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Variational inequalities with time-dependent constraints in Lp
Nonlinear Analysis 73 (2010) 390–398
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Variational inequalities with time-dependent constraints in Lp Masahiro Kubo Nagoya Institute of Technology, Nagoya, 466-8555, Japan
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Article history: Received 7 November 2009 Accepted 24 March 2010 MSC: 35K85 35K90
abstract We propose an abstract variational inequality with time-dependent constraints in a Banach space with a uniformly convex dual and establish the existence, uniqueness, and regularity of the solution thereof. The abstract result is applied to concrete variational inequalities with time-dependent constraints, thereby obtaining the Lp -regularity of solutions. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Parabolic variational inequalities Abstract parabolic evolution equations
1. Introduction Let X be a real Banach space whose dual X ∗ is uniformly convex, and H be a real Hilbert space such that X ⊂ H with continuous and dense embedding. We study a variational inequality: u(t ) ∈ K (t ) for all t ∈ (0, T ) and
(u0 (t ) + A(t )u(t ) − f (t ), u(t ) − z ) ≤ 0
(1.1)
for all z ∈ K (t ) and a.e. t ∈ (0, T ). Here A(t ) and K (t ) are a time-dependent m-accretive operator in X and a time-dependent closed and convex set in H, respectively; f is a given X -valued function; (·, ·) is the inner product of H; and u is the X -valued function to be found. Parabolic variational inequalities were introduced by Lions and Stampacchia [1] in a framework of evolution equations in a Hilbert space and with a time-independent constraint K (t ) ≡ K . Two directions have been pursued since the pioneering work of Lions and Stampacchia. One is the Lp (or Banach space) framework, and the other is time-dependent constraints. Both kinds of generalizations are important with respect to both applications and theoretical interests [2,3]. Brézis [4] studied the problem (1.1) in Lp with time-independent constraints by applying Kato’s theory [5] of nonlinear evolution equations in Banach spaces with uniformly convex duals. Abstract evolution equations applicable to variational inequalities with time-dependent constraints are of course well developed in a Hilbert space framework. We refer the reader to, for instance, [6–11]. To the best of our knowledge, there are no abstract results for parabolic evolution equations that can be applied to variational inequalities in Lp with time-dependent constraints. For concrete problems, we refer the reader to [3,12–15], amongst others. This paper aims to fill the gap and establish a unified and abstract framework of nonlinear evolution equations in a Banach space with a uniformly convex dual to deal with variational inequalities in Lp with time-dependent constraints. We formulate the variational inequality (1.1) in the form of the following evolution equation: u0 (t ) + A(t )u(t ) + B(t )u(t ) 3 f (t )
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.03.028
(1.2)
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
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with a time-dependent m-accretive operator B(t ). Furthermore, we impose the initial condition u(0) = u0
(1.3)
with a given initial value u0 . To derive the variational inequality (1.1) from (1.2), we set B(t ) := ∂ IK (t ) |X ×X , where ∂ IK (t ) : H → H is the subdifferential of the indicator function IK (t ) of K (t ) and the notation |X ×X denotes that the graph of B(t ) is the intersection of that of ∂ IK (t ) and X × X . Our main result (Theorem 2.3) confirms the existence and uniqueness of a solution to {(1.2) and (1.3)} satisfying B(·)u ∈ Lp (0, T ; X ). Then, with the help of an Lp -estimate for the parabolic operator d/dt + A(t ), we obtain Lp -estimates of the solution (Corollary 2.4, Propositions 5.3 and 5.6). See Remark 5.5 for a comparison with the method of Brézis [4]. We note that our result can be applied to problems in which (Remark 5.4)
∩0≤t ≤T D(A(t ) + B(t )) = ∅. The Banach space theory of abstract nonlinear evolution equations established thus far (e.g., [16,17,5]) is not applicable to such problems. The main theorem (Theorem 2.3) and its corollary (Corollary 2.4) are stated in Section 2 and proved in Section 3 by assuming uniform estimates of approximate solutions (Proposition 3.1) to be proved in Section 4. The abstract results in Section 2 are applied to concrete problems in Lp with time-dependent constraints in Section 5. Combining the time dependence of A(t ) and B(t ), we can deal with a variety of problems with time-dependent constraints. For instance, variational inequalities in a non-cylindrical domain are, to the best of our knowledge, dealt with in an Lp -framework for the first time in this paper (Section 5.3). Our idea is to combine time-dependent subdifferential evolution equations [6–10] with the Lp -theory of variational inequalities with time-independent constraints [4,18]. In the basic assumptions given below, we assume that the operators A(t ) and B(t ) are m-accretive in both X and H. In the main assumptions stated in the next section, we assume that the mappings t 7→ A(t ) and t 7→ B(t ) are smooth in H (Condition (A)) and X (Condition (B)-(ii)), respectively, in similar ways to [6–10], and that operators A(t ) and B(t ) are compatible in an appropriate sense (Condition (C)), which is implicit in [4,18]. Notation and basic assumptions The norms of X , X ∗ , and H are denoted by | · |X , | · |X ∗ , and | · |H , respectively. The inner product of H and the duality between X ∗ and X are denoted by (·, ·) and h·, ·i, respectively. Let p ≥ 2 and F : X → X ∗ be the duality map with gauge ξ (r ) = r p−1 ; that is [4,18], p−1
p
Fz := {z ∗ ∈ X |hz ∗ , z i = |z |X , |z ∗ |X ∗ = |z |X
}.
∗
By the uniform convexity of X , the duality map F is single-valued and uniformly continuous on each bounded set in X ([17, Lemma 1.2], [19, Proposition II.8.8]). This property is used to prove a key lemma of our argument (see Lemma 4.2). Let AX (t ) and AH (t ), t ∈ [0, T ], be time-dependent families of m-accretive operators in X and H, respectively, such that AX (t ) = AH (t )|X ×X . In other words, the graph of AX (t ) is the intersection of that of AH (t ) and X × X . Assume that AH (t ) = ∂ϕ t , where ∂ϕ t is the subdifferential of a proper, l.s.c. (lower-semicontinuous) and convex function ϕ t : H → R ∪ {+∞} such that ϕ t (z ) ≥ 0 for all t ∈ [0, T ] and z ∈ H. Let BX (t ) and BH (t ) be time-dependent m-accretive operators in X and H, respectively, such that BX (t ) = BH (t )|X ×X . Assume that BH (t ) = ∂ IK (t ) , where ∂ IK (t ) is the subdifferential of the indicator function IK (t ) of a closed and convex set K (t ) in H for all t ∈ [0, T ]. For λ > 0, the Yosida approximations of these operators are denoted by AX ,λ (t ), AH ,λ (t ), BX ,λ (t ), and BH ,λ (t ), that is, AX ,λ (t ) := AH ,λ (t ) := BX ,λ (t ) := BH ,λ (t ) :=
1 IX − JAX ,λ (t ) ,
JAX ,λ (t ) := IX + λAX ,λ (t )
1 IH − JAH ,λ (t ) ,
JAH ,λ (t ) := IH + λAH ,λ (t )
1
JBX ,λ (t ) := IX + λBX ,λ (t )
λ
λ λ
,
−1
,
−1
IX − JBX ,λ (t ) ,
1
λ
−1
IH − JBH ,λ (t ) ,
JBH ,λ (t ) := IH + λBH ,λ (t )
,
−1
,
where IX and IH are the identity maps on X and H, respectively. Note that we have AX ,λ (t ) = AH ,λ (t )|X and BX ,λ (t ) = BH ,λ (t )|X . Furthermore, set IKt ,λ (z ) :=
λ p
p
BX ,λ (t )z X =
1 pλ
p−1
z − JB
X ,λ
p (t )z X
and let ϕλt be the regularization of ϕ t for λ > 0. It is well-known that its subdifferential ∂ϕλt coincides with the Yosida approximation AH ,λ (t ) of AH (t ) = ∂ϕ t .
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M. Kubo / Nonlinear Analysis 73 (2010) 390–398
We assume that the mappings t 7→ ϕ t (z ) and t 7→ K (t ) are continuous in the Mosco sense [20], and hence, the function t 7→ ϕλt (z ) and the mappings t 7→ AH ,λ (t )z and t 7→ BH ,λ (t )z are continuous for all λ > 0 and z ∈ H. We also assume that the mappings t 7→ AX ,λ (t )z and t 7→ BX ,λ (t )z are continuous for all λ > 0 and z ∈ X . Then, the function t 7→ IKt ,λ (z ) is also continuous for all λ > 0 and z ∈ X . We refer the reader to [19] for the notions and elementary properties of duality mappings, accretive operators, subdifferentials and Yosida approximations. 2. Main theorem The main assumptions for the given data are listed below. (A) The function t 7→ ϕλt (z ) is of bounded variation and its positive variation is absolutely continuous on [0, T ] for all λ > 0 and z ∈ H. There are functions pA ∈ L2 (0, T ) and qA ∈ L1 (0, T ) such that for all λ > 0 and z ∈ H, the following inequality holds for a.e. t ∈ (0, T ): d dt
1/2 AH ,λ (t )z + qA (t ) ϕ t (z ) + 1 . ϕλt (z ) ≤ pA (t ) ϕλt (z ) + 1 λ H
(B) (i) For all t ∈ [0, T ], λ > 0, z , y ∈ X the following inequalities hold:
hFBX ,λ (t )z , y − z i ≤ IKt ,λ (y) − IKt ,λ (z ),
(2.1)
hFBX ,λ (t )z , JBX ,λ (t )y − JBX ,λ (t )z i ≤ 0.
(2.2)
(ii) The function t 7→ IKt ,λ (z ) is of bounded variation and its positive variation is absolutely continuous on [0, T ] for all λ > 0 and z ∈ X . There are functions pB ∈ Lp (0, T ) and qB ∈ L1 (0, T ) such that for all λ > 0 and z ∈ X , the following inequality holds for a.e. t ∈ (0, T ): d dt
p−1
IKt ,λ (z ) ≤ pB (t ) BX ,λ (t )z X
+ qB (t ). ∗
(C) There are functions rX ∈ Lp (0, T ) and rH ∈ Lp (0, T ), p−1 + (p∗ )−1 = 1, such that for all t ∈ [0, T ], λ > 0 and z ∈ X the following inequalities hold:
(BX ,λ (t )z , AX ,λ (t )z ) ≥ −rH (t ) BX ,λ (t )z X , p−1 hFBX ,λ (t )z , AX ,λ (t )z i ≥ −rX (t ) BX ,λ (t )z X .
(2.3) (2.4)
(D) u0 ∈ D(AX (0)) ∩ K (0), f ∈ L (0, T ; X ). p
We denote by (P) the problem {(1.2) and (1.3)} and give two alternative solutions thereof. Definition 2.1. A function u : [0, T ] → H is called an H-solution of (P) iff the following conditions are satisfied. (a) u ∈ W 1,2 (0, T ; H ) and sup0≤t ≤T ϕ t (u(t )) < +∞. (b) There are functions u∗ , u∗∗ ∈ L2 (0, T ; H ) such that u∗ (t ) ∈ AH (t )u(t ) and u∗∗ (t ) ∈ BH (t )u(t ) for a.e. t ∈ (0, T ), and the following equation is satisfied: u0 (t ) + u∗ (t ) + u∗∗ (t ) = f (t )
in H a.e. t ∈ (0, T ).
(c) u(0) = u0 in H. Definition 2.2. A function u : [0, T ] → X is called an X -solution of (P) iff the following conditions are satisfied. (a) u ∈ W 1,p (0, T ; X ) and sup0≤t ≤T ϕ t (u(t )) < +∞. (b) There are functions u∗ , u∗∗ ∈ Lp (0, T ; X ) such that u∗ (t ) ∈ AX (t )u(t ) and u∗∗ (t ) ∈ BX (t )u(t ) for a.e. t ∈ (0, T ), and the following equation is satisfied: u0 (t ) + u∗ (t ) + u∗∗ (t ) = f (t )
in X a.e. t ∈ (0, T ).
(c) u(0) = u0 in X . Now we state our main theorem. Theorem 2.3. Assume that Conditions (A)–(D) are satisfied. Then, there exists a unique H-solution u of problem (P) such that u∗∗ ∈ Lp (0, T ; X ). The following corollary is useful for deriving Lp -regularity.
(2.5)
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
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Corollary 2.4. Assume that Conditions (A)–(D) and (E) below are satisfied. (E) For each v0 ∈ D(AX (0)) and g ∈ Lp (0, T ; X ), there exists a unique function v ∈ W 1,p (0, T ; X ) such that
v 0 (t ) + AX (t )v(t ) 3 g (t ) in X a.e. t ∈ (0, T ), v(0) = v0 in X . Then, there exists a unique X -solution u of problem (P). We can derive the corollary from Theorem 2.3 using Condition (E) for v0 = u0 and g := f − u∗∗ and the monotonicity of AH (t ). In applying the main theorem to concrete problems, Condition (E) refers to the Lp -solvability of the parabolic PDE related to the operator A(t ). Theorem 2.3 is proved in the next two sections. 3. Proof of Theorem 2.3 For each λ > 0, we consider the following approximate problem for (P): u0λ (t ) + AX ,λ (t )uλ (t ) + BX ,λ (t )uλ (t ) = f (t ) in X a.e. t ∈ (0, T ),
(3.1)
uλ (0) = u0 .
(3.2)
We denote by (P)λ the problem {(3.1) and (3.2)}. It is well-known that the Yosida approximations AX ,λ (t ) and BX ,λ (t ) are Lipschitz continuous operators in X . Therefore, by a standard argument, there exists a unique solution uλ ∈ W 1,p (0, T ; X ) of (P)λ . Moreover, we have the following uniform estimate, which is proved in Section 4. Proposition 3.1. There exists a constant M0 > 0 such that for all λ > 0 the following holds:
|u0λ |L2 (0,T ;H ) + sup ϕλt (uλ (t )) + |BX ,λ (·)uλ |Lp (0,T ;X ) ≤ M0 . 0 ≤t ≤T
By a standard argument of monotonicity (cf. [10, (4.13), (4.14)]), we have 1 2
|uλ (t ) − uµ (t )|2H ≤
λ−µ
Z
2
t
|AH ,µ uµ |2H − |AH ,λ uλ |2H + |BH ,µ uµ |2H − |BH ,λ uλ |2H dτ .
0
Therefore, by Proposition 3.1, there is a sequence uλn such that u0λn → u0 uλn → u
weakly in L2 (0, T ; H ), in C ([0, T ]; H ),
AH ,λn (·)uλ (= AX ,λn (·)uλ ) → u∗
weakly in L2 (0, T ; H ),
BH ,λn (·)uλ (= BX ,λn (·)uλ ) → u
weakly in Lp (0, T ; X )
∗∗
for some u, u∗ , and u∗∗ . Then, by the demi-closedness of m-accretive operators, we can show that the limit function u is a unique H-solution of (P) with the property (2.5). Thus, the proof of Theorem 2.3 will be complete when Proposition 3.1 is proved, which is accomplished in the next section. 4. Proof of uniform estimates We prepare two lemmas derived from Conditions (A) and (B)-(ii), respectively. Lemma 4.1 ([6, Lemma 2], [7, Proposition 1], [10, Proposition 3.2], [9, Lemma 1.2.5], [11, Lemma 3.4]). For each v ∈ W 1,1 (0, T ; H ) and λ > 0, the function t 7→ ϕλt (v(t )) is of bounded variation, its positive variation is absolutely continuous on [0, T ], and the following inequality holds for a.e. t ∈ (0, T ): d dt
1/2 AH ,λ (t )v(t ) + qA (t ) ϕ t (v(t )) + 1 . ϕλt (v(t )) − (AH ,λ (t )v(t ), v 0 (t )) ≤ pA (t ) ϕλt (v(t )) + 1 λ H
Lemma 4.2. For each v ∈ W 1,1 (0, T ; X ) and λ > 0, the function t 7→ IKt ,λ (v(t )) is of bounded variation, its positive variation is absolutely continuous on [0, T ], and the following inequality holds for a.e. t ∈ (0, T ): d dt
p−1
IKt ,λ (v(t )) − hFBX ,λ (t )v(t ), v 0 (t )i ≤ pB (t ) BX ,λ (t )v(t ) X
+ qB (t ).
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M. Kubo / Nonlinear Analysis 73 (2010) 390–398
Lemma 4.2 is proved at the end of this section. For a proof of Lemma 4.1 we refer the reader to the cited references. Proof of Proposition 3.1. For simplicity, we write u for uλ . Since u(t ) ∈ X , we have AH ,λ (t )u(t ) = AX ,λ (t )u(t ). Therefore, multiplying (3.1) by AH ,λ (t )u(t ), we obtain
(u0 (t ), AH ,λ (t )u(t )) + |AH ,λ (t )u(t )|2H + (BX ,λ (t )u(t ), AH ,λ (t )u(t )) = (f (t ), AH ,λ (t )u(t )). Then, applying Lemma 4.1 and using Condition (2.3) in (C), we have d
1/2 AH ,λ (t )u(t ) ϕλt (u(t )) + |AH ,λ (t )u(t )|2H − rH (t ) BX ,λ (t )u(t ) X ≤ pA (t ) ϕλt (u(t )) + 1 H dt t + qA (t ) ϕλ (u(t )) + 1 + (f (t ), AH ,λ (t )u(t )). Hence, by the Schwarz and Hölder inequalities, we conclude that d dt
p 2p 1 1 ∗ ϕλt (u(t )) + |AH ,λ (t )u(t )|2H ≤ (pA (t )2 + qA (t )) ϕλt (u(t )) + 1 + |f (t )|2H + p BX ,λ (t )u(t ) X + ∗ rH (t )p . (4.1) 2
2 p
p
Next, noting that F (−z ) = −F (z ), we have by (3.1) F (u0 (t ) + AX ,λ (t )u(t ) − f (t )) + F (BX ,λ (t )u(t )) = 0. Multiplying u0 (t ) + AX ,λ (t )u(t ) − f (t ), we obtain
|u0 (t ) + AX ,λ (t )u(t ) − f (t )|pX + hF (BX ,λ (t )u(t )), u0 (t )i + hF (BX ,λ (t )u(t )), AX ,λ (t )u(t )i = hF (BX ,λ (t )u(t )), f (t )i. Therefore, applying Lemma 4.2 and using Condition (2.4) in (C), we have d
p
dt
p−1
IKt ,λ (u(t )) + |u0 (t ) + AX ,λ (t )u(t ) − f (t )|X − rX (t ) BX ,λ (t )u(t ) X
p−1 p−1 ≤ pB (t ) BX ,λ (t )u(t ) X + qB (t ) + BX ,λ (t )u(t ) X |f (t )|X . Hence, by Hölder’s inequality (p−1 + (p∗ )−1 = 1) and Eq. (3.1), we conclude that d dt
IKt ,λ (u(t )) +
1 p
1 |u0 (t ) + AX ,λ (t )u(t ) − f (t )|pX ≤ P (t ) := (pB (t ) + |f (t )|X + rX (t ))p + qB (t ) .
(4.2)
p
Now, adding inequalities (4.1) and (4.2), and using Eq. (3.1), we obtain d dt
ϕλt (u(t )) +
d dt
IKt ,λ (u(t )) +
1 2
1 1 0 |AH ,λ (t )u(t )|2H + 1 − p |u (t ) + AX ,λ (t )u(t ) − f (t )|pX ≤ Q (t )ϕλt (u(t )) + R(t ), 2
p
where Q (t ) := pA (t )2 + qA (t ),
2p ∗ R(t ) := Q (t ) + |f (t )|2H + ∗ rH (t )p + P (t ). p
Hence, by Gronwall’s lemma we see that there is a constant M0 > 0 independent of λ > 0, such that p
sup ϕλt (u(t )) + sup IKt ,λ (u(t )) + |AH ,λ (·)u|2L2 (0,T ;H ) + |u0 + AX ,λ (·)u|Lp (0,T ;X ) ≤ M0 .
0≤t ≤T
0≤t ≤T
Thus, Proposition 3.1 is proved and the proof of Theorem 2.3 is completed.
Proof of Lemma 4.2. Applying Conditions (B)-(i) for t := t + h, z := v(t + h), y := v(t ) and (B)-(ii) for z := v(t ), we obtain for all 0 ≤ t < t + h ≤ T , IKt +,λh (v(t + h)) − IKt ,λ (v(t )) = IKt +,λh (v(t + h)) − IKt +,λh (v(t )) + IKt +,λh (v(t )) − IKt ,λ (v(t ))
≤ hFBX ,λ (t + h)v(t + h), v(t + h) − v(t )i +
t +h
Z t
n
p−1
pB (τ ) BX ,λ (τ )v(t ) X
o + qB (τ ) dτ .
From this inequality, we see that the function t 7→ (v(t )) is of bounded variation and its positive variation is absolutely continuous. Furthermore, dividing both sides of this inequality by h and letting h ↓ 0, we obtain the desired inequality. Here, we need the continuity of the duality map F . IKt ,λ
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
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5. Obstacle problems in L p 5.1. Lemmas The following lemmas are useful for verifying Conditions (A) and (2.3) in (C), respectively. Lemma 5.1 ([9–11]). Assume that the family ϕ t of convex functions satisfies the following condition: (F) There are functions pˆ A ∈ W 1,2 (0, T ) and qˆ A ∈ W 1,1 (0, T ) such that for all 0 ≤ s < t ≤ T and z ∈ D(ϕ s ) there exists z˜ ∈ D(ϕ t ) satisfying the following inequalities:
1/2 |˜z − z |H ≤ |ˆpA (t ) − pˆ A (s)| ϕ s (z ) + 1 , ϕ t (˜z ) − ϕ s (z ) ≤ |ˆqA (t ) − qˆ A (s)| ϕ s (z ) + 1 . Then, the family A(t ) of operators satisfies Condition (A). Lemma 5.2 ([21, Lemma 5.1]). Let A be a positive self-adjoint operator on H, and let K be a closed convex set in H. Assume that there exists g ∈ H such that
(I + λA)−1 (z + λg ) ∈ K for all λ > 0 and z ∈ K . Then, we have
(∂ IK ,λ (z ), Aλ z ) ≥ (∂ IK ,λ (z ), (I + λA)−1 g ), where Aλ and ∂ IK ,λ are Yosida approximations of A and of the subdifferential ∂ Iλ of the indicator function IK of K , respectively. We refer to the cited references for the proofs of these lemmas. 5.2. Time-dependent bi-lateral constraints p Let Ω ⊂ RN (N ≥ 1) be a bounded set with smooth boundary ∂ Ω and set X := LP (Ω ) (p ≥ 2) and H := L2 (Ω ). Let 1 2 N aij ∈ C ([0, T ] × Ω ) (i, j = 1, . . . , N ) be such that aij = aji and for a constant α > 0 : i,j aij ξi ξj ≥ α|ξ | (ξ ∈ R ). Define
a differential operator L(t ) by (∂i := ∂∂x ) i L(t )z := −
X
∂i aij (t )∂j z .
i,j
Now define ϕ : H → R ∪ {+∞} by t
Z X 1 aij (t , x)∂i z (x)∂j z (x)dx, ϕ t (z ) := 2 Ω i,j +∞,
if z ∈ H01 (Ω ), otherwise.
Then, we have AH (t )z := ∂ϕ t (z ) = L(t )z
for z ∈ D(AH (t )) = H 2 (Ω ) ∩ H01 (Ω )
and AX (t )z := AH (t )|X ×X = L(t )z
1,p
for z ∈ D(AX (t )) = W 2,p (Ω ) ∩ W0 (Ω ).
It is well-known that the operators AH (t ) and AX (t ) are m-accretive in H and X , respectively. We can verify Condition (A) with the aid of Lemma 5.1. In fact, we can take z˜ = z in verifying Condition (F) (see [7]). Next, let ψi ∈ W 1,p (0, T ; X ) ∪ Lp (0, T ; W 2,p (Ω )) (i = 1, 2) be such that
ψ1 ≤ ψ2 in (0, T ) × Ω ,
ψ1 ≤ 0 ≤ ψ2 on (0, T ) × ∂ Ω
and set K (t ) := {z ∈ H | ψ1 (t ) ≤ z ≤ ψ2 (t ) in Ω }. Then, set BH (t ) := ∂ IK (t ) and BX (t ) := BH (t )|X ×X , where ∂ IK (t ) is the subdifferential of the indicator function IK (t ) of the closed and convex set K (t ) in H. We have for all λ > 0 and z ∈ X BX ,λ (t )z = z1,λ + z2,λ
and
JBX ,λ (t )z = z − λz1,λ − λz2,λ ,
(5.1)
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M. Kubo / Nonlinear Analysis 73 (2010) 390–398
where 1 z1,λ := − [z − ψ1 ]− ,
z2,λ :=
λ
1
λ
[z − ψ2 ]+ .
Also note that for all z ∈ X [4,18,19] Fz = |z |p−2 z and therefore, FBX ,λ (t )z = Fz1,λ + Fz2,λ ,
IKt ,λ (z ) = IKt ,λ (z1,λ ) + IKt ,λ (z2,λ ). With these identities and by a standard argument of obstacle problems, we can verify Conditions (2.1) and (2.2) in (B)-(i). To verify Condition (B)-(ii), we calculate for a.e. t ∈ (0, T ) d dt
IKt ,λ (z ) =
d
dt
λ p
|z1,λ |pX +
λ p
|z2,λ |pX
= hFz1,λ , −ψ1 (t )i + hFz2,λ , −ψ20 (t )i 0
≤ |z1,λ |pX−1 |ψ10 (t )|X + |z2,λ |pX−1 |ψ20 (t )|X ≤ pB (t )|BX ,λ (t )z |pX−1 , where pB (t ) := |ψ10 (t )|X + |ψ20 (t )|X . To verify inequality (2.3) in (C), we use Lemma 5.2 with A = AH (t ), z = zi,λ (see (5.1)), K = Ki (t ), and g = gi (i = 1, 2), where K1 (t ) := {z ∈ H | z ≥ ψ1 (t ) in Ω }, gi := L(t )ψi (t )
K2 (t ) := {z ∈ H | z ≤ ψ2 (t ) in Ω },
(i = 1, 2).
Finally, inequality (2.4) in (C) can be verified as follows (we write A(t ) and Aλ (t ) for AX (t ) and AX ,λ (t ), respectively). Set z1 := z ∨ ψ1 (t )
and
z2 := z ∧ ψ2 (t ).
Then for i = 1, 2
hFzλ,i , Aλ (t )z i = λhFzλ,i , Aλ (t )zλ,i i + hFzλ,i , Aλ (t )zi i = λhFzλ,i , Aλ (t )zλ,i i +
1
λ
hFzλ,i , zi − (I + λA(t ))−1 (zi + λgi (t ))i + hFzλ,i , (I + λA(t ))−1 gi (t )i
≥ hFzλ,i , (I + λA(t ))−1 gi (t )i. Here we use the accretivity of Aλ (t ) : hFz , Aλ (t )z i ≥ 0, and the property
(5.2)
(I + λA(t ))−1 (zi + λgi (t )) ∈ Ki (t ), which follows from the property zi ∈ Ki (t ).
(5.3)
Therefore, given that 1 ,p
u0 ∈ W 2,p (Ω ) ∩ W0 (Ω ) ∩ K (0),
f ∈ Lp (0, T ; Lp (Ω )),
we can derive the following proposition from Corollary 2.4 with the aid of the Lp -regularity theory of the parabolic operator ∂/∂ t + L(t ) [22, Chap. IV, Theorem 9.1]. Proposition 5.3. Under the above assumptions, there exists a unique solution u of the following problem: 1,p
(a) u ∈ W 1,p (0, T ; X ) ∩ C ([0, T ]; W0 (Ω )) ∩ Lp (0, T ; W 2,p (Ω )). (b) u(t ) ∈ K (t ) for all t ∈ [0, T ] and
(u0 (t ) + L(t )u(t ) − f (t ), u(t ) − z ) ≤ 0 for all z ∈ K (t ) and a.e. t ∈ (0, T ). (c) u(0) = u0 in X . Remark 5.4. When the obstacles are chosen in such a way that ψ1 (s) > ψ2 (t ) in E for some s, t ∈ [0, T ] and a set E ⊂ Ω with positive measure, we have K (s) ∩ K (t ) = ∅. Remark 5.5. In Brézis [4, Chap. II], problems with time-independent data L, ψ1 and ψ2 are dealt with, by first obtaining the property u0 ∈ L∞ (0, T ; X ) by applying Kato’s theory [5], then deriving a spatial Lp -estimate by the theory of elliptic variational inequality in [4, Chap. I] (or [18]) with the aid of an Lp -estimate for the elliptic operator L (cf. [4, (I.1.13)]). We note that our theory does not need such a condition as f 0 ∈ L1 (0, T ; Lp (Ω )), which was assumed in [4] to apply [5].
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
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5.3. Problems in a non-cylindrical domain Let Ω (t ) be a domain with smooth boundary such that Ω (t ) ⊂ Ω and where there is a family Φ t : Ω → Ω (t ∈ [0, T ]) of C 2 -diffeomorphisms satisfying Φ t (Ω (0)) = Ω (t ) and ∂t Φ t , ∂i Φ t , ∂i ∂j Φ t ∈ C ([0, T ] × Ω ). Now define ϕ t : H → R ∪ {+∞} by
ϕ t (z ) :=
Z 1 2
X Ω ( t ) i ,j
aij (t )∂i z (x)∂j z (x)dx,
+∞,
if z |Ω (t ) ∈ H01 (Ω (t )) and z = 0 in Ω \ Ω (t ), otherwise.
We can verify by a standard argument of subdifferentials that (AH (t ) := ∂ϕ t ) D(AH (t )) = {z ∈ H | z |Ω (t ) ∈ H 2 (Ω ) ∩ H01 (Ω ), z = 0 in Ω \ Ω (t )}, z ∗ ∈ AH (t )z if and only if z ∗ = L(t )z in Ω (t ) (z ∈ D(AH (t ))). Also, we can verify Condition (A) by applying Lemma 5.1 with z˜ := z ◦ Φ s ◦ Φ t (see [10,9,23]). Defining AX (t ) := AH (t )|X ×X , we can see with the aid of the Lp -solvability of the operator L(t ) in Ω (t ) that AX (t ) is m-accretive in X and that D(AX (t )) = {z ∈ D(AH (t ))| z |Ω (t ) ∈ W 2,p (Ω (t ))}. Next, let ψ ∈ W 1,p (0, T ; X ) ∩ Lp (0, T ; W 2,p (Ω )) be such that
ψ(t ) ≤ 0 in Ω \ Ω (t ) and set K (t ) := {z ∈ H | z ≥ ψ(t ) in Ω }. Now define BH (t ) := ∂ IK (t ) and BX (t ) := BH (t )|X ×X . Then, we have for all λ > 0 and z ∈ H 1 BH ,λ (t )z = zλ := − [z − ψ]− .
λ
With this identity and by a standard argument of obstacle problems as in Section 5.2, we can verify Condition (B). In addition, inequalities (2.3) and (2.4) in (C) can be verified, respectively, by applying Lemma 5.2 with z := zλ and g := L(t )ψ(t ) and by an argument similar to that of (5.2) and (5.3) (note that (I + λA(t ))−1 is linear, whereas A(t ) is not). Now, suppose that u0 ∈ D(AX (0)) ∩ K (0),
f ∈ Lp (0, T ; Lp (Ω )).
Transforming the problem into the cylindrical domain (0, T ) × Ω (0) using the diffeomorphism Φ t , we can derive the following proposition from Corollary 2.4 with the aid of [22, Chap. IV, Theorem 9.1]. Proposition 5.6. Under the above assumptions, there is a unique solution u of the following problem: 1 ,p
(a) u ∈ W 1,p (0, T ; X ) ∩ C ([0, T ]; W0 (Ω )) and T
Z 0
|u(t )|pW 2,p (Ω (t )) dt < +∞.
(b) u(t ) ∈ K (t ) and u(t ) = 0 in Ω \ Ω (t ) for all t ∈ [0, T ], and
Z Ω (t )
(u0 (t ) + L(t )u(t ) − f (t ))(u(t ) − z )dx ≤ 0
for all z ∈ K (t ) and a.e. t ∈ (0, T ). (c) u(0) = u0 in X .
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
J.L. Lions, G. Stampacchia, Variational inequalities, Commun. Pure Appl. Math. 20 (1967) 493–519. G. Duvaut, J.L. Lions, Inequalities in Mechanics and Physics, in: Grundlehren Math. Wiss., vol. 219, Springer, Berlin, Heidelberg, New York, 1976. A. Friedman, Variational principles and free-boundary problems, in: Pure Appl. Math, Wiley, New York, 1982. H. Brézis, Problèmes unilatéraux, J. Math. Pure Appl. IX. Ser. 51 (1972) 1–168. T. Kato, Accretive operators and nonlinear evolution equations in Banach spaces, in: Proc. Sympos. Pure Math. 18 Part 1, Nonlinear Funct. Anal. (1970) 138–161. H. Attouch, P. Bénilan, A. Damlamian, C. Picard, Equations d’évolution avec condition unilatérale, C. R. Acad. Sci. Paris Ser. A 279 (1974) 607–609. H. Attouch, A. Damlamian, Problemes d’evolution dans les Hilberts et applications, J. Math. Pure Appl. IX. Ser. 54 (1975) 53–74. N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math. 22 (1975) 304–331. N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Ed., Chiba Univ. Part II 30 (1981) 1–87. Y. Yamada, On evolution equations generated by subdifferential operators, J. Fac. Sci., Univ. Tokyo, Sect. IA 23 (1976) 491–515. M. Kubo, Characterization of a class of evolution operators generated by time-dependent subdifferentials, Funkc. Ekvacioj 32 (1989) 301–321. F. Donati, A penalty method approach to strong solutions of some nonlinear parabolic unilateral problems, Nonlinear Anal. 6 (1982) 585–597. S. Carl, V.K. Le, D. Motreanu, Evolutionary variational–hemivariational inequalities: existence and comparison results, J. Math. Anal. Appl. 345 (2008) 545–558. S. Carl, V.K. Le, D. Motreanu, Nonsmooth Variational Problems and their Inequalities: Comparison Principles and Applications, in: Springer Monographs in Mathematics, Springer, New York, 2007. S. Carl, V.K. Le, Sub–supersolution method for quasilinear parabolic variational inequalities, J. Math. Anal. Appl. 293 (2004) 269–284. M.G. Crandall, A. Pazy, Nonlinear evolution equations in Banach spaces, Israel J. Math. 11 (1972) 57–94. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Japan 19 (1967) 508–520. H. Brézis, G. Stampacchia, Sur la régularité de la solution d’inéquations elliptiques, Bull. Soc. Math. France 96 (1968) 153–180. R.E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, in: Math. Surveys Monogr., vol. 49, Amer. Math. Soc., Providence, 1997. H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, London, Melbourne, 1984. M. Kubo, Second order evolution equations with time-dependent subdifferentials, J. Evol. Equ. 7 (2007) 701–717. O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ural’tseva, Linear and Quasi-linear Equations of Parabolic Type, in: Transl. Math. Monogr., vol. 23, Amer. Math. Soc., Providence, 1968. N. Kenmochi, I. Pawlow, Parabolic–elliptic free boundary problems with time-dependent obstacles, Japan J. Appl. Math. 5 (1988) 87–121.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Variational inequalities with time-dependent constraints in Lp Masahiro Kubo Nagoya Institute of Technology, Nagoya, 466-8555, Japan
article
info
Article history: Received 7 November 2009 Accepted 24 March 2010 MSC: 35K85 35K90
abstract We propose an abstract variational inequality with time-dependent constraints in a Banach space with a uniformly convex dual and establish the existence, uniqueness, and regularity of the solution thereof. The abstract result is applied to concrete variational inequalities with time-dependent constraints, thereby obtaining the Lp -regularity of solutions. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Parabolic variational inequalities Abstract parabolic evolution equations
1. Introduction Let X be a real Banach space whose dual X ∗ is uniformly convex, and H be a real Hilbert space such that X ⊂ H with continuous and dense embedding. We study a variational inequality: u(t ) ∈ K (t ) for all t ∈ (0, T ) and
(u0 (t ) + A(t )u(t ) − f (t ), u(t ) − z ) ≤ 0
(1.1)
for all z ∈ K (t ) and a.e. t ∈ (0, T ). Here A(t ) and K (t ) are a time-dependent m-accretive operator in X and a time-dependent closed and convex set in H, respectively; f is a given X -valued function; (·, ·) is the inner product of H; and u is the X -valued function to be found. Parabolic variational inequalities were introduced by Lions and Stampacchia [1] in a framework of evolution equations in a Hilbert space and with a time-independent constraint K (t ) ≡ K . Two directions have been pursued since the pioneering work of Lions and Stampacchia. One is the Lp (or Banach space) framework, and the other is time-dependent constraints. Both kinds of generalizations are important with respect to both applications and theoretical interests [2,3]. Brézis [4] studied the problem (1.1) in Lp with time-independent constraints by applying Kato’s theory [5] of nonlinear evolution equations in Banach spaces with uniformly convex duals. Abstract evolution equations applicable to variational inequalities with time-dependent constraints are of course well developed in a Hilbert space framework. We refer the reader to, for instance, [6–11]. To the best of our knowledge, there are no abstract results for parabolic evolution equations that can be applied to variational inequalities in Lp with time-dependent constraints. For concrete problems, we refer the reader to [3,12–15], amongst others. This paper aims to fill the gap and establish a unified and abstract framework of nonlinear evolution equations in a Banach space with a uniformly convex dual to deal with variational inequalities in Lp with time-dependent constraints. We formulate the variational inequality (1.1) in the form of the following evolution equation: u0 (t ) + A(t )u(t ) + B(t )u(t ) 3 f (t )
E-mail address: [email protected]. 0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.03.028
(1.2)
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
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with a time-dependent m-accretive operator B(t ). Furthermore, we impose the initial condition u(0) = u0
(1.3)
with a given initial value u0 . To derive the variational inequality (1.1) from (1.2), we set B(t ) := ∂ IK (t ) |X ×X , where ∂ IK (t ) : H → H is the subdifferential of the indicator function IK (t ) of K (t ) and the notation |X ×X denotes that the graph of B(t ) is the intersection of that of ∂ IK (t ) and X × X . Our main result (Theorem 2.3) confirms the existence and uniqueness of a solution to {(1.2) and (1.3)} satisfying B(·)u ∈ Lp (0, T ; X ). Then, with the help of an Lp -estimate for the parabolic operator d/dt + A(t ), we obtain Lp -estimates of the solution (Corollary 2.4, Propositions 5.3 and 5.6). See Remark 5.5 for a comparison with the method of Brézis [4]. We note that our result can be applied to problems in which (Remark 5.4)
∩0≤t ≤T D(A(t ) + B(t )) = ∅. The Banach space theory of abstract nonlinear evolution equations established thus far (e.g., [16,17,5]) is not applicable to such problems. The main theorem (Theorem 2.3) and its corollary (Corollary 2.4) are stated in Section 2 and proved in Section 3 by assuming uniform estimates of approximate solutions (Proposition 3.1) to be proved in Section 4. The abstract results in Section 2 are applied to concrete problems in Lp with time-dependent constraints in Section 5. Combining the time dependence of A(t ) and B(t ), we can deal with a variety of problems with time-dependent constraints. For instance, variational inequalities in a non-cylindrical domain are, to the best of our knowledge, dealt with in an Lp -framework for the first time in this paper (Section 5.3). Our idea is to combine time-dependent subdifferential evolution equations [6–10] with the Lp -theory of variational inequalities with time-independent constraints [4,18]. In the basic assumptions given below, we assume that the operators A(t ) and B(t ) are m-accretive in both X and H. In the main assumptions stated in the next section, we assume that the mappings t 7→ A(t ) and t 7→ B(t ) are smooth in H (Condition (A)) and X (Condition (B)-(ii)), respectively, in similar ways to [6–10], and that operators A(t ) and B(t ) are compatible in an appropriate sense (Condition (C)), which is implicit in [4,18]. Notation and basic assumptions The norms of X , X ∗ , and H are denoted by | · |X , | · |X ∗ , and | · |H , respectively. The inner product of H and the duality between X ∗ and X are denoted by (·, ·) and h·, ·i, respectively. Let p ≥ 2 and F : X → X ∗ be the duality map with gauge ξ (r ) = r p−1 ; that is [4,18], p−1
p
Fz := {z ∗ ∈ X |hz ∗ , z i = |z |X , |z ∗ |X ∗ = |z |X
}.
∗
By the uniform convexity of X , the duality map F is single-valued and uniformly continuous on each bounded set in X ([17, Lemma 1.2], [19, Proposition II.8.8]). This property is used to prove a key lemma of our argument (see Lemma 4.2). Let AX (t ) and AH (t ), t ∈ [0, T ], be time-dependent families of m-accretive operators in X and H, respectively, such that AX (t ) = AH (t )|X ×X . In other words, the graph of AX (t ) is the intersection of that of AH (t ) and X × X . Assume that AH (t ) = ∂ϕ t , where ∂ϕ t is the subdifferential of a proper, l.s.c. (lower-semicontinuous) and convex function ϕ t : H → R ∪ {+∞} such that ϕ t (z ) ≥ 0 for all t ∈ [0, T ] and z ∈ H. Let BX (t ) and BH (t ) be time-dependent m-accretive operators in X and H, respectively, such that BX (t ) = BH (t )|X ×X . Assume that BH (t ) = ∂ IK (t ) , where ∂ IK (t ) is the subdifferential of the indicator function IK (t ) of a closed and convex set K (t ) in H for all t ∈ [0, T ]. For λ > 0, the Yosida approximations of these operators are denoted by AX ,λ (t ), AH ,λ (t ), BX ,λ (t ), and BH ,λ (t ), that is, AX ,λ (t ) := AH ,λ (t ) := BX ,λ (t ) := BH ,λ (t ) :=
1 IX − JAX ,λ (t ) ,
JAX ,λ (t ) := IX + λAX ,λ (t )
1 IH − JAH ,λ (t ) ,
JAH ,λ (t ) := IH + λAH ,λ (t )
1
JBX ,λ (t ) := IX + λBX ,λ (t )
λ
λ λ
,
−1
,
−1
IX − JBX ,λ (t ) ,
1
λ
−1
IH − JBH ,λ (t ) ,
JBH ,λ (t ) := IH + λBH ,λ (t )
,
−1
,
where IX and IH are the identity maps on X and H, respectively. Note that we have AX ,λ (t ) = AH ,λ (t )|X and BX ,λ (t ) = BH ,λ (t )|X . Furthermore, set IKt ,λ (z ) :=
λ p
p
BX ,λ (t )z X =
1 pλ
p−1
z − JB
X ,λ
p (t )z X
and let ϕλt be the regularization of ϕ t for λ > 0. It is well-known that its subdifferential ∂ϕλt coincides with the Yosida approximation AH ,λ (t ) of AH (t ) = ∂ϕ t .
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M. Kubo / Nonlinear Analysis 73 (2010) 390–398
We assume that the mappings t 7→ ϕ t (z ) and t 7→ K (t ) are continuous in the Mosco sense [20], and hence, the function t 7→ ϕλt (z ) and the mappings t 7→ AH ,λ (t )z and t 7→ BH ,λ (t )z are continuous for all λ > 0 and z ∈ H. We also assume that the mappings t 7→ AX ,λ (t )z and t 7→ BX ,λ (t )z are continuous for all λ > 0 and z ∈ X . Then, the function t 7→ IKt ,λ (z ) is also continuous for all λ > 0 and z ∈ X . We refer the reader to [19] for the notions and elementary properties of duality mappings, accretive operators, subdifferentials and Yosida approximations. 2. Main theorem The main assumptions for the given data are listed below. (A) The function t 7→ ϕλt (z ) is of bounded variation and its positive variation is absolutely continuous on [0, T ] for all λ > 0 and z ∈ H. There are functions pA ∈ L2 (0, T ) and qA ∈ L1 (0, T ) such that for all λ > 0 and z ∈ H, the following inequality holds for a.e. t ∈ (0, T ): d dt
1/2 AH ,λ (t )z + qA (t ) ϕ t (z ) + 1 . ϕλt (z ) ≤ pA (t ) ϕλt (z ) + 1 λ H
(B) (i) For all t ∈ [0, T ], λ > 0, z , y ∈ X the following inequalities hold:
hFBX ,λ (t )z , y − z i ≤ IKt ,λ (y) − IKt ,λ (z ),
(2.1)
hFBX ,λ (t )z , JBX ,λ (t )y − JBX ,λ (t )z i ≤ 0.
(2.2)
(ii) The function t 7→ IKt ,λ (z ) is of bounded variation and its positive variation is absolutely continuous on [0, T ] for all λ > 0 and z ∈ X . There are functions pB ∈ Lp (0, T ) and qB ∈ L1 (0, T ) such that for all λ > 0 and z ∈ X , the following inequality holds for a.e. t ∈ (0, T ): d dt
p−1
IKt ,λ (z ) ≤ pB (t ) BX ,λ (t )z X
+ qB (t ). ∗
(C) There are functions rX ∈ Lp (0, T ) and rH ∈ Lp (0, T ), p−1 + (p∗ )−1 = 1, such that for all t ∈ [0, T ], λ > 0 and z ∈ X the following inequalities hold:
(BX ,λ (t )z , AX ,λ (t )z ) ≥ −rH (t ) BX ,λ (t )z X , p−1 hFBX ,λ (t )z , AX ,λ (t )z i ≥ −rX (t ) BX ,λ (t )z X .
(2.3) (2.4)
(D) u0 ∈ D(AX (0)) ∩ K (0), f ∈ L (0, T ; X ). p
We denote by (P) the problem {(1.2) and (1.3)} and give two alternative solutions thereof. Definition 2.1. A function u : [0, T ] → H is called an H-solution of (P) iff the following conditions are satisfied. (a) u ∈ W 1,2 (0, T ; H ) and sup0≤t ≤T ϕ t (u(t )) < +∞. (b) There are functions u∗ , u∗∗ ∈ L2 (0, T ; H ) such that u∗ (t ) ∈ AH (t )u(t ) and u∗∗ (t ) ∈ BH (t )u(t ) for a.e. t ∈ (0, T ), and the following equation is satisfied: u0 (t ) + u∗ (t ) + u∗∗ (t ) = f (t )
in H a.e. t ∈ (0, T ).
(c) u(0) = u0 in H. Definition 2.2. A function u : [0, T ] → X is called an X -solution of (P) iff the following conditions are satisfied. (a) u ∈ W 1,p (0, T ; X ) and sup0≤t ≤T ϕ t (u(t )) < +∞. (b) There are functions u∗ , u∗∗ ∈ Lp (0, T ; X ) such that u∗ (t ) ∈ AX (t )u(t ) and u∗∗ (t ) ∈ BX (t )u(t ) for a.e. t ∈ (0, T ), and the following equation is satisfied: u0 (t ) + u∗ (t ) + u∗∗ (t ) = f (t )
in X a.e. t ∈ (0, T ).
(c) u(0) = u0 in X . Now we state our main theorem. Theorem 2.3. Assume that Conditions (A)–(D) are satisfied. Then, there exists a unique H-solution u of problem (P) such that u∗∗ ∈ Lp (0, T ; X ). The following corollary is useful for deriving Lp -regularity.
(2.5)
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
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Corollary 2.4. Assume that Conditions (A)–(D) and (E) below are satisfied. (E) For each v0 ∈ D(AX (0)) and g ∈ Lp (0, T ; X ), there exists a unique function v ∈ W 1,p (0, T ; X ) such that
v 0 (t ) + AX (t )v(t ) 3 g (t ) in X a.e. t ∈ (0, T ), v(0) = v0 in X . Then, there exists a unique X -solution u of problem (P). We can derive the corollary from Theorem 2.3 using Condition (E) for v0 = u0 and g := f − u∗∗ and the monotonicity of AH (t ). In applying the main theorem to concrete problems, Condition (E) refers to the Lp -solvability of the parabolic PDE related to the operator A(t ). Theorem 2.3 is proved in the next two sections. 3. Proof of Theorem 2.3 For each λ > 0, we consider the following approximate problem for (P): u0λ (t ) + AX ,λ (t )uλ (t ) + BX ,λ (t )uλ (t ) = f (t ) in X a.e. t ∈ (0, T ),
(3.1)
uλ (0) = u0 .
(3.2)
We denote by (P)λ the problem {(3.1) and (3.2)}. It is well-known that the Yosida approximations AX ,λ (t ) and BX ,λ (t ) are Lipschitz continuous operators in X . Therefore, by a standard argument, there exists a unique solution uλ ∈ W 1,p (0, T ; X ) of (P)λ . Moreover, we have the following uniform estimate, which is proved in Section 4. Proposition 3.1. There exists a constant M0 > 0 such that for all λ > 0 the following holds:
|u0λ |L2 (0,T ;H ) + sup ϕλt (uλ (t )) + |BX ,λ (·)uλ |Lp (0,T ;X ) ≤ M0 . 0 ≤t ≤T
By a standard argument of monotonicity (cf. [10, (4.13), (4.14)]), we have 1 2
|uλ (t ) − uµ (t )|2H ≤
λ−µ
Z
2
t
|AH ,µ uµ |2H − |AH ,λ uλ |2H + |BH ,µ uµ |2H − |BH ,λ uλ |2H dτ .
0
Therefore, by Proposition 3.1, there is a sequence uλn such that u0λn → u0 uλn → u
weakly in L2 (0, T ; H ), in C ([0, T ]; H ),
AH ,λn (·)uλ (= AX ,λn (·)uλ ) → u∗
weakly in L2 (0, T ; H ),
BH ,λn (·)uλ (= BX ,λn (·)uλ ) → u
weakly in Lp (0, T ; X )
∗∗
for some u, u∗ , and u∗∗ . Then, by the demi-closedness of m-accretive operators, we can show that the limit function u is a unique H-solution of (P) with the property (2.5). Thus, the proof of Theorem 2.3 will be complete when Proposition 3.1 is proved, which is accomplished in the next section. 4. Proof of uniform estimates We prepare two lemmas derived from Conditions (A) and (B)-(ii), respectively. Lemma 4.1 ([6, Lemma 2], [7, Proposition 1], [10, Proposition 3.2], [9, Lemma 1.2.5], [11, Lemma 3.4]). For each v ∈ W 1,1 (0, T ; H ) and λ > 0, the function t 7→ ϕλt (v(t )) is of bounded variation, its positive variation is absolutely continuous on [0, T ], and the following inequality holds for a.e. t ∈ (0, T ): d dt
1/2 AH ,λ (t )v(t ) + qA (t ) ϕ t (v(t )) + 1 . ϕλt (v(t )) − (AH ,λ (t )v(t ), v 0 (t )) ≤ pA (t ) ϕλt (v(t )) + 1 λ H
Lemma 4.2. For each v ∈ W 1,1 (0, T ; X ) and λ > 0, the function t 7→ IKt ,λ (v(t )) is of bounded variation, its positive variation is absolutely continuous on [0, T ], and the following inequality holds for a.e. t ∈ (0, T ): d dt
p−1
IKt ,λ (v(t )) − hFBX ,λ (t )v(t ), v 0 (t )i ≤ pB (t ) BX ,λ (t )v(t ) X
+ qB (t ).
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Lemma 4.2 is proved at the end of this section. For a proof of Lemma 4.1 we refer the reader to the cited references. Proof of Proposition 3.1. For simplicity, we write u for uλ . Since u(t ) ∈ X , we have AH ,λ (t )u(t ) = AX ,λ (t )u(t ). Therefore, multiplying (3.1) by AH ,λ (t )u(t ), we obtain
(u0 (t ), AH ,λ (t )u(t )) + |AH ,λ (t )u(t )|2H + (BX ,λ (t )u(t ), AH ,λ (t )u(t )) = (f (t ), AH ,λ (t )u(t )). Then, applying Lemma 4.1 and using Condition (2.3) in (C), we have d
1/2 AH ,λ (t )u(t ) ϕλt (u(t )) + |AH ,λ (t )u(t )|2H − rH (t ) BX ,λ (t )u(t ) X ≤ pA (t ) ϕλt (u(t )) + 1 H dt t + qA (t ) ϕλ (u(t )) + 1 + (f (t ), AH ,λ (t )u(t )). Hence, by the Schwarz and Hölder inequalities, we conclude that d dt
p 2p 1 1 ∗ ϕλt (u(t )) + |AH ,λ (t )u(t )|2H ≤ (pA (t )2 + qA (t )) ϕλt (u(t )) + 1 + |f (t )|2H + p BX ,λ (t )u(t ) X + ∗ rH (t )p . (4.1) 2
2 p
p
Next, noting that F (−z ) = −F (z ), we have by (3.1) F (u0 (t ) + AX ,λ (t )u(t ) − f (t )) + F (BX ,λ (t )u(t )) = 0. Multiplying u0 (t ) + AX ,λ (t )u(t ) − f (t ), we obtain
|u0 (t ) + AX ,λ (t )u(t ) − f (t )|pX + hF (BX ,λ (t )u(t )), u0 (t )i + hF (BX ,λ (t )u(t )), AX ,λ (t )u(t )i = hF (BX ,λ (t )u(t )), f (t )i. Therefore, applying Lemma 4.2 and using Condition (2.4) in (C), we have d
p
dt
p−1
IKt ,λ (u(t )) + |u0 (t ) + AX ,λ (t )u(t ) − f (t )|X − rX (t ) BX ,λ (t )u(t ) X
p−1 p−1 ≤ pB (t ) BX ,λ (t )u(t ) X + qB (t ) + BX ,λ (t )u(t ) X |f (t )|X . Hence, by Hölder’s inequality (p−1 + (p∗ )−1 = 1) and Eq. (3.1), we conclude that d dt
IKt ,λ (u(t )) +
1 p
1 |u0 (t ) + AX ,λ (t )u(t ) − f (t )|pX ≤ P (t ) := (pB (t ) + |f (t )|X + rX (t ))p + qB (t ) .
(4.2)
p
Now, adding inequalities (4.1) and (4.2), and using Eq. (3.1), we obtain d dt
ϕλt (u(t )) +
d dt
IKt ,λ (u(t )) +
1 2
1 1 0 |AH ,λ (t )u(t )|2H + 1 − p |u (t ) + AX ,λ (t )u(t ) − f (t )|pX ≤ Q (t )ϕλt (u(t )) + R(t ), 2
p
where Q (t ) := pA (t )2 + qA (t ),
2p ∗ R(t ) := Q (t ) + |f (t )|2H + ∗ rH (t )p + P (t ). p
Hence, by Gronwall’s lemma we see that there is a constant M0 > 0 independent of λ > 0, such that p
sup ϕλt (u(t )) + sup IKt ,λ (u(t )) + |AH ,λ (·)u|2L2 (0,T ;H ) + |u0 + AX ,λ (·)u|Lp (0,T ;X ) ≤ M0 .
0≤t ≤T
0≤t ≤T
Thus, Proposition 3.1 is proved and the proof of Theorem 2.3 is completed.
Proof of Lemma 4.2. Applying Conditions (B)-(i) for t := t + h, z := v(t + h), y := v(t ) and (B)-(ii) for z := v(t ), we obtain for all 0 ≤ t < t + h ≤ T , IKt +,λh (v(t + h)) − IKt ,λ (v(t )) = IKt +,λh (v(t + h)) − IKt +,λh (v(t )) + IKt +,λh (v(t )) − IKt ,λ (v(t ))
≤ hFBX ,λ (t + h)v(t + h), v(t + h) − v(t )i +
t +h
Z t
n
p−1
pB (τ ) BX ,λ (τ )v(t ) X
o + qB (τ ) dτ .
From this inequality, we see that the function t 7→ (v(t )) is of bounded variation and its positive variation is absolutely continuous. Furthermore, dividing both sides of this inequality by h and letting h ↓ 0, we obtain the desired inequality. Here, we need the continuity of the duality map F . IKt ,λ
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
395
5. Obstacle problems in L p 5.1. Lemmas The following lemmas are useful for verifying Conditions (A) and (2.3) in (C), respectively. Lemma 5.1 ([9–11]). Assume that the family ϕ t of convex functions satisfies the following condition: (F) There are functions pˆ A ∈ W 1,2 (0, T ) and qˆ A ∈ W 1,1 (0, T ) such that for all 0 ≤ s < t ≤ T and z ∈ D(ϕ s ) there exists z˜ ∈ D(ϕ t ) satisfying the following inequalities:
1/2 |˜z − z |H ≤ |ˆpA (t ) − pˆ A (s)| ϕ s (z ) + 1 , ϕ t (˜z ) − ϕ s (z ) ≤ |ˆqA (t ) − qˆ A (s)| ϕ s (z ) + 1 . Then, the family A(t ) of operators satisfies Condition (A). Lemma 5.2 ([21, Lemma 5.1]). Let A be a positive self-adjoint operator on H, and let K be a closed convex set in H. Assume that there exists g ∈ H such that
(I + λA)−1 (z + λg ) ∈ K for all λ > 0 and z ∈ K . Then, we have
(∂ IK ,λ (z ), Aλ z ) ≥ (∂ IK ,λ (z ), (I + λA)−1 g ), where Aλ and ∂ IK ,λ are Yosida approximations of A and of the subdifferential ∂ Iλ of the indicator function IK of K , respectively. We refer to the cited references for the proofs of these lemmas. 5.2. Time-dependent bi-lateral constraints p Let Ω ⊂ RN (N ≥ 1) be a bounded set with smooth boundary ∂ Ω and set X := LP (Ω ) (p ≥ 2) and H := L2 (Ω ). Let 1 2 N aij ∈ C ([0, T ] × Ω ) (i, j = 1, . . . , N ) be such that aij = aji and for a constant α > 0 : i,j aij ξi ξj ≥ α|ξ | (ξ ∈ R ). Define
a differential operator L(t ) by (∂i := ∂∂x ) i L(t )z := −
X
∂i aij (t )∂j z .
i,j
Now define ϕ : H → R ∪ {+∞} by t
Z X 1 aij (t , x)∂i z (x)∂j z (x)dx, ϕ t (z ) := 2 Ω i,j +∞,
if z ∈ H01 (Ω ), otherwise.
Then, we have AH (t )z := ∂ϕ t (z ) = L(t )z
for z ∈ D(AH (t )) = H 2 (Ω ) ∩ H01 (Ω )
and AX (t )z := AH (t )|X ×X = L(t )z
1,p
for z ∈ D(AX (t )) = W 2,p (Ω ) ∩ W0 (Ω ).
It is well-known that the operators AH (t ) and AX (t ) are m-accretive in H and X , respectively. We can verify Condition (A) with the aid of Lemma 5.1. In fact, we can take z˜ = z in verifying Condition (F) (see [7]). Next, let ψi ∈ W 1,p (0, T ; X ) ∪ Lp (0, T ; W 2,p (Ω )) (i = 1, 2) be such that
ψ1 ≤ ψ2 in (0, T ) × Ω ,
ψ1 ≤ 0 ≤ ψ2 on (0, T ) × ∂ Ω
and set K (t ) := {z ∈ H | ψ1 (t ) ≤ z ≤ ψ2 (t ) in Ω }. Then, set BH (t ) := ∂ IK (t ) and BX (t ) := BH (t )|X ×X , where ∂ IK (t ) is the subdifferential of the indicator function IK (t ) of the closed and convex set K (t ) in H. We have for all λ > 0 and z ∈ X BX ,λ (t )z = z1,λ + z2,λ
and
JBX ,λ (t )z = z − λz1,λ − λz2,λ ,
(5.1)
396
M. Kubo / Nonlinear Analysis 73 (2010) 390–398
where 1 z1,λ := − [z − ψ1 ]− ,
z2,λ :=
λ
1
λ
[z − ψ2 ]+ .
Also note that for all z ∈ X [4,18,19] Fz = |z |p−2 z and therefore, FBX ,λ (t )z = Fz1,λ + Fz2,λ ,
IKt ,λ (z ) = IKt ,λ (z1,λ ) + IKt ,λ (z2,λ ). With these identities and by a standard argument of obstacle problems, we can verify Conditions (2.1) and (2.2) in (B)-(i). To verify Condition (B)-(ii), we calculate for a.e. t ∈ (0, T ) d dt
IKt ,λ (z ) =
d
dt
λ p
|z1,λ |pX +
λ p
|z2,λ |pX
= hFz1,λ , −ψ1 (t )i + hFz2,λ , −ψ20 (t )i 0
≤ |z1,λ |pX−1 |ψ10 (t )|X + |z2,λ |pX−1 |ψ20 (t )|X ≤ pB (t )|BX ,λ (t )z |pX−1 , where pB (t ) := |ψ10 (t )|X + |ψ20 (t )|X . To verify inequality (2.3) in (C), we use Lemma 5.2 with A = AH (t ), z = zi,λ (see (5.1)), K = Ki (t ), and g = gi (i = 1, 2), where K1 (t ) := {z ∈ H | z ≥ ψ1 (t ) in Ω }, gi := L(t )ψi (t )
K2 (t ) := {z ∈ H | z ≤ ψ2 (t ) in Ω },
(i = 1, 2).
Finally, inequality (2.4) in (C) can be verified as follows (we write A(t ) and Aλ (t ) for AX (t ) and AX ,λ (t ), respectively). Set z1 := z ∨ ψ1 (t )
and
z2 := z ∧ ψ2 (t ).
Then for i = 1, 2
hFzλ,i , Aλ (t )z i = λhFzλ,i , Aλ (t )zλ,i i + hFzλ,i , Aλ (t )zi i = λhFzλ,i , Aλ (t )zλ,i i +
1
λ
hFzλ,i , zi − (I + λA(t ))−1 (zi + λgi (t ))i + hFzλ,i , (I + λA(t ))−1 gi (t )i
≥ hFzλ,i , (I + λA(t ))−1 gi (t )i. Here we use the accretivity of Aλ (t ) : hFz , Aλ (t )z i ≥ 0, and the property
(5.2)
(I + λA(t ))−1 (zi + λgi (t )) ∈ Ki (t ), which follows from the property zi ∈ Ki (t ).
(5.3)
Therefore, given that 1 ,p
u0 ∈ W 2,p (Ω ) ∩ W0 (Ω ) ∩ K (0),
f ∈ Lp (0, T ; Lp (Ω )),
we can derive the following proposition from Corollary 2.4 with the aid of the Lp -regularity theory of the parabolic operator ∂/∂ t + L(t ) [22, Chap. IV, Theorem 9.1]. Proposition 5.3. Under the above assumptions, there exists a unique solution u of the following problem: 1,p
(a) u ∈ W 1,p (0, T ; X ) ∩ C ([0, T ]; W0 (Ω )) ∩ Lp (0, T ; W 2,p (Ω )). (b) u(t ) ∈ K (t ) for all t ∈ [0, T ] and
(u0 (t ) + L(t )u(t ) − f (t ), u(t ) − z ) ≤ 0 for all z ∈ K (t ) and a.e. t ∈ (0, T ). (c) u(0) = u0 in X . Remark 5.4. When the obstacles are chosen in such a way that ψ1 (s) > ψ2 (t ) in E for some s, t ∈ [0, T ] and a set E ⊂ Ω with positive measure, we have K (s) ∩ K (t ) = ∅. Remark 5.5. In Brézis [4, Chap. II], problems with time-independent data L, ψ1 and ψ2 are dealt with, by first obtaining the property u0 ∈ L∞ (0, T ; X ) by applying Kato’s theory [5], then deriving a spatial Lp -estimate by the theory of elliptic variational inequality in [4, Chap. I] (or [18]) with the aid of an Lp -estimate for the elliptic operator L (cf. [4, (I.1.13)]). We note that our theory does not need such a condition as f 0 ∈ L1 (0, T ; Lp (Ω )), which was assumed in [4] to apply [5].
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5.3. Problems in a non-cylindrical domain Let Ω (t ) be a domain with smooth boundary such that Ω (t ) ⊂ Ω and where there is a family Φ t : Ω → Ω (t ∈ [0, T ]) of C 2 -diffeomorphisms satisfying Φ t (Ω (0)) = Ω (t ) and ∂t Φ t , ∂i Φ t , ∂i ∂j Φ t ∈ C ([0, T ] × Ω ). Now define ϕ t : H → R ∪ {+∞} by
ϕ t (z ) :=
Z 1 2
X Ω ( t ) i ,j
aij (t )∂i z (x)∂j z (x)dx,
+∞,
if z |Ω (t ) ∈ H01 (Ω (t )) and z = 0 in Ω \ Ω (t ), otherwise.
We can verify by a standard argument of subdifferentials that (AH (t ) := ∂ϕ t ) D(AH (t )) = {z ∈ H | z |Ω (t ) ∈ H 2 (Ω ) ∩ H01 (Ω ), z = 0 in Ω \ Ω (t )}, z ∗ ∈ AH (t )z if and only if z ∗ = L(t )z in Ω (t ) (z ∈ D(AH (t ))). Also, we can verify Condition (A) by applying Lemma 5.1 with z˜ := z ◦ Φ s ◦ Φ t (see [10,9,23]). Defining AX (t ) := AH (t )|X ×X , we can see with the aid of the Lp -solvability of the operator L(t ) in Ω (t ) that AX (t ) is m-accretive in X and that D(AX (t )) = {z ∈ D(AH (t ))| z |Ω (t ) ∈ W 2,p (Ω (t ))}. Next, let ψ ∈ W 1,p (0, T ; X ) ∩ Lp (0, T ; W 2,p (Ω )) be such that
ψ(t ) ≤ 0 in Ω \ Ω (t ) and set K (t ) := {z ∈ H | z ≥ ψ(t ) in Ω }. Now define BH (t ) := ∂ IK (t ) and BX (t ) := BH (t )|X ×X . Then, we have for all λ > 0 and z ∈ H 1 BH ,λ (t )z = zλ := − [z − ψ]− .
λ
With this identity and by a standard argument of obstacle problems as in Section 5.2, we can verify Condition (B). In addition, inequalities (2.3) and (2.4) in (C) can be verified, respectively, by applying Lemma 5.2 with z := zλ and g := L(t )ψ(t ) and by an argument similar to that of (5.2) and (5.3) (note that (I + λA(t ))−1 is linear, whereas A(t ) is not). Now, suppose that u0 ∈ D(AX (0)) ∩ K (0),
f ∈ Lp (0, T ; Lp (Ω )).
Transforming the problem into the cylindrical domain (0, T ) × Ω (0) using the diffeomorphism Φ t , we can derive the following proposition from Corollary 2.4 with the aid of [22, Chap. IV, Theorem 9.1]. Proposition 5.6. Under the above assumptions, there is a unique solution u of the following problem: 1 ,p
(a) u ∈ W 1,p (0, T ; X ) ∩ C ([0, T ]; W0 (Ω )) and T
Z 0
|u(t )|pW 2,p (Ω (t )) dt < +∞.
(b) u(t ) ∈ K (t ) and u(t ) = 0 in Ω \ Ω (t ) for all t ∈ [0, T ], and
Z Ω (t )
(u0 (t ) + L(t )u(t ) − f (t ))(u(t ) − z )dx ≤ 0
for all z ∈ K (t ) and a.e. t ∈ (0, T ). (c) u(0) = u0 in X .
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