fellowship within the 7th European Community Framework Programme under GrantAgreement No. 295118, the International Project co-financed by the Ministry of Science andHigher Education of Republic of Poland under grant no. W111/7.PR/2012 and the NationalScience Center of Poland under Maestro Advanced Project no. DEC-2012/06/A/ST1/00262. † Corresponding author email: [email protected]
Introduction
In this paper we study the following inclusion problem ( u ′′ ( t ) + A ( t, u ′ ( t )) + B ( t, u ( t )) + γ ∗ M ( γu ′ ( t )) ∋ f ( t ) u (0) = u , u ′ (0) = v (1.1)and we deal with the existence of its solution in an appropriate function space.In the above problem, A and B are single-valued, nonlinear operators, M is amultivalued term, γ is a linear, continuous and compact operator and γ ∗ denotesits adjoint operator. Our goal is to generalize the result obtained in [6], wherethe second order equation has been studied. In our case, because of the presenceof multivalued term M , we need to apply a technique taken from set valuedanalysis. Moreover, in [6], the operator A is assumed to be hemicontinuousand monotone with respect to the second variable. In comparison to [6], weassume that it is only pseudomonotone which allows to deal with a larger classof operators. On the other hand, it forces to use more advanced approach.Similarly as in [6], the operator B is assumed to be a nonlinear perturbation ofa linear principal part B . Using the idea presented in [6], we start with thefollowing numerical scheme τ n +1 + τ n (cid:18) u n +1 − u n τ n +1 − u n − u n − τ n (cid:19) + A (cid:18) t n , u n +1 − u n τ n +1 (cid:19) + B ( t n , u n ) + γ ∗ M (cid:18) γ u n +1 − u n τ n +1 (cid:19) ∋ f n , n = 1 , ..., N − , (1.2)with initial condition. We obtain a solution { u n } applying an existence result fora corresponding elliptic inclusion in each fixed time step. To this end, we use asurjectivity result for a pseudomonotone, coercive multivalued operator. Havingthe solution of the time-semidiscrete problem (1.2), we construct a sequence u τ of piecewise constant functions in order to approximate a solution of (1.1) andsequences v τ and ˆ v τ of piecewise constant and piecewise linear functions in orderto approximate its time derivative. First, using a priori bounds in reflexivefunctional spaces, we obtain a weak limit for the approximate sequences. Thenwe pass to convergence analysis in order to prove that the limit function satisfies(1.1).This kind of approach, known also as the Rothe method, has been usedfor solving many types of evolution partial differential equations or variationalinequalities. We refer to [14] as for a basic handbook concerning this subject.The Rothe method for evolution inclusion has been applied first in [7] and thendeveloped in [3, 4, 5, 8, 9, 11, 12, 13].There are two main difficulties arising in our problem, both concern theanalysis of convergence. The first issue comes from the fact that the operator A is assumed to be pseudomonotone with respect to the second variable, which isa relatively weak assumption in comparison to [6]. Moreover, we have to providean analogous property of its Nemytskii operator A . To this end, we use Lemma3.3, which requires to know that the considered sequence of piecewise constant2unctions is bounded in space M p,q (0 , T ; W, W ∗ ), thus, in particular that theyhave a bounded total variation.The second main difficulty appears when passing to the limit with multival-ued term. In this part, we use the Aubin-Celina convergence theorem. However,to do this, we need to have a strong convergence of the sequence γv τ in an appro-priate space, where the functions are piecewise constant, and in a consequence,their time derivatives are not regular enough to apply classical Lions-Aubincompactness results in our case. Thus, we apply more general result of Lemma2.5, which requires only that functions v τ have bounded total variations, in-stead of bounded time derivative in a space of type L q with respect to time.We remark that the compactness of the operator γ is a crucial assumption,which allows to use Lemma 2.5. In examples γ is either the compact embedding W ,p (Ω) ⊂ L p (Ω) or the trace operator γ : W ,p (Ω) → L p ( ∂ Ω).The rest of the paper is organized as follows. In Section 2, we introducebasic definitions and recall some useful results. In Section 3, we formulate anabstract problem and establish assumptions on its data. In Section 4, we statea discrete problem and obtain a priori estimates on its solution. In Section 5,we study convergence of solutions of the discrete problem to a solution of exactone. Finally, in Section 6, we show two examples for which our main result isapplicable.
In this section we introduce the basic definitions and recall results used in thesequel. We start with the definition of a pseudomonotone operator in bothsingle valued and multivalued case.
Definition 2.1.
Let X be a real Banach space. A single valued operator A : X → X ∗ is called pseudomonotone, if for any sequence { v n } ∞ n =1 ⊂ X suchthat v n → v weakly in X and lim sup n →∞ h Av n , v n − v i we have h Av, v − y i lim inf n →∞ h Av n , v n − y i for every y ∈ X . Definition 2.2.
Let X be a real Banach space. The multivalued operator A : X → X ∗ is called pseudomonotone if the following conditions hold:1) A has values which are nonempty, weakly compact and convex,2) A is upper semicontinuous from every finite dimensional subspace of X into X ∗ furnished with weak topology,3) if { v n } ∞ n =1 ⊂ X and { v ∗ n } ∞ n =1 ⊂ X ∗ are two sequences such that v n → v weakly in X , v ∗ n ∈ A ( v n ) for all n > and lim sup n →∞ h v ∗ n , v n − v i ,then for every y ∈ X there exists u ( y ) ∈ A ( v ) such that h u ( y ) , v − y i lim inf n →∞ h v ∗ n , v n − y i . Now we recall two important results concerning properties of pseudomono-tone operators. 3 roposition 2.3.
Assume that X is a reflexive Banach space and A , A : X → X ∗ are pseudomonotone operators. Then A + A : X → X ∗ is a pseudomono-tone operator. Theorem 2.4.
Let X be a reflexive Banach space and let A : X → X ∗ be apseudomonotone and coercive operator. Then A is surjective, i.e. R ( A ) = X ∗ . Let X be a Banach space and let I = (0 , T ) be a time interval. We introducethe space BV ( I ; X ) of functions of bounded total variation on I . Let π denoteany finite partition of I by a family of disjoint subintervals { σ i = ( a i , b i ) } suchthat I = ∪ ni =1 σ i . Let F be the family of all such partitions. Then for a function x : I → X we define its total variation by k x k BV ( I ; X ) = sup π ∈F (cid:26) X σ i ∈ π k x ( b i ) − x ( a i ) k X (cid:27) . As a generalization of the above definition, for 1 q < ∞ , we define a seminorm k x k qBV q ( I ; X ) = sup π ∈F (cid:26) X σ i ∈ π k x ( b i ) − x ( a i ) k qX (cid:27) and the space BV q ( I ; X ) = { x : I → X ; k x k BV q ( I ; X ) < ∞} . For 1 p ∞ , 1 q < ∞ and Banach spaces X and Z such that X ⊂ Z , weintroduce the following space M p,q ( I ; X, Z ) = L p ( I ; X ) ∩ BV q ( I ; Z ) . Then M p,q ( I ; X, Z ) is also a Banach space with the norm given by k · k L p ( I ; X ) + k · k BV q ( I ; Z ) .Finally, we recall a compactness result, which will be used in the sequel. Forits proof, we refer to [7]. Proposition 2.5.
Let p, q < ∞ . Let X ⊂ X ⊂ X be real Banach spacessuch that X is reflexive, the embedding X ⊂ X is compact and the embedding X ⊂ X is continuous. Then the embedding M p,q (0 , T ; X ; X ) ⊂ L p (0 , T ; X ) is compact. In this section we formulate an abstract problem and give a list of assumptionsconcerning the data of the problem. For a Banach space X by X ∗ we denote itstopological dual, by h· , ·i X ∗ × X ∗ × X ∗ × X the duality pairings for the pair ( X, X ∗ )and by i XY : X → Y we will denote the embedding operators of X into Y provided that X ⊆ Y . 4irst we introduce appropriate spaces. Let ( W, k · k W ) be a reflexive Banachspace densely and continuously embedded in a reflexive Banach space ( V, k ·k V ),and let ( V, k · k V ) be densely and continuously embedded in a Hilbert space( H, ( · , · ) , | · | ). We also assume that the embedding W ⊆ H is compact. We have W ⊆ V ⊆ H ⊆ V ∗ ⊆ W ∗ , where V ∗ and W ∗ denote the dual spaces to V and W , respectively. Let ( U, k·k U )be a Banach space such that there exists a compact mapping γ : W → U .For T > p > W = L p (0 , T ; W ), V = L p (0 , T ; V ), H = L (0 , T ; H ), U = L p (0 , T ; U ). We knot that their dual spaces are W ∗ = L q (0 , T ; W ∗ ), V ∗ = L q (0 , T ; V ∗ ), U ∗ = L q (0 , T ; U ∗ ), respectively (where p + q =1). We identify the space H with its dual and denote by ( · , · ) H the scalar productin H .We are concerned with the following problem. Problem P . Find u ∈ W with u ′ ∈ W and u ′′ ∈ W ∗ such that u ′′ ( t ) + A ( t, u ′ ( t )) + B ( t, u ( t )) + γ ∗ M ( γu ′ ( t )) ∋ f ( t ) a.e. t ∈ (0 , T ) , (3.1) u (0) = u , u ′ (0) = v . (3.2)A solution of Problem P will be understood in the following sense. Definition 3.1.
The function u ∈ W is said to be a solution of Problem P if u ′ ∈ W , u ′′ ∈ W ∗ , u satisfies (3.2) , and there exists a function η ∈ U ∗ such that u ′′ ( t ) + A ( t, u ′ ( t )) + B ( t, u ( t )) + γ ∗ η ( t ) = f ( t ) a.e. t ∈ (0 , T ) , (3.3) η ( t ) ∈ M ( γu ′ ( t )) a.e. t ∈ (0 , T ) . (3.4)We impose the following assumptions on the data of Problem P . H ( A ) : A : [0 , T ] × W → W ∗ is such that(i) for all v ∈ W , the mapping t → A ( t, v ) is continuous,(ii) k A ( t, v ) k W ∗ β A (cid:0) k v k p − W (cid:1) for a.e. t ∈ (0 , T ), all v ∈ W with β A > h A ( t, v ) , v i W ∗ × W > µ A k v k pW − β | u | − λ for all v ∈ W with µ A > β, λ ∈ R ,(iv) v → A ( t, v ) is pseudomonotone for all t ∈ [0 , T ].We assume that B : [0 , T ] × V → W ∗ has a decomposition B ( t, v ) = B ( v ) + C ( t, v ), where H ( B ) : B : ∈ L ( V, V ∗ ) is symmetric and strongly positive, with constants µ B , β B > h B v, v i > µ B k v k V , k B v k β B k v k V .H ( C ) : C : [0 , T ] × V → W ∗ is such that5i) for all v ∈ V , the function t → C ( t, v ) is continuous,(ii) k C ( t, v ) k W ∗ β C (1 + k v k q V ) for a.e. t ∈ (0 , T ), all v ∈ V with β C > k C ( t, v ) − C ( t, w ) k W ∗ α (max( k v k V , k w k V )) | v − w | q for all t ∈ [0 , T ], all v, w ∈ V , where α : R + → R + is a monotonically increasing function. H ( M ) : M : U → U ∗ is such that(i) for all u ∈ U , M ( u ) is a nonempty, closed and convex set,(ii) M is upper semicontinuous in ( s - U × w - U ∗ )-topology,(iii) k η k U ∗ c M (1 + k w k p − U ) for all w ∈ U , all η ∈ M ( w ). H ( f ) f ∈ W ∗ . H ( γ ): γ : W → U is linear, continuous and compact and its Nemytskii operator γ : M p,q (0 , T ; W, W ∗ ) → L q (0 , T ; U ∗ ) is compact. H : µ A > c M k γ k p L ( W,U ) .Now, we provide a result concerning pseudomonotonicity of the superposi-tion γ ∗ M ( γ ). Lemma 3.2.
Let the multivalued operator M : U → U ∗ satisfy assumption H ( M ) and the operator γ : W → U be linear, continuous and compact. Thenthe operator W ∋ v → γ ∗ M ( γu ) ∈ W ∗ is pseudomonotone. The proof of Lemma 3.2 can be obtained similarly as the proof of Proposi-tion 5.6 in [2].We complete this section with a lemma, which will play a crucial role in theconvergence of numerical scheme presented below.
Lemma 3.3.
Let A : [0 , T ] × W → W ∗ be an operator satisfying hypotheses H ( A ) and A : W → W ∗ be a Nemytskii operator corresponding to A defined by ( A v )( t ) = A ( t, v ( t )) for all t ∈ [0 , T ] , all v ∈ W . Assume that { v n } ⊂ W is asequence bounded in M p,q (0 , T ; W, W ∗ ) and such that v n → v weakly in W and lim sup n →∞ hA v n , v n − v i W ∗ ×W . Then A v n → A v weakly in W ∗ . The proof of Lemma 3.3 can be obtained by standard techniques, cf. Lemma2 in [9]. 6
Discrete problem
In this section we consider a discrete problem corresponding to Problem P .For N ∈ N we consider an arbitrary fixed time grid0 = t < t . . . t n − < t Nn = T, τ n = t n − t n − for n = 1 , . . . , N. We define the following discretization parameters τ n + = τ n + τ n +1 , t n + = t n + 12 τ n +1 for n = 1 , . . . , N − ,r n +1 = τ n +1 τ n for n = 1 , . . . , N − ,γ n := max (cid:18) , r n − r n − (cid:19) for n = 2 , . . . , N,τ max = max n =1 ,...,N τ n , r max := max n =2 ,...,N r n , r min := max n =2 ,...,N r n ,c γ := max n =3 ,...,N γ n τ n , σ ( τ ) = 12 N − X j =1 ( τ j +1 − τ j ) τ j +1 + τ j . We also define f n = τ n + 12 R t n + 12 t n − f ( t ) dt for n = 1 , ..., N − u τ , v τ , whose convergence to u and v will be specified later.The discrete problem reads as follows. Problem P τ . Find sequences { u n } Nn =0 ⊂ W and { v n } Nn =0 ⊂ W such that v n = 1 τ n +1 ( u n +1 − u n ) , n = 0 , , . . . , N − , (4.1)1 τ n + ( v n − v n − ) + A ( t n , v n ) + B ( t n , u n ) + γ ∗ η n = f n ,n = 1 , , . . . , N − , (4.2) η n ∈ M ( γv n ) , (4.3) u = u τ , v = v τ . (4.4)In what follows, we formulate a theorem concerning existence of solution toProblem P τ . Theorem 4.1.
Under hypotheses H ( A ) , H ( B ) , H ( C ) , H ( M ) , H and τ max < β there exist sequences { u n } Nn =0 and { v n } Nn =0 being a solution to Problem P τ .Proof. We define the multivalued operator T : W → W ∗ by T v = 1 τ n + v + A ( t n , v ) + γ ∗ M ( γv ) , for v ∈ W. T is coercive. Let v ∈ W and z ∈ T v . Thus we have z = τ n + 12 v + A ( t n , v ) + γ ∗ η with η ∈ M ( γv ). Using hypotheses H ( A ), H ( B ), H ( C ) and H ( M ), we estimate h z, v i W ∗ × W = 1 τ n + ( v, v ) H + h A ( t n , v ) , v i W ∗ × W + h η, γv i U ∗ × U > τ n + | v | H + µ A k v k pW − β | v | H − λ − k η k U ∗ k γv k U > (cid:18) τ n + − β (cid:19) | v | H + µ A k v k pW − λ − c M (cid:16) k γv k p − U (cid:17) k γv k U > (cid:18) τ n + − β (cid:19) | v | H + µ A k v k pW − λ − c M k γ k p L ( W,U ) k v k p − c M k γ k L ( W,U ) k v k W > (cid:18) τ n + − β (cid:19) | v | H + (cid:16) µ A − c M k γ k p L ( W,U ) (cid:17) k v k pW − λ − c M k γ k L ( W,U ) k v k W . Using H and inequality τ max < β , we see that T is coercive. From H ( A )( iv )and Lemma 3.2, we conclude that operator T is pseudomonotone as a sum ofthree pseudomonotone operators. This allows to use Theorem 2.4 to concludethat T is surjective, and as a result, we can establish the existence of v n for agiven v , . . . , v n − in Problem P τ . Moreover, using u n = u + n − X j =0 ( u j +1 − u j )= u + n − X j =0 τ j +1 v j := L ( v n ) , n = 0 , , . . . , N, we can recover the sequence u , u , ..., u n . This completes the proof.The next lemma concerns a priori estimate for solution of Problem P τ . Inwhat follows, we denote by c a constant independent on τ , which can vary fromline to line. The dependence of c on the other data or parameter will be specifiedif needed. Lemma 4.2 ( A priori estimate) . Let hypotheses H ( A ) , H ( B ) , H ( C ) , H ( M ) , H hold and the time grid satisfy the following constraint τ max < min ( (cid:16) µ A − c M k γ k p L ( W,U ) (cid:17) β B k i W V k L ( W,V ) , β ) . (4.5) Then, for n = 1 , , . . . , N − , we have k u n +1 k V + | v n | + n X j =1 | v j − v j − | + n X j =1 τ j + k v j k pW + n X j =1 τ j + k η j k qU ∗ c (cid:18) k u k V + | v | + τ k v k V + n X j =1 τ j + k f j k qW ∗ (cid:19) , (4.6) where c = c ( r min , r max , c γ , T ) > . Moreover n X j =1 τ j + (cid:13)(cid:13)(cid:13)(cid:13) τ j + ( v j − v j − ) (cid:13)(cid:13)(cid:13)(cid:13) qW ∗ c. (4.7) Proof.
We test (4.2) by v n and calculate1 τ j + ( v n − v n − , v n ) = 12 τ j + (cid:0) | v n | − | v n − | + | v n − v n − | (cid:1) (4.8) h A ( t n , v n ) , v n i W ∗ × W > µ A k v n k pW − β | v n | − λ (4.9) h B ( t n , u n ) , v n i W ∗ × W = h B ( u n ) , v n i + h C ( t, u n ) , v n i . (4.10)We introduce the inner product h· , ·i B : V × V → R by h u, v i B := h B u, v i V ∗ × V for u, v ∈ V and the corresponding norm k u k B = p h u, u i B for u ∈ V. Note that the norms k u k B and k u k V are equivalent since µ B k u k V k u k B β B k v k V for all u ∈ V . We have h B u n , v n i V ∗ × V = h B Lv n , v n i V ∗ × V = (cid:10) B Lv n , τ n +1 ( Lv n +1 − Lv n ) (cid:11) V ∗ × V = 12 τ n +1 (cid:0) k Lv n +1 k B − k Lv n k B − τ n +1 k v n k B (cid:1) = 12 τ n +1 ( k u n +1 k B − k u n k B − τ n +1 k v n k B ) . (4.11)From hypotheses H ( C ), using Young inequality, for any fixed ε >
0, we findthat |h C ( t n , u n ) , v n i W ∗ × W | k C ( t n , u n ) k W ∗ k v n k W ε k v n k pW + c ( ε ) k C ( t n , u n ) k qW ∗ ε k v n k pW + c ( ε )(1 + k u n k V ) . We come to the multivalued term |h γ ∗ η n , v n i W ∗ × W | = |h η n , γv n i U ∗ × U | k η n k U ∗ k γv n k U c M (1 + k γv n k p − U ) k γv n k ( c M k γ k p L ( W,U ) + ε ) k v n k pW + c ( ε ) c qM k γ k q L ( W,U ) h f n , v n i ε k v n k pW + c ( ε ) k f n k qW ∗ . (4.12)We test (4.2) with v n , apply (4.8)-(4.12), replace n with j and multiply by2 τ j + to obtain | v j | − | v j − | + | v j − v j − | + 2 τ j + ( µ A − c M k γ k p L ( W,U )) − ε ) k v k pW − τ j + β | v j | + τ j + τ j +1 ( k u j +1 k B − k u j k B ) − τ j +1 τ j + k v j k B − τ j + c ( ε ) k u j k V λτ j + + 2 τ j + c ( ε ) + 2 τ j + c ( ε ) c qM k γ k q L ( W,U ) +2 τ j + c ( ε ) k f j k qW ∗ . (4.13)We sum up (4.13) for j = 1 , . . . , n , to obtain | v n | + n X j =1 | v j − v j − | + 2 n X j =1 τ j + (cid:16) µ A − c M k γ k p L ( W,U ) − ε (cid:17) k v j k pW + 12 (cid:18) r n +1 (cid:19) k u n +1 k B + 12 n X j =2 (cid:18) r j − r j +1 (cid:19) k u j k B | v | + 2 β n X j =1 τ j + | v j | + 12 (cid:18) r (cid:19) k u k B + n X j =1 τ j + τ j +1 k v j k B +2 c ( ε ) n X j =1 τ j + k u j k V + 2 c ( ε ) n X j =1 τ j + k f j k qW ∗ + cT. (4.14)Note that n X j =1 τ j + τ j +1 k v k B n X j =1 τ j + τ j +1 β B k i W V k L ( W,V ) (1 + k v j k pW ) β B k i W V k L ( W,V ) n X j =1 τ j + τ j +1 + β B k i W V k L ( W,V ) n X j =1 τ j + τ j +1 k v j k pW cT + β B k i W V k L ( W,V ) τ max n X j =1 τ j + k v j k pW (4.15)and 12 n X j =2 (cid:18) r j − r j +1 (cid:19) k u j k B = − n X j =2 (cid:18) r j +1 − r j (cid:19) k u j k B = − n X j =2 τ j +1 γ j +1 τ j +1 k u j k B > − n X j =2 τ j +1 c γ β B k u j k V (4.16)From τ j +1 τ j = r j +1 is follows that τ j τ j +1 r min . Thus2 c ( ε ) n X j =1 τ j + k u j k V = c ( ε ) n X j =1 ( τ j + τ j +1 ) k u j k V c ( ε ) n X j =1 (cid:18) τ j +1 r min + τ j +1 (cid:19) k u j k V = c ( ε ) (cid:18) r min + 1 (cid:19) n X j =1 τ j +1 k u j k V , (4.17)2 β n X j =1 τ j + | v j | βτ max | v n | + β (cid:18) r min + 1 (cid:19) n − X j =1 τ j +1 | v j | (4.18)and µ B (cid:18) r n +1 (cid:19) k u n +1 k V (cid:18) r n +1 (cid:19) k u n +1 k B . (4.19)Using (4.15)-(4.19) in (4.14), we get(1 − βτ max ) | v n | + 12 µ B (cid:18) r n +1 (cid:19) k u n +1 k B + n X j =1 | v j − v j − | + n X j =1 τ j + h µ A − c M k γ k p L ( W,U ) − ε ) − β B k i W V k L ( W,V ) τ max i k u j k pW | v | + 12 (cid:18) r (cid:19) β B k τ v + u k V + β (cid:18) r min + 1 (cid:19) n − X j =1 τ j +1 | v j | + cT + (cid:20) c ( ε ) (cid:18) r min + 1 (cid:19) + 12 c γ β B (cid:21) n X j =1 τ j +1 k u j k V +2 c ( ε ) n X j =1 τ j + k f j k qW ∗ . Using H and (4.5), we see that for ε > H ( M )( iii ), gives(4.5).As for (4.6), we use (4.2) and get (cid:13)(cid:13)(cid:13)(cid:13) τ j + ( v j − v j − ) (cid:13)(cid:13)(cid:13)(cid:13) qW ∗ c ( k A ( t j , v j ) k qW ∗ + k B ( t j , u j ) k qW ∗ + k γ ∗ η j k qW ∗ + k f j k qW ∗ ) , (4.20)with a positive constant c . From growth conditions on A , B and C , we estimate k A ( t j , v j ) k qW ∗ c (1 + k v j k pW ) , (4.21) k B ( t j , v j ) k qW ∗ c (1 + k u j k V ) , (4.22) k γ ∗ η j k qW ∗ k γ k q L ( W,U ) k η j k qU ∗ . (4.23)11sing (4.21)-(4.23) in (4.20), multiplying (4.20) by τ j + and summing up with j = 1 , . . . , n , we have n X j =1 τ j + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) τ j + ( v j − v j − ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) qW ∗ c (cid:18) n X j =1 τ j + k v j k pW + n X j =1 τ j + k u j k V + n X j =1 τ j + k η k qU ∗ + n X j =1 τ j + k f j k qW ∗ (cid:19) . (4.24)Finally, using (4.6), we get (4.7) from (4.24). This completes the proof of thelemma.Now, we use the solution { u n } Nn =0 , { v n } Nn =0 of (4.1)-(4.4) to define piecewiseconstant and piecewise linear functions whose convergence will be studied innext section. u τ ( t ) := t ∈ [0 , t ] u n for t ∈ ( t n − , t n + ] n = 1 , , . . . , N −
10 for t ∈ ( t N − , t N ] ,v τ ( t ) := v for t ∈ [0 , t ] v n for t ∈ ( t n − , t n + ] n = 1 , , . . . , N − v N for t ∈ ( t N − , t N ] , ˆ v τ ( t ) := v for t ∈ [0 , t ] v n + t − t n + 12 τ n + 12 ( v n − v n − ) for t ∈ ( t n − , t n + ] n = 1 , , . . . , N − v N − for t ∈ ( t N − , t N ] ,η τ ( t ) := η for t ∈ [0 , t ] η n for t ∈ ( t n − , t n + ] n = 1 , , . . . , N − η N for t ∈ ( t N − , t N ] ,f τ ( t ) := t ∈ [0 , t ] f n for t ∈ ( t n − , t n + ] n = 1 , , . . . , N −
10 for t ∈ ( t N − , t N ] . Note that the above functions depend on the parameter N . However, for thesake of simplicity, we omit the symbol N in their notation.It is well known (see Remark 8.15 in [14]) that f τ → f in W ∗ as N → ∞ . (4.25) In this section we study the behaviour of sequences u τ , v τ , ˆ u τ , η τ and f τ withrespect to the increasing number of time grids N . In what follows, all con-vergences, unless it is specified differently, will be understood with respect to12 → ∞ . In particular, we impose the following, additional assumptions. H ( τ ) :(1) τ max → τ max Dτ min with a constant D > N ,(3) σ τ → .H :(1) u τ → u in V ,(2) v τ → v in H ,(3) sup N ∈ N τ max k v τ k V < ∞ .We introduce the integral operator K : V → V defined by( Kw )( t ) : = Z t w ( s ) ds for all w ∈ V for t ∈ [0 , T ] . Lemma 5.1 (Convergences) . Under hypotheses H ( A ) , H ( B ) , H ( M ) , H ( γ ) , H ( τ ) , H , and H , there exists u ∈ C (0 , T ; V ) and v ∈ W with v ′ ∈ W ∗ suchthat u = u + Kv and for a subsequence, we have(a) u τ → u weakly ∗ in L ∞ (0 , T ; V ) ,(b) v τ → v weakly in W and weakly ∗ in L ∞ (0 , T ; H ) ,(c) ˆ v τ → v weakly in W and weakly ∗ in L ∞ (0 , T ; H ) ,(d) ˆ v ′ τ → v ′ weakly in W ∗ ,(e) Kv τ → Kv weakly ∗ in L ∞ (0 , T ; W ) ,(f ) u + Kv τ − u τ → in L r (0 , T ; V ) for r ∈ [1 , ∞ ) ,(g) η τ → η weakly in U ∗ ,(h) ˆ v τ → v in L r (0 , T ; H ) for r ∈ [1 , ∞ ) ,(i) v τ → v in L r (0 , T ; H ) for r ∈ [1 , ∞ ) ,(j) u τ → u in L r (0 , T ; H ) for r ∈ [1 , ∞ ) ,(k) u + Kv τ → u w C (0 , T ; H ) ,(l) v τ is bounded in M p,q (0 , T ; W ; W ∗ ) . roof. By estimates (4.6) and (4.7), we easily get k u τ k L ∞ (0 ,T ; V ) ≤ c, (5.1) k v τ k W + k v τ k L ∞ (0 ,T ; H ) ≤ c, (5.2) k ˆ v τ k W + k ˆ v τ k L ∞ (0 ,T ; H ) ≤ c, (5.3) k ˆ v ′ τ k W ∗ ≤ c, (5.4) k η τ k U ∗ ≤ c, (5.5)The convergences ( a )-( d ) and ( g ) follow from (5.1)-(5.4) and (5.5), respectively.However, we need to show that limits obtained in ( b ) and ( c ) coincide. Notethat k ˆ v τ − v τ k H = N − X j =1 Z t j + 12 t j − (cid:12)(cid:12)(cid:12)(cid:12) t − t j + τ j + ( v j − v j − ) (cid:12)(cid:12)(cid:12)(cid:12) dt = N − X j =1 (cid:12)(cid:12)(cid:12)(cid:12) v j − v j − τ j + (cid:12)(cid:12)(cid:12)(cid:12) Z t j + 12 t j − ( t − t j + ) dt = 13 N − X j =1 (cid:12)(cid:12)(cid:12)(cid:12) v j − v j − τ j + (cid:12)(cid:12)(cid:12)(cid:12) τ j + τ max N − X j =1 (cid:12)(cid:12) v j − v j − (cid:12)(cid:12) → . (5.6)So ˆ v τ − v τ → H . Now since v τ − ˆ v τ → v − ˆ v weakly in W , it follows that v τ − ˆ v τ → v − ˆ v weakly in H . From the uniqueness of the limit we obtain v = ˆ v .Now we prove ( e ). Let g ∈ L (0 , T ; W ∗ ). Then h g, Kv τ − Kv i L (0 ,T ; W ∗ ) × L ∞ (0 ,T ; W ) = Z T (cid:28) g ( t ) , Z t ( v τ ( s ) − v ( s )) ds (cid:29) W ∗ × W dt = Z T Z t h g ( t ) , v τ ( s ) − v ( s ) i W ∗ × W ds dt = Z T Z Ts h g ( s ) , v τ ( s ) − v ( s ) i W ∗ × W dt ! ds = Z T *Z Ts g ( t ) dt, v τ ( s ) − v ( s ) + W ∗ × W ds = Z T h G ( s ) , v τ ( s ) − v ( s ) i W ∗ × W ds = h G, v τ − v i W ∗ ×W → , where G ( s ) = R Ts g ( t ) dt for s ∈ [0 , T ]. Since G ∈ L ∞ (0 , T ; W ∗ ), we have G ∈ W ∗ . This proves ( e ). 14o prove ( f ) we first estimate integrals Z t k u + v t k V dt Z t ( k u k V + t k v k V ) dt Z t k u k V + t k v k V ) dt = 2 t k u k V + 23 τ k v k V cτ max ( k u k V + k v k V ) . (5.7)For n = 1 , ..., N −
1, we have I n = Z t n + 12 t n − k u + ( Kv τ )( t ) − u τ ( t ) k V dt = Z t n + 12 t n − (cid:13)(cid:13)(cid:13)(cid:13) u + v τ + n − X j =1 τ j + v j + ( t − t n − ) v n − u − n − X j =1 τ j +1 v j (cid:13)(cid:13)(cid:13)(cid:13) V dt = Z t n + 12 t n − (cid:13)(cid:13)(cid:13)(cid:13) u − u + v τ + n − X j =1 ( τ j + − τ j +1 ) v j + ( t − t n − ) v n (cid:13)(cid:13)(cid:13)(cid:13) V dt c Z t n + 12 t n − (cid:18) k u − u k V + τ k v k V + (cid:18) n − X j =1 | τ j + − τ j +1 |k v j k V (cid:19) +( t − t n − ) k v n (cid:13)(cid:13) V (cid:19) dt c (cid:20) τ n + k u − u k V + τ n + τ k v k V + τ n + (cid:18) n − X j =1 | τ j + − τ j +1 |k v j k V (cid:19) + 13 τ n + k v n k V (cid:21) . Hence N − X n =1 I n c (cid:20) T k u − u k V + τ max k v k V + T (cid:18) N − X j =1 | τ j + − τ j +1 |k v j k V (cid:19) + τ max N − X n =1 τ n + k v n k V (cid:21) . (5.8)Finally, we estimate the integral Z Tt N − (cid:13)(cid:13)(cid:13)(cid:13) u + v τ + N − X j =1 τ j + v j (cid:13)(cid:13)(cid:13)(cid:13) V dt τ max (cid:18) k u k V + τ max k v k V + (cid:18) N − X j =1 τ j + k v j k V (cid:19) (cid:19) τ max (cid:18) k u k V + τ max k v k V + N N − X j =1 τ j + k v j k V (cid:19) τ max (cid:18) k u k V + τ max k v k V + DT N − X j =1 τ j + k v j k V (cid:19) . (5.9)Now, using Cauchy-Schwartz inequality, we estimate (cid:18) N − X j =1 | τ j + − τ j +1 |k v j k V (cid:19) = (cid:18) N − X j =1 (cid:12)(cid:12)(cid:12)(cid:12) τ j − τ j +1 (cid:12)(cid:12)(cid:12)(cid:12) k v j k V (cid:19) = (cid:18) N − X j =1 τ j − τ j +1 p τ j + p τ j + k v j k V (cid:19) (cid:18) N − X j =1 ( τ j − τ j +1 ) τ j + (cid:19)(cid:18) N − X j =1 τ j + k v j k V (cid:19) = σ ( τ ) N − X j =1 τ j + k v j k V . (5.10)Since p >
2, we have s s p for all s ∈ R . Thus, we have N − X j =1 τ j + k v j k V k i W V k L ( W,V ) (cid:18) T + N − X j =1 τ j + k v j k pW (cid:19) . (5.11)Therefore, by (5.7)-(5.11) and hypothesis H ( τ ) we obtain u + Kv τ − u τ → L (0 , T ; V ). Since v τ is bounded in W it is also bounded in V , so Kv τ in boundedin L ∞ (0 , T ; V ). Moreover u τ is bounded in L ∞ (0 , T ; V ). So u + Kv τ − u τ isbounded in L ∞ (0 , T ; V ). For r >
2, we have k u + Kv τ − u τ k rL r (0 ,T ; V ) = Z T k u + Kv τ ( t ) − u τ ( t ) k r − V k u + Kv τ ( t ) − u τ ( t ) k V dt k u + Kv τ − u τ k L ∞ (0 ,T ; V ) k u + Kv τ − u τ k L (0 ,T ; V ) → . Therefore u + Kv τ − u τ → L r (0 , T ; V ) for all r ∈ [1 , ∞ ), which completesthe proof of ( f ).From ( a ), ( e ), ( f ) and uniqueness of the weak limit in L (0 , T ; V ) we obtain u = u + Kv (5.12)and, in particular, u ∈ C (0 , T ; V ). From ( c ), ( d ), compactness of embedding W ⊂ H and the Lions-Aubin lemma, we haveˆ v τ → v in L p (0 , T ; H ) . Again, since ˆ v τ is bounded in L ∞ (0 , T ; H ) it follows thatˆ v τ → v in L r (0 , T ; H ) , for all r ∈ [1; ∞ ) , h ).From (5.6) and ( h ) we have v τ → v in L (0 , T ; H ), and also v τ → v in L r (0 , T ; H ) with r ∈ [1 , ∞ ], since v τ is bounded in L ∞ (0 , T ; H ).Thus ( i ) holds. Now, using ( i ), we calculate, k Kv τ − Kv k C (0 ,T ; H ) = max t ∈ [0 ,T ] (cid:13)(cid:13)(cid:13)(cid:13) Z t v τ ( s ) ds − Z t v ( s ) ds (cid:13)(cid:13)(cid:13)(cid:13) H max t ∈ [0 ,T ] Z t k v τ ( s ) − v ( s ) k H ds = Z T k v τ ( s ) − v ( s ) k H ds = k v τ − v k L (0 ,T ; H ) → . (5.13)From (5.12) we have k u − u τ k L r (0 ,T ; H ) = k u + Kv − u τ k L r (0 ,T ; H ) = k u + Kv τ − u τ + Kv − Kv τ k L r (0 ,T ; H ) k u + Kv τ − u τ k L r (0 ,T ; H ) + k Kv − Kv τ k L r (0 ,T ; H ) k u + Kv τ − u τ k L r (0 ,T ; H ) + T r k Kv − Kv τ k C (0 ,T ; H ) . (5.14)Combining ( f ), (5.13) and (5.14), we obtain ( j ).Moreover, by (5.12) we have k u + Kv τ − u k C (0 ,T ; H ) = k Kv τ − Kv k C (0 ,T ; H ) .This together with (5.13) gives ( k ).It remains to show ( l ). Taking into account (5.2), it is enough to estimatethe seminorm k v τ k BV q (0 ,T ; W ∗ ) . Since the function v τ is piecewise constant, theseminorm will be measured by means of jumps between elements of sequence { v kτ } Nk =1 . Namely, let { m i } ni =0 ⊂ { , ..., N } be an increasing sequence of numberssuch that m = 0, m n = N and k v τ k qBV q (0 ,T ; W ∗ ) = n X i =1 k v m i τ − v m i − τ k qW ∗ . (5.15)In what follows, we estimate n X i =1 k v m i τ − v m i − τ k qW ∗ n X i =1 (cid:18) ( m i − m i − ) q − m i X k = m i − +1 k v kτ − v k − τ k qW ∗ (cid:19) (cid:18) n X i =1 ( m i − m i − ) q − (cid:19)(cid:18) n X i =1 m i X k = m i − +1 k v kτ − v k − τ k qW ∗ (cid:19) N q − N X k =1 k v kτ − v k − τ k qW ∗ = N q − τ qj + N X k =1 (cid:13)(cid:13)(cid:13)(cid:13) v kτ − v k − τ τ j + (cid:13)(cid:13)(cid:13)(cid:13) qW ∗ N q − τ q − max τ j + N X k =1 (cid:13)(cid:13)(cid:13)(cid:13) v kτ − v k − τ τ j + (cid:13)(cid:13)(cid:13)(cid:13) qW ∗ N q − D q − τ q − min τ j + N X k =1 (cid:13)(cid:13)(cid:13)(cid:13) v kτ − v k − τ τ j + (cid:13)(cid:13)(cid:13)(cid:13) qW ∗ CT q − τ j + N X k =1 (cid:13)(cid:13)(cid:13)(cid:13) v kτ − v k − τ τ j + (cid:13)(cid:13)(cid:13)(cid:13) qW ∗ . (5.16)We now combine (4.7) with (5.15) and (5.16) to see that k v τ k qBV q (0 ,T ; W ∗ ) isbounded. This completes the proof of the lemma.Now we formulate the existence theorem which is the main result of thepaper. Theorem 5.2.
Let hypotheses H ( A ) , H ( B ) , H ( C ) , H ( γ ) , H hold and u ∈ V, v ∈ H, f ∈ L q (0 , T ; W ∗ ) . Then Problem P has a solution such that u ∈ C ([0 , T ]; V ) .Proof. We define Nemytskii operators A : W → W ∗ , B : V → V ∗ , C : V → W ∗ and γ : W → U , by( A v )( t ) = A ( t, v ( t )) for all t ∈ [0 , T ] , all v ∈ W , ( B v )( t ) = B v ( t ) for all t ∈ [0 , T ] , all v ∈ V , ( C v )( t ) = C ( t, v ( t )) for all t ∈ [0 , T ] , all v ∈ V , ( γv )( t ) = γv ( t ) for all tt ∈ [0 , T ] , all v ∈ W . Moreover, we approximate the operators A and C by their piecewise constantinterpolates given by( A τ v )( t ) := A ( t , v ( t )) for t ∈ [0 , t ] A ( t n , v ( t )) for t ∈ ( t n − , t n + ] n = 1 , , . . . , N − A ( t N − , v ( t )) for t ∈ ( t N − , t N ] , ( C τ v )( t ) := C ( t , v ( t )) for t ∈ [0 , t ] C ( t n , v ( t )) for t ∈ ( t n − , t n + ] n = 1 , , . . . , N − C ( t N − , v ( t )) for t ∈ ( t N − , t N ] . Let u τ , v τ , ˆ v τ , η τ and f τ be the functions defined in Section 4. Now, Problem P τ is equivalent toˆ v ′ τ + A τ v τ + B u τ + C τ u τ + γη τ = f τ in L q (0 , T ; W ∗ ) , (5.17) η τ ( t ) ∈ M ( γv τ ( t )) for a.e. t ∈ (0 , T ) . (5.18)We will pass to the weak limit in W ∗ with (5.17). From Lemma 5.1 ( d ), wehave ˆ v ′ τ → v ′ weakly in W ∗ . (5.19)18ext, from Lemma 5.1( a ), we obtain u τ → u weakly in V . (5.20)Thus, by continuity of B , we also have B u τ → B u weakly in V ∗ . (5.21)Next we will show that C τ u τ → C u in W ∗ . First we will show that C τ u → C u in W ∗ . We will use Lebesgue dominated convergence theorem. We show thepointwise convergence, which follows from H ( C )( i ), namely kC τ u ( t ) − C u ( t ) k W ∗ = k C ( t n , u ( t )) − C ( t, u ( t )) k W ∗ → t ∈ (0 , T ) . Next, we show boundedness, as follows kC τ u ( t ) − C u ( t ) k qW ∗ = k C ( t n ) , u ( t ) − C ( t, u ( t )) k qW ∗ q − ( k C ( t n , u ( t )) k qW ∗ + k C ( t, u ( t )) k qW ∗ ) q − β C (cid:18) k u ( t ) k q V (cid:19) q q − β C (cid:0) k u ( t ) k V (cid:1) c (1 + k u ( t ) k V ) . The function t → c (1 + k u ( t ) k V ) is integrable, because u ∈ L (0 , T ; V ). By theLebesgue dominated convergence theorem, we have C τ u → C u in W ∗ . Fromhypotheses H ( C )( iii ) and Lemma 5.1, we obtain kC τ u τ − C τ u k q W ∗ = Z T k C ( t n , u τ ( t )) − C ( t n , u ( t )) k qW ∗ dt Z T ( α (max {k u τ ( t ) k V , k u ( t ) k V } )) q | u τ ( t ) − u ( t ) | dt α (cid:16) max n k u τ k qL ∞ (0 ,T ; V ) , k u k qL ∞ (0 ,T ; V ) o(cid:17) k u τ − u k L (0 ,T ; H ) → . Since kC τ u τ − C u k W ∗ kC τ u τ − C τ u k W ∗ + kC τ u − C u k W ∗ → C τ u τ → C u in W ∗ (5.22)By Lemma 5.1( g ) and the continuity of γ ∗ , we infer that γ ∗ η τ → γ ∗ η weakly in W ∗ . (5.23)It remains to show that A τ v τ → A v weakly in W ∗ . (5.24)In order to prove (5.24), we proceed in two steps. First, we show that A τ v τ − A v τ → W ∗ . (5.25)To this end, let w ∈ W . We define the function h τ ( t ) = h ( A τ v τ ) ( t ) − ( A v τ ) ( t ) , w ( t ) i W ∗ × W for t ∈ (0 , T )19nd note that hA τ v τ − A v τ , w i W ∗ ×W = R T h τ ( t ) dt . Let S ⊂ [0 , T ] be a set ofmeasure zero, such that the function w is well defined on the set [0 , T ] \ S . Let t ∈ [0 , T ] \ S and n ∈ N be such that t ∈ [ t n − , t n + ]. We estimate | h τ ( t ) | = |h A ( t n , v τ ( t )) − A ( t, v τ ( t )) , w ( t ) i W ∗ × W | k A ( t n , v τ ( t )) − A ( t, v τ ( t )) k W ∗ k w ( t ) k W . By hypothesis H ( τ ), it is clear that t n → t . Thus, by hypothesis H ( A )( i ),we have k A ( t n , v τ ( t )) − A ( t, v τ ( t )) k W ∗ →
0, so h τ ( t ) → t ∈ [0 , T ].Moreover, we have | h τ ( t ) | k A ( t n , v τ ( t )) − A ( t, v τ ( t )) k W ∗ k w ( t ) k W (cid:0) k A ( t n , v τ ( t )) k W ∗ + k A ( t, v τ ( t )) k W ∗ (cid:1) k w ( t ) k W (cid:0) β A + 2 β A k v τ ( t ) k p − W ∗ (cid:1) k w ( t ) k W = 2 β A k w ( t ) k W + 2 β A k v τ ( t ) k p − W k w ( t ) k W . By the H¨older inequality, the right hand side is integrable on [0 , T ], so we can useLebesgue dominated convergence theorem and we hA τ v τ − A v τ , w i W ∗ ×W → . Since the function w is arbitrary, we obtain (5.25).In the second step, we calculatelim sup hA v τ , v τ − v i W ∗ ×W lim sup hA τ v τ , v τ − v i W ∗ ×W + lim sup hA v τ − A τ v τ , v τ − v i W ∗ ×W lim sup hA τ v τ , v τ − v i W ∗ ×W + lim sup hA v τ − A τ v τ , v τ i W ∗ ×W + lim sup hA τ v τ − A v τ , v i W ∗ ×W . (5.26)Using (5.25), we have lim sup hA τ v τ − A v τ , v i W ∗ ×W = 0 . (5.27)Analogously as in the proof of (5.27), we show thatlim sup hA v τ − A τ v τ , v τ i W ∗ ×W = 0 . (5.28)From (5.17) we getlim sup hA τ v τ , v τ − v i W ∗ ×W = h f τ , v τ − v i W ∗ ×W + (ˆ v ′ τ , v − v τ ) H + hB u τ , v − v τ i V ∗ ×V − hC τ u τ , v τ − v i W ∗ ×W − h η τ , γv τ − γv i U ∗ ×U . (5.29)From (4.25) and Lemma 5.1(d), we have h f τ , v τ − v i W ∗ ×W → . (5.30)20oreover, we havelim sup(ˆ v ′ τ , v − v τ ) H = lim sup ((ˆ v ′ τ , v ) H − (ˆ v ′ τ , ˆ v τ ) H + (ˆ v ′ τ , ˆ v τ − v τ ) H ) lim(ˆ v ′ τ , v ) H − lim inf(ˆ v ′ τ , ˆ v τ ) H + lim sup(ˆ v ′ τ , ˆ v τ − v τ ) H = ( v ′ , v ) H − lim inf (cid:18) | ˆ v τ ( T ) | − | ˆ v τ (0) | (cid:19) + lim sup (cid:18) − N − X j =1 (cid:0) t j + − t j − (cid:1) (cid:19) (cid:0) | v ( T ) | − | v (0) | + lim sup | ˆ v τ (0) | − lim inf | ˆ v τ ( T ) | (cid:1) (cid:0) | v ( T ) | − lim inf | ˆ v τ ( T ) | + lim | v | − | v (0) | (cid:1) . From Lemma 5.1(c) and (d) and from continuity of the embedding { v ∈ L p (0 , T ; W ) | v ′ ∈ L q (0 , T ; W ∗ ) } ⊂ C (0 , T ; H ) , we have v τ → v that v τ ( t ) → v ( t ) weakly in H . From hypothesis H (2) and theuniqueness of the weak limit, we have v (0) = v and v → v in H. (5.31)The continuity of the norm implies | v τ (0) | → | v (0) | . Moreover, v τ ( T ) → v ( T )weakly in H and by the weak lower semicontinuity of norm, we have | v ( T ) | lim inf | v τ ( T ) | . Summarizing, we conclude thatlim sup(ˆ v ′ τ , v − v τ ) H . (5.32)Since the operator B : V → V ∗ defines the inner product on V and since( Kw ) ′ = w for all w ∈ L (0 , T ; V ), integrating by parts, we get hB Kw, w i V ∗ ×V = Z T hB ( Kw )( t ) , w ( t ) i V ∗ × V dt = Z T hB ( Kw )( t ) , ( Kw ) ′ ( t ) i V ∗ × V dt = 12 hB ( Kw )( T ) , ( Kw )( T ) i V ∗ × V − hB ( Kw )(0) , ( Kw )(0) i W ∗ × W = 12 k Kw ( T ) k B > . (5.33)By (5.12) and (5.33) we have h B u τ , v − v τ i V ∗ ×V = h B u, v − v τ i V ∗ ×V + h B ( u τ − u − Kv τ ) , v − v τ i V ∗ ×V − h B K ( v − v τ ) , v − v τ i V ∗ ×V h B u, v − v τ i V ∗ ×V + h B ( u τ − u − Kv τ ) , v − v τ i V ∗ ×V . (5.34)From Lemma 5.1( b ), it follows that v τ → v weakly in V . Thus h B u, v − v τ i V ∗ ×V → . f ), we also have h B ( u τ − u − Kv τ ) , v − v τ i V ∗ ×V → . Thus, it follows from (5.34), thatlim sup hB u τ , v − v τ i W ∗ ×W . (5.35)From (5.22) and Lemma 5.1( b ), we obtainlim hC τ u τ , v τ − v i W ∗ ×W = 0 . (5.36)From Lemma 5.1 ( g ), ( l ) and hypothesis H ( γ ), passing to a subsequence ifnecessary, we have h η τ , γv τ − γv i U ∗ ×U → . (5.37)Applying (5.32)-(5.37) in (5.29) we havelim sup hA τ v τ , v τ − v i W ∗ ×W . (5.38)From (5.27), (5.28), (5.38) in (5.26), we getlim sup hA v τ , v τ − v i W ∗ ×W . (5.39)Now, from Lemma 5.1( b ), ( l ), (5.39) and Lemma 3.3 we obtain A v τ → A v weakly in W ∗ . (5.40)By (5.40) and (5.25) we obtain (5.24). Using (5.19)-(5.24) we pass to the limitin (5.17) and obtain v ′ + A v + B u + C u + γ ∗ η = f. (5.41)Next, we pass to the limit with inclusion (5.18). From Lemma 5.1( l ) and hy-pothesis H ( γ ), we have that γv τ → γv in U and in consequence γv τ ( t ) → γv ( t ) in U, for a.e. t ∈ [0 , T ] . (5.42)From Lemma 5.1 ( g ), (5.42) and Aubin-Celina convergence theorem (cf. [1]),we get η ( t ) ∈ M ( γv ( t )) for a.e. t ∈ [0 , T ] . (5.43)Moreover, by Lemma 5.1 we have u = u + Kv , thus u (0) = u and u ′ = v .Combining it with (5.31), we see that u ′ (0) = v . This, together with (5.41) and(5.43) shows that u is a solution of Problem P . This completes the proof.22 Examples
In this section we consider two problems, for which the existence result obtainedin Theorem 5.2 is applicable.Let Ω be an open bounded subset of R N with a Lipschitz boundary ∂ Ω. Theboundary is divided in two parts Γ , Γ such that Γ ∪ Γ = ∂ Ω and the N − is positive. We denote by ν the outward unit vectornormal to ∂ Ω. Let p ≥ T > g : R → R , j : Γ → R , j : Ω → R , f : [0 , T ] × Ω → R and f : [0 , T ] × Ω → R be given. We formulatetwo problems. Problem P . Find u : [0 , T ] × Ω → R such that u ′′ − α div (cid:0) |∇ u ′ | p − ∇ u ′ (cid:1) + g ( u ′ ) − ∆ u + | u | δ u = f in Ω × (0 , T ) ,u = 0 on Γ , ∂u∂ν + ν · (cid:0) |∇ u ′ | p − ∇ u ′ (cid:1) = η ∈ ∂j ( γu ′ ) on Γ × (0 , T ) ,u (0) = u , u ′ (0) = v . Problem P . Find u : [0 , T ] × Ω → R such that u ′′ − α div (cid:0) |∇ u ′ | p − ∇ u ′ (cid:1) + g ( u ′ ) − ∆ u + | u | δ u + γ ∗ η = f in Ω × (0 , T ) ,η ∈ ∂j ( u ′ ) in Ω × (0 , T ) ,u = 0 on ∂ Ω × (0 , T ) ,u (0) = u , u ′ (0) = v . In the above problems ∂j i denotes the Clarke subdifferential of the function j i , i = 1 , α i > δ − p .We impose the following assumptions on the functions g , j and j . H ( g ) g : R → R is such that(i) g is continuous,(ii) inf s ∈ R g ( s ) s > −∞ ,(iii) | g ( s ) | c g (1 + | s | p − ) for all s ∈ R with c g > H ( j ) j : Γ × R → R is such that(i) j ( · , ξ ) is measurable for all ξ ∈ R and j ( · , ∈ L (Γ ),(ii) j ( x, · ) is locally Lipschitz for a.e. x ∈ Γ ,(iii) | η | c j (1 + | ξ | p − ) for all η ∈ ∂j ( x, ξ ), x ∈ Γ with c j > .H ( j ) j : Ω × R → R is such that(i) j ( · , ξ ) is measurable for all ξ ∈ R and j ( · , ∈ L (Ω),23ii) j ( x, · ) is locally Lipschitz for a.e. x ∈ Ω,(iii) | η | c j (1 + | ξ | p − ) for all η ∈ ∂j ( x, ξ ), x ∈ Ω with c j > . We introduce the spaces W = { v ∈ W ,p (Ω) , v = 0 on Γ } and W = W ,p (Ω)equipped with the norm k v k W = k v k W = (cid:18)Z Ω |∇ v ( x ) | p dx (cid:19) p . Moreover, we define spaces V = { v ∈ H (Ω) : v = 0 on Γ } , V = H (Ω), H = L (Ω), equipped with the norm k v k V = k v k V = (cid:18)Z Ω |∇ v ( x ) | dx (cid:19) . Finally, we take U = L p (Γ ) and U = L p (Ω). Next, we consider operators A : W → W ∗ , A : W → W ∗ , B : V → W ∗ and B : V → W ∗ defined by h A u, v i W ∗ × W = α Z Ω |∇ u | p − ∇ u · ∇ v dx + Z Ω g ( u ) · v dx for all u, v ∈ W , h A u, v i W ∗ × W = α Z Ω |∇ u | p − ∇ u · ∇ v dx + Z Ω g ( u ) · v dx for all u, v ∈ W , h B u, v i W ∗ × W = Z Ω ∇ u · ∇ vdx + Z Ω | u | δ u · v dx for all u ∈ V , v ∈ W , h B u, v i W ∗ × W = Z Ω ∇ u · ∇ vdx + Z Ω | u | δ u · v dx for all u ∈ V , v ∈ W . We define the spaces W = L p (0 , T ; W ), W = L p (0 , T ; W ), V = L p (0 , T ; V ), H = L (0 , T ; H ), U = L p (0 , T ; U ) and U = L p (0 , T ; U ).We need the following assumptions on the right hand side of Problems P and P . H ( f ): f ∈ W . H ( f ): f ∈ W .We define the functionals F ∈ W ∗ and F ∈ W ∗ by F ( v ) = Z Ω f · v dx for all v ∈ W , F ( v ) = Z Ω f · v dx for all v ∈ W . Now we introduce the notion of a weak solution of Problems P and P . Definition 6.1.
A function u ∈ W is said to be a weak solution of Problem P if u ′ ∈ W , u ′′ ∈ W ∗ and satisfies h u ′′ ( t ) + A u ′ ( t ) + B u ( t ) , v i W ∗ × W + R Γ η ( x ) v ( x ) d Γ = F ( v ) for a.e. t ∈ (0 , T ) , for all v ∈ W ,η ( x ) ∈ ∂j ( u ′ ( x )) for a.e. x ∈ Γ ,u (0) = u , u ′ (0) = v . efinition 6.2. A function u ∈ W is said to be a weak solution of Problem P if u ′ ∈ W , u ′′ ∈ W ∗ and satisfies h u ′′ ( t ) + A u ′ ( t ) + B u ( t ) , v i W ∗ × W + R Ω η ( x ) v ( x ) dx = F ( v ) for a.e. t ∈ (0 , T ) , for all v ∈ W ,η ( x ) ∈ ∂j ( u ′ ( x )) for a.e. x ∈ Ω ,u (0) = u , u ′ (0) = v . We remark that the weak formulations in Definitions 6.1 and 6.2 are obtainedfrom equations in Problems P and P , respectively, by multiplying them by atest function v ∈ W ( v ∈ W , respectively) and using the Green formula.In what follows we will deal with the existence of weak solutions of Prob-lems P and P . First we define two auxiliary functionals J : U → R and J : U → R given by J ( v ) = Z Γ j ( x, v ( x )) d Γ for all v ∈ U ,J ( v ) = Z Ω j ( x, v ( x )) dx for all v ∈ U . Next, we define the multifunctions M : U → U ∗ and M : U → U ∗ givenby M i ( v ) = ∂J i ( v ) for all v ∈ U i , i = 1 ,
2. Finally, let γ : W → U denote thetrace operator and γ : W → U the embedding operator. Now, we formulatetwo auxiliary problems. Problem P . Find u ∈ W with u ′ ∈ W and u ′′ ∈ W ∗ such that u ′′ ( t ) + A ( u ′ ( t )) + B ( u ( t )) + γ ∗ M ( γ u ′ ( t )) ∋ f ( t ) a.e. t ∈ [0 , T ] ,u (0) = u , u ′ (0) = v . Problem P Find u ∈ W with u ′ ∈ W and u ′′ ∈ W ∗ such that u ′′ ( t ) + A ( u ′ ( t )) + B ( u ( t )) + γ ∗ M ( γ u ′ ( t )) ∋ f ( t ) a.e. t ∈ [0 , T ] ,u (0) = u , u ′ (0) = v . Remark 6.3.
By the properties of Clarke subdifferential of functionals J i , itfollows that each solution of Problem P i is also a solution of Problem P i , i = 1 , . We recall that the following Poincare inequalities hold Z Ω | v ( x ) | p dx ≤ ˜ c Z Ω |∇ v ( x ) | p dx for all v ∈ W , (6.1) Z Ω | v ( x ) | p dx ≤ ˜ c Z Ω |∇ v ( x ) | p dx for all v ∈ W (6.2)25ith ˜ c , ˜ c >
0. Let us define the following constants c A = max (cid:8) c q ˜ c p | Ω | q , α + c g ˜ c pq (cid:9) ,c A = max (cid:8) c q ˜ c p | Ω | q , α + c g ˜ c pq (cid:9) ,c M = c j p max (cid:8) , | Γ | q (cid:9) ,c M = c j p max (cid:8) , | Ω | q (cid:9) , where | Γ | and | Ω | denote the surface measure of Γ and the Lebesgue measureof Ω, respectively. Now we formulate lemmata containing the properties of theoperators A , A , M and M . Lemma 6.4.
If assumption H ( g ) holds, then operator A satisfies(i) k A u k W ∗ ≤ c A (1 + k u k p − W ) for all u ∈ W ,(ii) h A u, u i W ∗ × W ≥ α k u k pW + inf s ∈ R g ( s ) s | Ω | for all u ∈ W ,(iii) A is pseudomonotone.Proof. Condition ( i ) follows from H ( g )( iii ) and (6.1). Condition ( ii ) followsdirectly from the definition of A and H ( g )( ii ). Finally, for the pseudomono-tonicity of A , we refer to Chapter 2 of [14]. Lemma 6.5.
If assumption H ( g ) holds, then operator A satisfies(i) k A u k W ∗ ≤ c A (1 + k u k p − W ) for all u ∈ W ,(ii) h A u, u i W ∗ × W ≥ α k u k pW + inf s ∈ R g ( s ) s | Ω | for all u ∈ W ,(iii) A is pseudomonotone. The proof of Lemma 6.5 is analogous to the proof of Lemma 6.4.
Lemma 6.6.
If assumption H ( j ) holds, then operator M satisfies(i) for all u ∈ U , M ( u ) is a nonempty, closed and convex set,(ii) M is upper semicontinuous in ( s - U × w - U ∗ ) -topology,(iii) k η k U ∗ c M (1 + k w k p − U ) for all w ∈ U , all η ∈ M ( w ) . Lemma 6.7.
If assumption H ( j ) holds, then operator M satisfies(i) for all u ∈ U , M ( u ) is a nonempty, closed and convex set,(ii) M is upper semicontinuous in ( s - U × w - U ∗ ) -topology,(iii) k η k U ∗ c M (1 + k w k p − U ) for all w ∈ U , all η ∈ M ( w ) . Let γ i : W i → U i be Nemytskii operator corresponding to γ i defined by( γ i v )( t ) = γ i v ( t ) for all v ∈ W i , i = 1 ,
2. The following lemmata deal with theproperties of γ and γ . 26 emma 6.8. The Nemytskii operator γ : M p,q (0 , T ; W , W ∗ ) → L q (0 , T ; U ∗ ) is compact.Proof. Let ε ∈ (0 , ). Then the embedding i : W → W − ε,p (Ω) is compact.The trace operator ˜ γ : W − ε,p (Ω) → W − ε,p ( ∂ Ω) is linear and continuous and,finally, the embedding j : W − ε,p ( ∂ Ω) → L p ( ∂ Ω) = U is also linear and con-tinuous. Thus γ = j ◦ ˜ γ ◦ i is linear, continuous and compact. Moreover,the spaces V ⊂ W − ε,p (Ω) ⊂ V ∗ satisfy assumptions of Proposition 2.5 sothe embedding M p,q (0 , T ; V , V ∗ ) ⊂ L p (0 , T ; W − ε,p (Ω)) is compact. Since theembedding L p (0 , T ; W − ε,p (Ω)) ⊂ U is continuous the Nemytskii operator cor-responding to γ is compact. Lemma 6.9.
The Nemytskii operator γ : M p,q (0 , T ; W , W ∗ ) → L q (0 , T ; U ∗ ) is compact.Proof. Use directly Proposition 2.5 to the triple of spaces W , U ∗ and W ∗ .Now we impose additional assumptions on the constants of the problems. H : α > c M k γ k p L ( W ,U ) , H : α > c M k γ k p L ( W ,U ) .We are in a position to formulate the existence results for Problems P and P . Theorem 6.10.
Let assumptions H ( g ) , H ( j ) , H ( f ) , H ( H ( j ) , H ( f ) , H ,respectively) hold and u ∈ V, v ∈ H . Then Problem P (Problem P , respec-tively) admits a weak solution.Proof. We apply Theorem 5.2 to Problems P and P . To this end we observethat Lemmata 6.4 and 6.5 imply that operators A and A satisfy assumptionscorresponding to H ( A ). It is also clear that both operators B and B canbe represented as a sum of linear term B and nonlinear one C , which satisfyassumptions corresponding to H ( B ) and H ( C ) (see example in [6]). Moreover,Lemmata 6.6 and 6.7 provide that the multivalued operators M and M satisfyassumptions analogous to H ( M ). Similarly, Lemmata 6.8 and 6.9 guaranty thatassumption H ( γ ) is fulfilled in case of operators γ and γ . Finally, assumptions H and H are analogous to assumption H of Theorem 5.2. Including assump-tions H ( f ) and H ( f ), we are in a position to use Theorem 5.2 and obtain theexistence of solution to Problems P and P . From Remark 6.3 and Definitions6.1 and 6.2, we get that Problems P and P admit weak solutions. References [1] J.P. Aubin, H. Frankowska, Set-Valued Analysis, Birkh¨auser, Boston,Basel, Berlin (1990). 272] K. Bartosz, Numerical Methods for Evolution Hemivariational Inequalities,Chapter 5 in W. Han, S. Mig´orski, M. Sofonea,
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