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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Evolutionary quasi-variational and variational inequalities with constraints on the derivatives

Fernando MirandaORCID iD: https://orcid.org/0000-0002-7624-4816 / José Francisco RodriguesORCID iD: https://orcid.org/0000-0001-8438-0749 / Lisa SantosORCID iD: https://orcid.org/0000-0003-0286-1616
Published Online: 2018-10-09 | DOI: https://doi.org/10.1515/anona-2018-0113

Abstract

This paper considers a general framework for the study of the existence of quasi-variational and variational solutions to a class of nonlinear evolution systems in convex sets of Banach spaces describing constraints on a linear combination of partial derivatives of the solutions. The quasi-linear operators are of monotone type, but are not required to be coercive for the existence of weak solutions, which is obtained by a double penalization/regularization for the approximation of the solutions. In the case of time-dependent convex sets that are independent of the solution, we show also the uniqueness and the continuous dependence of the strong solutions of the variational inequalities, extending previous results to a more general framework.

Keywords: Evolutionary quasi-variational inequalities; gradient constraints; constraints on the derivatives

MSC 2010: 35R35; 35K87; 35K92; 47J20; 49J40

1 Introduction

While variational inequalities where introduced in 1964, by Fichera and Stampacchia in the framework of minimization problems with obstacle constraints, the first evolutionary variational inequality was solved in the seminal paper of Lions and Stampacchia [24], which was followed by many other works, including the extension to pseudo-monotone operators by Brézis in 1968 [7] (see also [23, 33]). Quasi-variational inequalities were introduced later by Bensoussan and Lions in 1973 to describe impulse control problems [5] and were developed for several other mathematical models with free boundaries (see, for instance, [25, 3]), mainly as implicit unilateral problems of obstacle type, in which the constraints depend on the solution.

The first physical models with gradient constraints formulated with quasi-variational inequalities of evolution type were proposed by Prighozhin, in [29] and [28], respectively for the sandpile growth and for the magnetization of type-II superconductors. This last model has motivated a first existence result for stationary problems in [21], including other applications in elastoplasticity and in electrostatics, and, in [31], in the parabolic framework for the p-Laplacian with an implicit gradient constraint, which was later extended to quasi-variational solutions for first-order quasilinear equations in [32], always in the scalar cases.

In this work, we consider weak solutions 𝒖=𝒖(x,t) to a class of quasi-variational inequalities associated with evolution equations or systems of the type

t𝒖+L*𝒂(L𝒖)+𝒃(𝒖)=𝒇(1.1)

formally in the unsaturated region of the scalar constraint

|L𝒖|G[𝒖],

i.e. in the domain {(x,t):|L𝒖(x,t)|<G[𝒖](x,t)}, with a nonlocal positive and compact operator G, where t𝒖 denotes the partial time derivative, L is a linear partial differential operator in x with bounded coefficients and L* is its formal dual. Here the monotone vector fields 𝒂 and 𝒃 are of power-type growth, and the boundary value problems may be coercive or not. However, in the region {(x,t):|L𝒖(x,t)|=G[𝒖](x,t)} equation (1.1), in general, does not hold unless an extra term is added, raising interesting open questions. The general form of L covers, in particular, the gradient, the Laplacian and higher-order operators, the curl, the symmetric part of the Jacobian or classes of smooth vector fields such as those of Hörmander type. Weak quasi-variational solutions, which in general are non-unique and do not have the time derivative in the dual space of the solution, are obtained by the passages to the limit of two vanishing parameters, one for an appropriate approximation/penalization of the constraint on L𝒖 and a second one for a coercive regularization, as in [32]. This method allows the application of the Schauder fixed point theorem to a general regularized two parameters variational equation of type (1.1) and extends considerably the work [2].

When the constraint G, which may depend on time and space, is independent of the solution, i.e. G=g(x,t), the problem becomes a variational one with the solution belonging to a time dependent convex set of a suitable Banach space. In this case, if the vector fields 𝒂 and 𝒃 are monotone, there holds uniqueness of the weak solution. Under additional assumptions on the data, we show the existence, uniqueness and continuous dependence of the stronger solution of the corresponding evolution variational inequality, when the time derivative is actually an L2 function. Here our method is adapted to gradient-type constraints and it develops and extends the pioneer work of [34], which was continued in [35], extended to a p-curl system in [27] and to thick flows by [30] (see also [26]). Although variational inequalities with time dependent convex sets have been studied in several works (see, for instance, [17, 20] and the references therein), for the case of a convex with gradient constraint only a few results have been stated, namely in [6], as an application of abstract theorems, which assumptions are difficult to verify and, in general, require stronger hypotheses.

Recently, other approaches to evolutionary quasi-variational problems with gradient constraint have been developed by Kenmochi and co-workers in [16, 11, 18, 19], using variational evolution inclusions in Hilbert spaces with subdifferentials with a nonlocal dependence on parameters, and by Hintermüller and Rautenberg in [13], using the pseudo-monotonicity and the C0-semigroup approach of Brézis-Lions, and in [14], using contractive iteration arguments that yield uniqueness results and numerical approximation schemes in interesting but special situations. Although the elegant and abstract approach of [13] yields the existence of weak quasi-variational solutions under general stability conditions of Mosco type and a general scheme for the numerical approximation of a solution, the required assumptions for the existence theory are somehow more restrictive than ours, in particular in what concerns the required strong coercive condition. Other recent results on evolutionary quasi-variational inequalities can be found in [36, 20], both in more abstract frameworks and oriented to unilateral-type problems and, therefore, with limited interest to constraints on the derivatives of the solutions. Recently, in [15], a semidiscretization in time, with monotone non-decreasing data, was used to obtain non-decreasing in time solutions to quasi-variational inequalities with gradient constraints, including an interesting numerical scheme.

This paper is organized as follows: In Section 2, we state our framework and the main results on the existence of weak quasi-variational solutions and on the well-posedness of the strong variational solutions. In Section 3, we illustrate the nonlocal constraint operator G and the linear partial differential operator L with several examples of applications. Section 4 deals with the approximated problem and a priori estimates. The proof of the existence of the weak quasi-variational solutions is given in Section 5 and, finally, in Section 6 we show the uniqueness and the continuous dependence on the data in the variational inequality case.

2 Assumptions and main results

Let Ω be a bounded open subset of d with a Lipschitz boundary, d2 and for t(0,T] denote Qt=Ω×(0,t). For a real vector function 𝒖=𝒖(x,t)=(u1,,um), (x,t)QT, and a multi-index α=(α1,,αd), with α1,,αd0 and α1++αd=|α|, we denote

αui=|α|uix1α1xdαd,

the partial derivatives of ui. Given real numbers a,b, we set ab=max{a,b} and ab=min{a,b}.

We introduce now several assumptions which will be important to set the functional framework of our problem.

Assumption 2.1.

For p[1,], let L be a linear differential operator of order s1, given by

L:𝑽pLp(Ω)such that𝑽p={𝒖Lp(Ω)m:L𝒖Lp(Ω)}

is endowed with the graph norm, ,m.

In general, the operator L can have the form

(L𝒖)j=|α|sk=1mλα,kjαuk,

where j=1,,, α=(α1,,αd)0d is a multi-index and each λα,kj is in L(Ω), but we shall consider mainly the following four illustrative examples with constant coefficients, although we can consider also their generalizations with variable coefficients as in the fifth example:

  • (i)

    Lu=u (gradient of u, m=1, =d).

  • (ii)

    Lu=Δu (Laplacian of u, m==1).

  • (iii)

    L𝒖=×𝒖 (curl of 𝒖, d=m==3).

  • (iv)

    L𝒖=D𝒖=12(𝒖+𝒖T) (symmetric part of the Jacobian of 𝒖, d=m, =m2).

  • (v)

    𝑳u=(X1u,,Xu), where Xj=i=1dαjixi, where αji are appropriate scalar real functions (subelliptic gradient of u, m=1, 1j, 1id).

Assumption 2.2.

Let 𝒂:QT× and 𝒃:QT×mm be Carathéodory functions, i.e. they are measurable functions in the variables (x,t) for all 𝝃 and 𝜼m, respectively, and they are continuous in the variables 𝝃 and 𝜼m for a.e. (x,t)QT. Suppose, additionally, that 𝒂 and 𝒃 satisfy the following structural conditions: for all 𝝃,𝝃 and 𝜼,𝜼m and a.e. (x,t)QT,

|𝒂(x,t,𝝃)|a*|𝝃|p-1,(2.1)(𝒂(x,t,𝝃)-𝒂(x,t,𝝃))(𝝃-𝝃)0,|𝒃(x,t,𝜼)|b*|𝜼|(p-1)1,(𝒃(x,t,𝜼)-𝒃(x,t,𝜼))(𝜼-𝜼)0,

where a* and b* are positive constants and 1<p<.

Assumption 2.3.

For a given p(1,), we work with a closed subspace 𝕏p of 𝑽p such that 𝕏pL2(Ω)m and 𝒗𝕏p:=L𝒗Lp(Ω) is a norm in 𝕏p equivalent to the norm induced from 𝑽p.

Remark 2.4.

For simplicity, in this work we consider a functional framework where we suppose the Poincaré and Sobolev-type inequalities to be valid, as in the Dirichlet problems of the five examples. However, our approach is still valid for more general frameworks to include Neumann and mixed-type boundary conditions.

Assumption 2.5.

There exists a Hilbert subspace of L2(Ω)m such that (𝕏p,,𝕏p) is a Gelfand triple and the inclusion of 𝕏p into is compact for the given p, 1<p<.

From now on, we set

𝒱p=Lp(0,T;𝕏p),=Lp(0,T;),1p,

and we observe that Lp(0,T;𝕏p)=𝒱p, with p=pp-1 for 1<p<.

By well-known embedding theorems on Sobolev–Bochner spaces (see, for instance, [33, Chapter 7]), we have

𝒴p={𝒗𝒱p:t𝒗𝒱p}𝒞([0,T];)(2.2)

and Assumption 2.5 implies, by the Aubin–Lions lemma, that the embedding 𝒴p is also compact for 1<p<.

Assumption 2.6.

We consider a nonlinear continuous functional G:L1(QT) such that its restriction to 𝒱p is compact with values in 𝒞([0,T];L(Ω)), i.e. G:𝒱p𝒞([0,T];L(Ω)) is compact. In addition, we assume

0<g*G[𝒖](x,t)g*for all 𝒖𝒱p, for all t[0,T] and a.e. xΩ,

for given constants g* and g*.

Since G is compact in 𝒱p, in particular for any sequence {𝒗n}n weakly convergent to 𝒗 in 𝒱p, there exists a subsequence, still denoted by {𝒗n}n, such that {G[𝒗n]}n converges uniformly to G[𝒗] in 𝒞([0,T];L(Ω)).

For 𝒗𝒱p and a.e. t(0,T), we define the nonempty convex set for G[𝒗](t)L(Ω):

𝕂G[𝒗](t)={𝒘𝕏p:|L𝒘|G[𝒗](t)},(2.3)

where || is the Euclidean norm in and we denote 𝒘𝕂G[𝒗] if and only if 𝒘(t)𝕂G[𝒗](t) for a.e. t(0,T).

For 1p<, we denote the duality pairing between 𝕏p and 𝕏p by ,p and we consider the quasi-variational inequality associated with (1.1) and (2.3). Find 𝒖𝒱p satisfying

{𝒖𝕂G[𝒖],0Tt𝒗,𝒗-𝒖p+QT𝒂(L𝒖)L(𝒗-𝒖)+QT𝒃(𝒖)(𝒗-𝒖)QT𝒇(𝒗-𝒖)-12Ω|𝒗(0)-𝒖0|2,for all 𝒗𝒴p such that 𝒗𝕂G[𝒖].(2.4)

Theorem 2.7.

Suppose that Assumptions 2.12.6 are satisfied, 𝐟L2(QT)m and 𝐮0KG[𝐮0]. Then the quasi-variational inequality (2.4) has a weak solution 𝐮VpL(0,T;L2(Ω)m).

We note that by Assumption 2.6 the solutions have bounded L𝒖 but, in general, this may not imply that 𝒖 is itself bounded. We also observe that, by insufficient regularity in time, we could not guarantee that the weak solution u satisfies the initial condition in the classical sense, but only in the generalized sense (2.4) as in [7, 23].

We consider a positive bounded function g:QT+ and the special case of the convex set (2.3) with

G[𝒗](x,t)=g(x,t)for a.e. t(0,T).

So,

𝒗𝕂gif and only if𝒗(t)𝕂g(t)={𝒗𝕏p:|L𝒗|g(t)} for a.e. t(0,T).(2.5)

In this case, the convex set being independent of the solution, the problem becomes variational and the weak solution of Theorem 2.7 is unique by the following theorem.

Theorem 2.8.

The variational inequality (2.4) with a fixed convex Kg, as in (2.5), for a given strictly positive function g=g(x,t)C([0,T];L(Ω)), 𝐮0Kg(0) and 𝐟L2(QT)m, has at most one solution provided XpH and one of the monotonicity conditions is strict, i.e.

(𝒃(x,t,𝜼)-𝒃(x,t,𝜼))(𝜼-𝜼)>0for 𝜼𝜼,

or

(𝒂(x,t,𝝃)-𝒂(x,t,𝝃))(𝝃-𝝃)>0for 𝝃𝝃,

and Assumption 2.3 holds.

We can now introduce the strong formulation of the corresponding variational inequality. Find

𝒘𝒱pH1(0,T;L2(Ω)m)

satisfying, for all t(0,T],

{𝒘𝕂g,𝒘(0)=𝒘0,Qtt𝒘(𝒗-𝒘)+Qt𝒂(L𝒘)L(𝒗-𝒘)+Qt𝒃(𝒘)(𝒗-𝒘)Qt𝒇(𝒗-𝒘),for all 𝒗𝕂g𝒱p.(2.6)

We observe that if 𝒘 is a strong solution to (2.6), it is also a weak solution to (2.4). Indeed, if we take 𝒗𝒴p𝕂g𝒞(0,T;L2(Ω)m) in (2.6) with t=T, since

0Tt𝒘-t𝒗,𝒗-𝒘p=12Ω|𝒗(0)-𝒘0|2-12Ω|𝒗(T)-𝒘(T)|212Ω|𝒗(0)-𝒘0|2,

we immediately conclude that 𝒘 also satisfies (2.4).

We consider also a stronger non-coercive framework with a potential vector field 𝒂 and a lower-order term 𝒃 with linear growth, by replacing Assumption 2.2 by the following.

Assumption 2.9.

Let 𝒂:QT× and 𝒃:QT×mm be Carathéodory functions, i.e. they are measurable functions in the variables (x,t) for all 𝝃 and 𝜼m, respectively, and they are continuous in the variables 𝝃 and 𝜼m, respectively, for a.e. (x,t)QT. Suppose, additionally, that there exists A:QT× such that, for all 𝝃 and a.e. (x,t)Q¯T, the function A is differentiable in t and in 𝝃, and

A=A(x,t,𝝃) is convex in 𝝃,𝝃A=𝒂,(2.7)0A(x,t,𝝃)a*|𝝃|p,|tA(x,t,𝝃)|A1+A2|𝝃|p,(2.8)

and 𝒃 satisfies the following structural conditions: for all 𝜼,𝜼m and a.e. (x,t)QT,

(𝒃(x,t,𝜼)-𝒃(x,t,𝜼))(𝜼-𝜼)0,|𝒃(x,t,𝜼)|b*|𝜼|,(2.9)

where a*, A1, A2 and b* are positive constants, 1<p<.

In the non-coercive case, we have the well-posedness result on the existence, uniqueness and continuous dependence of the (strong) variational solution (2.6). Under the additional strong monotonicity assumption, for instance for operators of p-Laplacian type, when 𝒂(𝝃)=|𝝃|p-2𝝃, the continuous dependence result in the coercive case also holds in the space 𝒱p.

Theorem 2.10.

Suppose that Assumptions 2.1, 2.3, 2.5 and 2.9 are satisfied and

𝒇L2(QT)m,𝒘0𝕂g(0),gW1,(0,T;L(Ω))with gg*>0.(2.10)

Then the variational inequality (2.6) has a unique solution 𝐰VpH1(0,T;L2(Ω)m).

Theorem 2.11.

Suppose that the assumptions of Theorem 2.10 hold and for i=1,2 let 𝐰i be the solution to the variational inequality (2.6) with data 𝐟i,𝐰i0,gi satisfying (2.10). Then there exists a positive constant C=C(T) such that

𝒘1-𝒘2L(0,T;L2(Ω)m)2C(𝒇1-𝒇2L2(QT)m2+𝒘10-𝒘20L2(Ω)m2+g1-g2L1(0,T;L(Ω))).(2.11)

If, in addition, 𝐚 satisfies

(𝒂(x,t,𝝃)-𝒂(x,t,𝝃))(𝝃-𝝃){a*|𝝃-𝝃|pif p2,a*(|𝝃|+|𝝃|)p-2|𝝃-𝝃|2if p<2,(2.12)

where a* is a positive constant depending on p, 1<p<, then there exists C*=C(a*,p,T)>0 such that

𝒘1-𝒘2L(0,T;L2(Ω)m)2+𝒘1-𝒘2𝒱p2pC*(𝒇1-𝒇2L2(QT)m2+𝒘10-𝒘20L2(Ω)m2+g1-g2L1(0,T;L(Ω))).(2.13)

Remark 2.12.

For strong solutions 𝒘𝕂gH1(0,T;L2(Ω)m), the variational inequality (2.6) is, for a.e. t(0,T), equivalent to

Ωt𝒘(t)(𝒛-𝒘(t))+Ω𝒂(t,L𝒘(t))L(𝒛-𝒘(t))+Ω𝒃(t,𝒘(t))(𝒛-𝒘(t))Ω𝒇(t)(𝒛-𝒘(t))for all 𝒛𝕂g(t),(2.14)

provided we assume g𝒞([0,T];L(Ω)), gg*>0. Indeed, for arbitrary δ>0, 0<δ<t<T-δ, for fixed t(0,T), we set

εδ=supt-δ<τ<t+δg(t)-g(τ)L(Ω)

and we may define

𝒗(τ)={𝟎if τ(t-δ,t+δ),g*g*+εδ𝒛if τ(t-δ,t+δ),

which is such that 𝒗𝒱p𝕂g whenever 𝒛𝕂g(t). Hence we can choose this 𝒗 as test function in (2.6) with t=T, divide by 2δ and let δ0 obtaining, by Lebesgue’s theorem, inequality (2.14) for a.e. t(0,T).

As a Corollary of Theorems 2.8 and 2.10, we can drop the differentiability in time of g and still obtain an existence and uniqueness result for the weak variational inequality (2.4) with 𝕂g, extending [18, Theorem 3.8].

Theorem 2.13.

Suppose that Assumptions 2.1, 2.3, 2.5 and 2.9 are satisfied and

𝒇L2(QT)m,g𝒞([0,T];L(Ω))with gg*>0,𝒘0𝕂g(0).

Then the variational inequality (2.4) for Kg has a unique weak solution 𝐰VpC([0,T];L2(Ω)m).

3 Applications with particular G and L

In this section, we present some examples of compact nonlocal operators G satisfying Assumption 2.6, and linear operators L satisfying Assumption 2.1.

3.1 Nonlocal compact operators

Here we are interested in two examples of compact operators G given in the form

G[𝒗]=g(x,t,𝜻(𝒗)(x,t))a.e. in QT,(3.1)

where g=g(x,t,𝜻):QT×m is a positive function, continuous in (x,t)QT and in 𝜻m, and 𝜻:𝒱p𝒞(Q¯T)m is a completely continuous mapping.

3.1.1 Regularization by integration in time

We define the compact operator by

𝜻(𝒗)(x,t)=0t𝒗(x,s)K(t,s)𝑑s for a.e. (x,t)QT,

where K=K(t,s) is a given kernel satisfying

K,tKL((0,T)×(0,T)).(3.2)

For simplicity, we assume here the existence of a constant g* and a real bounded function g* such that

0<g*g(x,t,𝝃)g*(M) for a.e. (x,t)QT and for all 𝝃:|𝝃|M.

We also assume that the embedding

𝕏p𝒞(Ω¯)mis compact,(3.3)

which, by the Rellich–Kondratchov theorem, is satisfied if 𝕏pWs,q(Ω)m with q>ds.

Suppose 𝒱p=Lp(0,T;𝕏p), p>1, and observe that, by assumption (3.2), not only 𝜻(𝒗)𝒱p but also t𝜻(𝒗)𝒱p, i.e.

𝜻(𝒗)W1,p(0,T;𝕏p).

Hence, by [4, Lemma 2.2], for instance the image by 𝜻 of a bounded subset of 𝒱p, being bounded in W1,p(0,T;𝕏p), by (3.3) is relatively compact in 𝒞(Q¯T)m. So 𝜻:𝒱p𝒞(Q¯T)m is a completely continuous mapping, and therefore G defined in (3.1) satisfies Assumption 2.6.

3.1.2 Coupling with a nonlinear parabolic equation

We may define the compact operator through the unique solution of the Cauchy-Dirichlet problem for the quasilinear parabolic scalar equation

tζ-𝒂(x,t,ζ)=φ𝒗in QT,(3.4)ζ=0on Ω×(0,T),ζ(0)=ζ0on Ω,(3.5)

where φ𝒗=φ(x,t) depends on 𝒗𝒱p, and the vector field 𝒂 satisfies (2.1) and (2.12) with p=2 and =d.

It is well known that for each φL2(QT) and ζ0L2(Ω), the weak solution

ζL(0,T;L2(Ω))L2(0,T;H01(Ω))

to (3.4), (3.5) and (2.6) depends continuously, in these spaces, on the variation of φ in the weak topology of L2(QT). Moreover, if ζ0𝒞γ(Ω¯) is Hölder continuous for some 0<γ<1 and φLq(QT) for q>d+22, the following estimate holds (see [22, p. 419]):

ζ𝒞λ(Q¯T)C(ζ0𝒞γ(Ω¯)+φLq(QT))

for some λ, 0<λγ<1, where C>0 is a constant independent of the data φ.

Now, for each 𝒗𝒱p with p>d+22 (p=2 if d=1) and given 𝝍L(QT)m and 𝜼L(QT), we may choose ζ=ζ(𝒗) in (3.1) as being the solution of (3.4)–(3.5), with a given ζ0𝒞γ(Ω¯) and

φ𝒗=φ0+𝝍𝒗+𝜼L𝒗Lp(QT)(3.6)

for some fixed φ0Lp(QT). Hence, by (3.6) and the Ascoli theorem, the mapping 𝒗φ𝒗ζ(𝒗) is completely continuous from 𝒱p into 𝒞(Q¯T). Indeed, if 𝒗n𝒗 in 𝒱p, {ζ(𝒗n)}n is bounded in

𝒞λ(Q¯T)L2(0,T;H01(Ω))

and, for some subsequence, ζ(𝒗n)ζ weakly in L2(0,T;H01(Ω)) and uniformly in Q¯T, for a

ζL2(0,T;H01(Ω))𝒞(Q¯T),

where we have ζ=ζ(𝒗) by monotonicity and uniqueness of the solution of (3.4)–(3.5). Then the whole sequence converges and the complete continuity of G=G[𝒗], from 𝒱p into 𝒞(Q¯T), is guaranteed by the assumptions.

We observe that, if 𝕏pWs,p(Ω)m, s=1,2,, in (3.4) we can also choose

φ𝒗=φ0+0|α|s𝝍αα𝒗,

with 𝝍αL(QT)m, provided p>d+22, and even more general terms involving linear combinations of the gradients of the 𝝍αα𝒗Lp(QT), 0|α|s, provided p>d+2.

3.2 Linear differential operators

In this subsection, we illustrate some concrete results for the operators L referred to as examples in Section 1 for convex sets of the type (2.3). For simplicity, in all the examples we consider the vector fields

𝒂(𝝃)=α|𝝃|p-2𝝃,𝝃,α=α(x,t)0 a.e. in QT,𝒃𝟎,

and we assume that the operator G satisfies Assumption 2.6.

3.2.1 A problem with gradient constraint

Corollary 3.1.

Let Ω be a bounded open subset of Rd with a Lipschitz boundary, let Vp=Lp(0,T;W01,p(Ω)) and let p>max{1,2dd+2}. Let further fL2(QT) and u0KG[u0]. Then the following quasi-variational inequality has a weak solution:

{u𝕂G[u],0Ttv,v-up+QTα|u|p-2u(v-u)QTf(v-u)-12Ω|v(0)-u0|2,for all v𝒴p such that v𝕂G[u].

Actually, with Lu=u and Vp=W1,p(Ω), Assumptions 2.12.6 are satisfied because the inclusion of 𝕏p=W01,p(Ω) into =L2(Ω) is compact for p>max{1,2dd+2}.

The degenerate case α0 corresponds to the variational model of sandpile growth where G models the slope of the pile (see [29]). In [29], Prigozhin introduces an operator G which is discontinuous in the height u of the sandpile and leads to a quasi-variational formulation that is still an open problem.

3.2.2 A problem with Laplacian constraint

Corollary 3.2.

Let Ω be a bounded open subset of Rd with a C1,1 boundary, and let Vp=Lp(0,T;W02,p(Ω)) with p>max{1,2dd+4}. Let fL2(QT) and u0KG[u0]. Then the following quasi-variational inequality has a weak solution:

{u𝕂G[u],0Ttv,v-up+QTα|Δu|p-2ΔuΔ(v-u)QTf(v-u)-12Ω|v(0)-u0|2,for all v𝒴p such that v𝕂G[u].

Here we choose

Vp={vLp(Ω):ΔvLp(Ω)},

i.e. the operator L is the Laplacian. The subspace 𝕏p=W02,p(Ω) is endowed with the norm

v𝕏p=ΔuLp(Ω),

which is equivalent to the usual norm of W2,p(Ω) because Δ is an isomorphism between 𝕏p and Lp(Ω). Besides, (𝕏p,L2(Ω),𝕏p) is a Gelfand triple and the inclusion 𝕏pL2(Ω) is compact because p>max{1,2dd+4}.

3.2.3 A problem with curl constraint

Corollary 3.3.

Let Ω be a bounded open subset of R3 with a Lipschitz boundary, and let p>65 and fL2(QT). Define

𝕏p={𝒗Lp(Ω)3:×𝒗Lp(Ω)3,𝒗=0,𝒗𝒏|Ω=0}

or

𝕏p={𝒗Lp(Ω)3:×𝒗Lp(Ω)3,𝒗=0,𝒗×𝒏|Ω=𝟎}.

If 𝐮0KG[𝐮0], the following quasi-variational inequality has a weak solution:

{𝒖𝕂G[𝒖],0Tt𝒗,𝒗-𝒖p+QTα|×𝒖|p-2×𝒗×(𝒗-𝒗)QT𝒇(𝒗-𝒖)-12Ω|𝒗(0)-𝒖0|2,for all 𝒗𝒴p such that 𝒗𝕂G[𝒖].

Here L𝒗=×𝒗 and 𝒱p={𝒗Lp(Ω)3:×𝒗Lp(Ω)3}. In both choices of 𝕏p, corresponding to different boundary conditions, it is well known that 𝕏p is a closed subspace of W1,p(Ω)3 and that the semi-norm ×Lp(Ω)3 is a norm equivalent to the one induced in 𝕏p by the usual norm in W1,p(Ω)3 (for details see [1]). Here 𝕏p is compactly embedded in ={𝒗L2(Ω)3:𝒗=0}.

This model is related to the Bean-type superconductivity variational inequality, which was solved in [27], with prescribed critical threshold G. If we let here this threshold be, for instance, dependent on the temperature ζ defined by (3.4) and (3.5) and we impose p>52, we obtain the existence of a weak solution to the corresponding thermal and electromagnetic coupled problem.

3.2.4 Non-Newtonian thick fluids – a problem with a constraint on D

Set

D𝒖=12(𝒖+𝒖T),Vp={𝒗Lp(Ω)d:D𝒗Lp(Ω)d2},𝕁={𝒗𝒟(Ω)d:𝒗=0},𝕏p=𝕁¯W1,p(Ω)dfor p>1,d2.

Let 𝒱p=Lp(0,T;Vp) and observe that 𝕏p is compactly embedded in

={𝒗L2(Ω)d:𝒗=0}=𝕁¯L2(Ω)d

if p>2dd+2, by the Sobolev and Korn inequalities. Hence, using the results of [30, 26] for the variational inequality for incompressible thick fluids in the simpler case of the Stokes flow, we obtain the following conclusion.

Corollary 3.4.

Let Ω be a bounded open subset of Rd with a Lipschitz boundary, and let d2, p>2dd+2, 𝐟L2(QT)d and 𝐮0KG[𝐮0]. Then the quasi-variational inequality

{𝒖𝕂G[𝒖],0Tt𝒗,𝒗-𝒖p+QTα|D𝒖|p-2D𝒖D(𝒗-𝒖)QT𝒇(𝒗-𝒖)-12Ω|𝒗(0)-𝒖0|2,for all 𝒗𝒴p such that 𝒗𝕂G[𝒖]

has a weak solution.

3.2.5 A problem with first-order vector fields constraint

Let Ωd, d2, be a connected bounded open set and let L=(X1,,X) be a family of Lipschitz vector fields on d that connect the space. We shall assume that the regularity of Ω and the structure of L support the following Sobolev–Poincaré compact embedding for p2:

𝕏pL2(Ω).(3.7)

This is the case of an Hörmander operator with

Xj=i=1dαijxi,j=1,,,

with αij𝒞(Ω¯) such that the Lie algebra generated by these vector fields has dimension d, when the set 𝕏p is the closure of 𝒟(Ω) in

Vp={vLp(Ω):XjvLp(Ω),j=1,,}with p2,

with the graph norm and Ω𝒞. Indeed, in this case, it is known (see [10, 12, 9]) that the extension of the Rellich–Kondratchov theorem,

𝕏p=𝒟(Ω)¯VpL2(Ω)is compact for p2,

holds, and so (𝕏p,L2(Ω),𝕏p) is a Gelfand triple with compact embeddings. For other classes of vector fields, namely associated with degenerate subelliptic operators, and a characterization of domains where (3.7) holds, see, for instance, [12, 8]. By the application of Theorem 2.7 we can now conclude the following existence result.

Corollary 3.5.

Suppose that Ω is a bounded open subset of Rd with a smooth boundary. Under assumption (3.7), if p2, fL2(QT) and u0KG[u0], the quasi-variational inequality

{u𝕂G[u],0Ttv,v-up+QTαj=1(i=1|Xiu|2)p-22XjuXj(v-u)QTf(v-u)-12Ω|v(0)-u0|2,for all v𝒴p such that v𝕂G[𝒖]

has a weak solution.

4 The approximated problem

In order to establish the existence of a solution to the quasi-variational inequality (2.4), we start by proving the existence of the solution to the problem of an approximated system of equations, defined for fixed 𝝋, δ(0,1) and ε(0,1). With this regularization and penalization of the quasi-variational inequality (2.4) with convex sets 𝕂G[𝝋](t), t[0,T], we apply a fixed point argument. Consider the following increasing continuous function kε:0+:

kε(s)={0if s0,esε-1if 0s1ε,e1ε2-1if s1ε.(4.1)

Observe that the function kεδ=δ+kε approximates the maximal monotone graph

k¯δ(s){{δ}if s<0,[δ,[if s=0.

We start with an auxiliary lemma.

Lemma 4.1.

Let ψ be a scalar real function defined in QT. Then, for p(1,), the operator

𝑻ε(x,t,𝝃)=kε(|𝝃|-ψ(x,t))|𝝃|p-2𝝃(4.2)

is monotone.

Proof.

To simplify, we omit the argument (x,t) and we denote kε(|𝝃|-ψ) simply by kε(|𝝃|). We may assume, without loss of generality, that |𝝃||𝝃|. Because 𝑺(𝝃)=|𝝃|p-2𝝃 is monotone and kε is a nonnegative increasing function, we have

(𝑻ε(𝝃)-𝑻ε(𝝃))(𝝃-𝝃)=(kε(|𝝃|)𝑺(𝝃)-kε(|𝝃|)𝑺(𝝃))(𝝃-𝝃)=kε(|𝝃|)(𝑺(𝝃)-𝑺(𝝃))(𝝃-𝝃)+(kε(|𝝃|)-kε(|𝝃|))|𝝃|p-2𝝃(𝝃-𝝃)(kε(|𝝃|)-kε(|𝝃|))|𝝃|p-1(|𝝃|-|𝝃|)0.

Proposition 4.2.

Suppose that Assumptions 2.1 to 2.5 are satisfied. By considering functions

𝒇L2(QT)m,𝝋,𝒖0𝕂G[𝝋](0),(4.3)

the problem that consists of finding 𝐮εδ,𝛗 such that

{t𝒖εδ,𝝋(t),𝝍p+Ω𝒂(t,L𝒖εδ,𝝋(t))L𝝍+Ω𝒃(t,𝒖εδ,𝝋(t))𝝍+Ω(δ+kε(|L𝒖εδ,𝝋(t)|-G[𝝋](t)))|L𝒖εδ,𝝋(t)|p-2L𝒖εδ,𝝋(t)L𝝍=Ω𝒇(t)𝝍for all 𝝍𝕏p for a.e. t(0,T),𝒖εδ,𝝋(0)=𝒖0(4.4)

has a unique solution 𝐮εδ,𝛗Vp with t𝐮εδ,𝛗Vp, i.e. 𝐮εδ,φYpC([0,T];H).

Proof.

The existence and uniqueness of the solution of problem (4.4) is a consequence of a general result for parabolic quasilinear operators of monotone type (see, for instance, [33, Theorem 8.9, p. 224 or Theorem 8.30, p. 243]). ∎

Proposition 4.3.

Suppose that Assumptions 2.12.5 are satisfied. Under assumption (4.3), the solution 𝐮εδ,𝛗 of the problem (4.4) verifies the following a priori estimates:

𝒖εδ,𝝋L(0,T;L2(Ω)m)C,(4.5)L𝒖εδ,𝝋Lp(QT)Cδ1/p,(4.6)t𝒖εδ,𝝋(𝒱p)Cεδ1/p,

where C and Cε are positive constants independent of 𝛗 and of δ, and C is also independent of ε.

Proof.

Using 𝒘=𝒖εδ,𝝋 as a test function in (4.4), we get, for a.e. t(0,T),

t𝒘(t),𝒘(t)p+Ω𝒂(t,L𝒘(t))L𝒘(t)+Ω𝒃(t,𝒘(t))𝒘(t)+Ω(δ+kε(|L𝒘(t)|-G[𝝋](t)))|L𝒘(t)|p=Ω𝒇(t)𝒘(t).

Set Qt=Ω×(0,t). Integrating the last equality between 0 and t, recalling the monotonicity of 𝒂, 𝒃 and of 𝑻ε defined in (4.2), and applying the Hölder and Young inequalities to the right-hand side of the above equation, we obtain the inequality

Ω|𝒘(t)|2+2δQt|L𝒘|p𝒘L2(Qt)m2+𝒇L2(Qt)m2+Ω|𝒖0|2.(4.7)

By the Gronwall inequality we conclude that

𝒘L(0,T;L2(Ω))eT(𝒇L2(QT)m2+𝒖0L2(Ω)m2),

and so we proved (4.5). From (4.7) we immediately obtain (4.6).

Next we prove that, given 𝝍𝕏p,

|QT𝒃(𝒘)𝝍|Cmax{𝒘Lp(QT)p-1,𝒘L(0,T;L2(Ω))}𝝍𝕏p,(4.8)

with C being a positive constant. We notice that, by Assumption 2.3, 𝕏pL2(Ω)m. So there exists a positive constant C such that, for all 𝒗𝕏p, we have

𝒗L2p(Ω)mC𝒗𝕏p.

We split the proof in two cases.

(i) 1<p<2. Then

|QT𝒃(𝒘)𝝍|b*0TΩ|𝒘(t)||𝝍|b*0T𝒘(t)L2(Ω)m𝝍L2(Ω)mCb*T𝒘L(0,T;L2(Ω)m)𝝍𝕏p.

(ii) p2. Then

|QT𝒃(𝒘)𝝍|b*QT|𝒘|p-1|𝝍|b*0T𝒘(t)Lp(Ω)mp-1𝝍Lp(Ω)mCb*T1p𝒘Lp(QT)mp-1𝝍𝕏p.

From the first equation of (4.4) we conclude that

|t𝒘(t),𝝍p|C1((a*+δ+e1ε2)L𝒘(t)Lp(Ω)p-1L𝝍Lp(Ω)+max{𝒘Lp(QT)mp-1,𝒘L(0,T;L2(Ω)m)}𝝍𝕏p+𝒇(t)L2(Ω)m𝝍L2(Ω)m).

Using again Assumption 2.3, we obtain

0T|t𝒘(t),𝝍p|p𝑑tC2((a*+e1ε2)pL𝒘Lp(QT)p+max{𝒘Lp(QT)mp,𝒘L(0,T;L2(Ω)m)p}+𝒇L2(QT)mp)𝝍𝕏pp,

concluding now easily that

t𝒘(t)𝒱pp=0Tsup𝝍Xp1|t𝒘(t),𝝍p|pdtCεδ.

Proposition 4.4.

Suppose that Assumptions 2.12.5 are verified. Assuming also (4.3), define the function S:HYp by S(𝛗)=𝐮εδ,𝛗, where 𝐮εδ,𝛗 is the unique solution of problem (4.4). Then S is continuous.

Proof.

Let us consider a sequence {𝝋n}n converging to 𝝋 in . Setting 𝒘n=𝒖εδ,𝝋n and 𝒘=𝒖εδ,𝝋, we need to prove that

𝒘n𝑛𝒘 in 𝒱pandt𝒘n𝑛t𝒘 in 𝒱p.

The argument is standard, but we present it here for the sake of completeness. Both functions 𝒘n and 𝒘 solve (4.4), so, for any 𝝍𝕏p,

t(𝒘n(t)-𝒘(t)),𝝍p+Ω(𝒂(t,L𝒘n(t))-𝒂(t,L𝒘(t)))L𝝍+Ω(𝒃(t,𝒘n(t))-𝒃(t,𝒘(t)))𝝍+δΩ(|L𝒘n(t)|p-2L𝒘n(t)-|L𝒘(t)|p-2L𝒘(t))L𝝍+Ω(kε(|L𝒘n(t)|-G[𝝋n](t))|L𝒘n(t)|p-2L𝒘n(t)-kε(|L𝒘(t)|-G[𝝋](t))|L𝒘(t)|p-2L𝒘(t))L𝝍=0.

Replacing 𝝍 by 𝒘n(t)-𝒘(t) in the last expression and integrating it over (0,t), we get

12Ω|𝒘n(t)-𝒘(t)|2+Qt(𝒂(L𝒘n)-𝒂(L𝒘))L(𝒘n-𝒘)+Qt(𝒃(𝒘n)-𝒃(𝒘))(𝒘n-𝒘)   +δΩ(|L𝒘n|p-2L𝒘n-|L𝒘|p-2L𝒘)L(𝒘n-𝒘)   +Qt(kε(|L𝒘n|-G[𝝋n])|L𝒘n|p-2L𝒘n-kε(|L𝒘|-G[𝝋n])|L𝒘|p-2L𝒘)L(𝒘n-𝒘)=Qt(kε(|L𝒘|-G[𝝋])-kε(|L𝒘|-G[𝝋n]))|L𝒘|p-2L𝒘L(𝒘n-𝒘).(4.9)

Using the monotonicity of 𝒂, 𝒃 and the operator 𝑻ε defined in (4.2), we can neglect the second, third and fifth terms of the inequality above.

In the case p2, we obtain, applying the Hölder and Young inequalities and denoting by Dp the constant related to the strongly monotone term in δ (see (2.12)),

12Ω|𝒘n(t)-𝒘(t)|2+δDpQt|L(𝒘n-𝒘)|pQt|kε(|L𝒘|-G[𝝋])-kε(|L𝒘|-G[𝝋n])||L𝒘|p-1|L(𝒘n-𝒘)|C1δp-1Qt|kε(|L𝒘|-G[𝝋])-kε(|L𝒘|-G[𝝋n])|p|L𝒘|p+δ2DpQt|L(𝒘n-𝒘)|p,

and therefore we get

𝒘n-𝒘L(0,T;L2(Ω)m)2+δDpL(𝒘n-𝒘)Lp(QT)p2C1δp-1Qt|kε(|L𝒘|-G[𝝋])-kε(|L𝒘|-G[𝝋n])|p|L𝒘|p.(4.10)

Consider now the case 1<p<2. From (4.9) we get again

12Ω|𝒘n(t)-𝒘(t)|2+δQt(|L𝒘n|p-2L𝒘n-|L𝒘|p-2L𝒘)L(𝒘n-L𝒘)Qt|kε(|L𝒘|-G[𝝋])-kε(|L𝒘|-G[𝝋n])||L𝒘|p-1|L(𝒘n-𝒘)|,

and, using also the coercive condition on δ (see (2.12)) and the Hölder inverse inequality, we obtain

12Ω|𝒘n(t)-𝒘(t)|2+δDp(L𝒘nLp(QT)p+L𝒘Lp(QT)p)p-2pL(𝒘n-𝒘)Lp(QT)2Qt|kε(|L𝒘|-G[𝝋])-kε(|L𝒘|-G[𝝋n])||L𝒘|p-1|L(𝒘n-𝒘)|.

Recalling, by (4.6),

L𝒘nLp(QT)pCpδ,L𝒘Lp(QT)pCpδ

and applying the Hölder and Young inequalities to the right-hand side, we obtain

𝒘n-𝒘L(0,T;L2(Ω)m)2+δ2pC2L(𝒘n-𝒘)Lp(QT)212C2δ2p(QT|kε(|L𝒘|-G[𝝋])-kε(|L𝒘n|-G[𝝋n])|p|L𝒘|p)2p+C2δ2p2L(𝒘n-𝒘)Lp(QT)2,

and so

𝒘n-𝒘L(0,T;L2(Ω)m)2+C2δ2p2L(𝒘n-𝒘)Lp(QT)212C2δ2p(QT|kε(|L𝒘|-G[𝝋])-kε(|L𝒘n|-G[𝝋n])|p|L𝒘|p)2p.(4.11)

Observe now that we have

|kε(|L𝒘|-G[𝝋])-kε(|L𝒘n|-G[𝝋n])|p|L𝒘|p(2e1ε2)p|L𝒘|p

a.e. in QT, and kε is a continuous function. By recalling that 𝝋n𝑛𝝋 in , Assumption 2.6 implies that

G[𝝋n]𝑛G[𝝋]

in L1(QT). Hence, at least for a subsequence,

G[𝝋n]𝑛G[𝝋]a.e. in QT

and, by the dominated convergence theorem,

kε(|L𝒘n|-G[𝝋n])𝑛kε(|L𝒘|-G[𝝋])in Lp(QT).

Therefore, the right-hand sides of (4.10) and (4.11) converge to zero a.e. when n.

By definition,

t(𝒘n-𝒘)𝒱pp=0T(sup𝝍𝕏p1t(𝒘n(t)-𝒘(t)),𝝍)p)p=0T(sup𝝍𝕏p1(Ω(𝒂(t,L𝒘(t))-𝒂(t,L𝒘n(t)))L𝝍+Ω(𝒃(t,𝒘(t))-𝒃(t,𝒘n(t)))𝝍+δΩ(|L𝒘(t)|p-2L𝒘(t)-|L𝒘n(t)|p-2L𝒘n(t))L𝝍+Ωkε(|L𝒘(t)|-G[𝝋](t))|L𝒘(t)|p-2L𝒘(t)L𝝍-Ωkε(|L𝒘n(t)|-G[𝝋n](t))|L𝒘n(t)|p-2L𝒘n(t)L𝝍))p.

But

(Ω(𝒃(t,𝒘(t))-𝒃(t,𝒘n(t)))𝝍)p((𝒃(t,𝒘(t))-𝒃(t,𝒘n(t)))L2(Ω)m𝝍L2(Ω)m)pC((𝒃(𝒘)-𝒃(𝒘n))L(0,T;(L2(Ω)m)𝝍𝕏p)p.

Applying the Hölder inequality, we conclude that

t(𝒘n-𝒘)𝒱pC((QT|𝒂(L𝒘)-𝒂(L𝒘n)|p)1p+(𝒃(𝒘)-𝒃(𝒘n))L(0,T;(L2(Ω)m)   +δ(QT||L𝒘n|p-2L𝒘n-|L𝒘|p-2L𝒘|p)1p   +(QT|kε(|L𝒘|-G[𝝋])|L𝒘|p-2L𝒘-kε(|L𝒘n|-G[𝝋n])|L𝒘n|p-2L𝒘n|p)1p),

and, arguing as before, we conclude the proof. ∎

Theorem 4.5.

Suppose that Assumptions 2.12.6 and (4.3) are verified. Let i be the inclusion of Yp into H and let S:HYp be the function defined in Proposition 4.4. Then the function iS has a fixed point in H. This fixed point solves the problem that consists of finding 𝐮εδYp such that

{t𝒖εδ(t),𝝍p+Ω𝒂(t,L𝒖εδ(t))L𝝍+Ω𝒃(t,𝒖εδ(t))𝝍+δΩ|L𝒖εδ(t)|p-2L𝒖εδ(t)L𝝍+Ωkε(|L𝒖εδ(t)|-G[𝒖εδ(t)])|L𝒖εδ(t)|p-2L𝒖εδ(t)L𝝍=Ω𝒇(t)𝝍for all 𝝍𝕏p𝒖εδ(0)=𝒖0.(4.12)

Proof.

We use the Schauder fixed point theorem. We already proved the continuity of S. By Assumption 2.5 and the Aubin–Lions lemma the embedding 𝒴p is compact for 1<p<, and so iS is completely continuous as a map of into itself. By Proposition 4.3, given 𝝋, we have

𝒖εδ,𝝋𝒱pCδ1/p,

where C is a constant independent of 𝝋 and δ (it may depend on ε). Because i is continuous, there exists C1 such that 𝒗C1𝒗𝒴p, and we get

iS(𝝋)CC1.

Then the image of iS is bounded, so we may apply the Schauder fixed point theorem, obtaining immediately the conclusion of the existence of a 𝒖εδ=iS(𝒖εδ) in 𝒴p. ∎

5 Weak solutions of the quasi-variational inequality

In this section, we prove the existence of a solution of the quasi-variational inequality (2.4) by taking suitable subsequences of solutions of (4.12) first as ε0 and then as δ0.

Firstly we collect the a priori estimates for the solution 𝒖εδ of problem (4.12) which are independent of ε.

Proposition 5.1.

Suppose that Assumptions 2.12.6 are verified. Assume that 𝐟L2(QT)m and 𝐮0K[G(𝐮0)]. Let 𝐮εδ be a solution of the approximated problem (4.12). Then there exists a positive constant C independent of ε and δ such that

𝒖εδL(0,T;L2(Ω)m)C,(5.1)𝒖εδ𝒱pCδ1/p,(5.2)𝒂(L𝒖εδ)𝒱pCδ1/p,(5.3)𝒃(𝒖εδ)𝒱pCδ1/p,(5.4)kε(|L𝒖εδ|-G[𝒖εδ])L1(QT)C.(5.5)

Proof.

The first two estimates are direct consequences of inequalities (4.5) and (4.6), respectively, taking 𝝋=𝒖εδ.

Given 𝝍𝒱p, we have

|QT𝒂(L𝒖εδ)L𝝍|a*QT|L𝒖εδ|p-1|L𝝍|a*L𝒖εδLp(QT)p-1L𝝍Lp(QT)a*Cp-1δ1/pL𝝍𝒱p,

proving (5.3).

From (4.8) we have

𝒃(𝒘)𝒱pCmax{𝒘Lp(QT)mp-1,𝒘L(0,T;L2(Ω)m)}.

But 𝒘L(0,T;L2(Ω)m) is uniformly bounded by (5.1), and

𝒘Lp(QT)mp-1C1L𝒘Lp(QT)mp-1C2δ1/p

by (5.2).

Choosing 𝒖εδ as test function in (4.12), and integrating between 0 and t, we get

12Ω|𝒖εδ(t)|2+Qt𝒂(L𝒖εδ)L𝒖εδ+Qt𝒃(𝒖εδ)𝒖εδ+δQt|L𝒖εδ|p+Qtkε(|L𝒖εδ|-G[𝒖εδ])|L𝒖εδ|p=Qt𝒇𝒖εδ+12Ω𝒖02.

Therefore, because 𝒂 and 𝒃 are monotone, 𝒂(𝟎)=𝟎, 𝒃(𝟎)=𝟎 and by using the Gronwall inequality, we obtain

Qtkε(|L𝒖εδ|-G[𝒖εδ])|L𝒖εδ|pC1,(5.6)

where C1 is a constant independent of ε and δ. As G[𝒖εδ]g*>0 and kε0 in {|L𝒖εδ|G[𝒖εδ]}, we obtain

QTkε(|L𝒖εδ|-G[𝒖εδ])={|L𝒖εδ|>G[𝒖εδ]}kε(|L𝒖εδ|-G[𝒖εδ]){|L𝒖εδ|>G[𝒖εδ]}kε(|L𝒖εδ|-G[𝒖εδ])|L𝒖εδ|p(g*)p.

Using (5.6), we obtain

kε(|L𝒖εδ|-G[𝒖εδ])L1(QT)C1(g*)p,

concluding the proof of (5.5). ∎

Lemma 5.2.

Let 𝐮εδ be a solution of the approximated problem (4.12). If 𝐮δ is the weak limit of a subsequence of {𝐮εδ}ε when ε0, then 𝐮δKG[𝐮δ].

Proof.

To prove that 𝒖δ belongs to the convex set 𝕂G[𝒖δ], we use arguments as in [27], which we adapt to our problem. We split QT in three sets:

{Aεδ={(x,t)QT:|L𝒖εδ(x,t)|-G[𝒖εδ](x,t)<ε},Bεδ={(x,t)QT:ε|L𝒖εδ(x,t)|-G[𝒖εδ](x,t)<1ε},Cεδ={(x,t)QT:|L𝒖εδ(x,t)|-G[𝒖εδ](x,t)1ε}.(5.7)

We recall that, by Assumption 2.6, the operator G is compact. So, as a subsequence of {𝒖εδ}ε (still denoted by {𝒖εδ}ε) converges weakly to 𝒖δ in 𝒱p, we obtain that G[𝒖εδ] converges to G[𝒖δ] strongly in 𝒞([0,T];L(Ω)) and

QT(|L𝒖δ|-G[𝒖δ])+=QTlim¯ε0((|L𝒖εδ|-G[𝒖εδ])1εε)lim¯ε0QT((|L𝒖εδ|-G[𝒖εδ])1εε)=lim¯ε0(Aεδε+Bεδ(|L𝒖εδ|-G[𝒖εδ])+Cεδ1ε).

We observe that

Aεδεε|QT|ε00,|Bεδ|=Bεδ1QTkε(|L𝒖εδ|-G[𝒖εδ])+1e1/εCe1/εε00Cεδ1ε1εQTkε(|L𝒖εδ|-G[𝒖εδ])e1/ε2-1Cε(e1/ε2-1)ε00.

Besides,

Bεδ(|L𝒖εδ|-G[𝒖εδ])|L𝒖εδ|-G[𝒖εδ]Lp(QT)|Bεδ|1pε00.

So,

QT(|L𝒖δ|-G[𝒖δ])+=0,

which means that

|L𝒖δ|G[𝒖δ]a.e. in QT.

Lemma 5.3.

Let 𝐯Vp=Lp(0,T;Xp) be such that 𝐯KG[𝐯], and let 𝐳KG[𝐯](0). Then there exists a regularizing sequence {𝐯n}n and a sequence of scalar functions {gn}n with the following properties:

  • (i)

    𝒗nL(0,T;𝕏p) and t𝒗nL(0,T;𝕏p).

  • (ii)

    𝒗n𝑛𝒗 strongly in 𝒱p.

  • (iii)

    lim¯n0Tt𝒗n,𝒗n-𝒗p0.

  • (iv)

    |L𝒗n|gn , where gn𝒞([0,T];L(Ω)) and gn𝑛G[𝒗] in 𝒞([0,T];L(Ω)).

Proof.

Let 𝒗n be the unique solution of the ordinary differential equation 𝒗n+1nt𝒗n=𝒗 with 𝒗n(0)=𝒛. The function 𝒗n has the following expression:

𝒗n(x,t)=e-nt0t𝒗(x,τ)nenτ𝑑τ+e-nt𝒛(x)

and it is well known that it satisfies (i), (ii) and (iii) (see [23, p. 274] or [33, p. 206]). Therefore, it follows

|L𝒗n(x,t)|e-nt0t|L𝒗(x,τ)|nenτ𝑑τ+e-nt|L𝒛(x)|e-nt0tG[𝒗](x,τ)nenτ𝑑τ+e-ntG[𝒗](x,0)for a.e. (x,t)QT.

Defining

gn(x,t)=e-nt0tG[𝒗](x,τ)nenτ𝑑τ+e-ntG[𝒗](x,0)=-0G~[𝒗](x,sn+t)es𝑑s,

where G~[𝒗](x,t) denotes the extension of G[𝒗]𝒞([0,T];L(Ω)) by G[𝒗](x,0) for t<0, we have

|gn(x,t)-G[𝒗](x,t)|-0|G~[𝒗](x,sn+t)-G[𝒗](x,t)|es𝑑s

and, by the uniform continuity of G[𝒗](t) in [0,T] with values in L(Ω), we also have the uniform convergence of gn𝑛G[𝒗] in 𝒞([0,T];L(Ω), concluding (iv). ∎

Proof of Theorem 2.7.

The boundedness of {𝒖εδ}ε in L(0,T;L2(Ω)m)𝒱p implies that there exists a subsequence, still denoted by {𝒖εδ}ε, converging weakly-* to a function 𝒖δ when ε0, in L(0,T;L2(Ω)m)𝒱p. So, by Lemma 5.2, 𝒖δ𝕂G[𝒖δ] and we may extract a subsequence of {𝒖δ}δ converging weakly-* to some 𝒖 in L(0,T;L2(Ω)m)𝒱p when δ0.

Observe that, for any measurable set ωQT,

ω|L𝒖|lim¯δ0ω|L𝒖δ|ωlimδ0G[𝒖δ]ωG[𝒖],

using Assumption 2.6. Consequently,

|L𝒖|G[𝒖]a.e. in QT,

and so 𝒖𝕂G[𝒖].

Step 1: The limit when ε0. From estimates (5.3), (5.4) and (5.2) in Proposition 5.1 there exist 𝝌δ𝒱p, 𝚼δ𝒱p and 𝚲δLp(QT) such that for subsequences,

𝒂(L𝒖εδ)ε0𝝌δin 𝒱p weak,𝒃(𝒖εδ)ε0𝚼δin 𝒱p weak.|L𝒖εδ|p-2L𝒖εδε0𝚲δin Lp(QT) weak.(5.8)

Define the operator 𝒜δ:𝒱p𝒱p by

𝒜δ𝒗,𝒘=QT(𝒂(L𝒗)+δ|L𝒗|-2L𝒗)L𝒘+QT𝒃(𝒗)𝒘.

Given 𝒗 belonging to the space 𝒴p defined in (2.2), we have

0Tt𝒖εδ,𝒗-𝒖εδp=-12Ω|𝒖εδ(T)-𝒗(T)|2+12Ω|𝒖0-𝒗(0)|2+0Tt𝒗,𝒗-𝒖εδp12Ω|𝒖0-𝒗(0)|2+0Tt𝒗,𝒗-𝒖εδp.

From now on, we denote kε(|L𝒖εδ|-G[𝒖εδ]) simply by kε, with no risk of confusion.

Using 𝒖εδ-𝒗 as test function in (4.12) and integrating between 0 and T, we obtain

𝒜δ𝒖εδ,𝒖εδ12Ω|𝒖0-𝒗(0)|2+0Tt𝒗,𝒗-𝒖εδp+𝒜δ𝒖εδ,𝒗+QTkε|L𝒖εδ|p-2L𝒖εδL(𝒗-𝒖εδ)-QT𝒇(𝒗-𝒖εδ).(5.9)

Hence, for all 𝒗𝒴p,

𝒜δ𝒖εδ,𝒖εδ-𝒖12Ω|𝒖0-𝒗(0)|2+0Tt𝒗,𝒗-𝒖εδp+𝒜δ𝒖εδ,𝒗-𝒖+QTkε|L𝒖εδ|p-2L𝒖εδL(𝒗-𝒖εδ)-QT𝒇(𝒗-𝒖εδ).(5.10)

Let 𝒖n be the regularizing sequence of 𝒖 defined in the previous lemma. Using 𝒖n as test function in (5.10), we get

𝒜δ𝒖εδ,𝒖εδ-𝒖0Tt𝒖n,𝒖n-𝒖εδp+𝒜δ𝒖εδ,𝒖n-𝒖+QTkε|L𝒖εδ|p-2L𝒖εδL(𝒖n-𝒖εδ)-QT𝒇(𝒖n-𝒖εδ).(5.11)

But

QTkε|L𝒖εδ|p-2L𝒖εδL(𝒖n-𝒖εδ)QTkε|L𝒖εδ|p-1(|L𝒖n|-|L𝒖εδ|)=QTkε|L𝒖εδ|p-1(|L𝒖n|-G[𝒖εδ])+QTkε|L𝒖εδ|p-1(G[𝒖εδ]-|L𝒖εδ|)QTkε|L𝒖εδ|p-1(|L𝒖n|-G[𝒖εδ]).(5.12)

In fact, the term kε|L𝒖εδ|p-1(G[𝒖εδ]-|L𝒖εδ|) is less than or equal to zero because when |L𝒖εδ|<G[𝒖εδ], then kε(|L𝒖εδ|-G[𝒖εδ])=0.

Then, recalling (5.5) and (5.6), we conclude that

QTkε|L𝒖εδ|p-1(|L𝒖n|-G[𝒖εδ])QTkε|L𝒖εδ|p-1(Gn-G[𝒖εδ])(QTkε)1p(QTkε|L𝒖εδ|p)1pGn-G[𝒖εδ]L(QT)CGn-G[𝒖εδ]L(QT).(5.13)

Going back to (5.11) and using (5.12) and the estimate above, we get

𝒜δ𝒖εδ,𝒖εδ-𝒖0Tt𝒖n,𝒖n-𝒖εδp+𝒜δ𝒖εδ,𝒖n-𝒖+CGn-G[𝒖εδ]L(QT)-QT𝒇(𝒖n-𝒖εδ).

So, noticing that G[𝒖εδ]ε0G[𝒖δ] in 𝒞([0,T];L(Ω)) and recalling (5), we obtain

lim¯ε0𝒜δ𝒖εδ,𝒖εδ-𝒖0Tt𝒖n,𝒖n-𝒖δp+0T𝝌δ+δ𝚲δ,L(𝒖n-𝒖)+0T𝚼δ,𝒖n-𝒖+CGn-G[𝒖δ]L(QT)-QT𝒇(𝒖n-𝒖δ).

Step 2: The limit when δ0. Because there exists a positive constant C independent of ε and δ such that

𝝌δ𝒱plim¯ε0𝒂(L𝒖εδ)𝒱pC,𝚼δ𝒱plim¯ε0𝒃(𝒖εδ)Lp(QT)mC,𝚲δLp(QT)lim¯ε0|L𝒖εδ|p-2L𝒖εδLp(QT)C,

we obtain, for a subsequence,

𝝌δδ0𝝌in 𝒱p,𝚼δδ0𝚼in 𝒱p,δ𝚲δδ00in Lp(QT).

Since G[𝒖δ]δ0G[𝒖] in L(QT), we obtain

lim¯δ0lim¯ε0𝒜δ𝒖εδ,𝒖εδ-𝒖0Tt𝒖n,𝒖n-𝒖p+0T𝝌,L(𝒖n-𝒖)+0T𝚼,𝒖n-𝒖+CGn-G[𝒖]L(QT)-QT𝒇(𝒖n-𝒖).

Letting n in the above inequality, using that 0Tt𝒖n,𝒖n-𝒖p0 and Gn𝑛G[𝒖] in 𝒞([0,T];L(Ω)), we conclude that

lim¯δ0lim¯ε0𝒜δ𝒖εδ,𝒖εδ-𝒖0.

If 𝒜:𝒱p𝒱p is defined by

𝒜𝒗,𝒘=QT𝒂(L𝒗)L𝒘+QT𝒃(𝒗)𝒘,

then

𝒜𝒖εδ,𝒖εδ-𝒖=𝒜δ𝒖εδ,𝒖εδ-𝒖-δQT|L𝒖εδ|p-2L𝒖εδL(𝒖εδ-𝒖)

yields

lim¯δ0lim¯ε0𝒜𝒖εδ,𝒖εδ-𝒖=lim¯δ0lim¯ε0𝒜δ𝒖εδ,𝒖εδ-𝒖0.

Step 3: Conclusion. Let aδ=lim¯ε0𝒜𝒖εδ,𝒖εδ-𝒖. As lim¯δ0aδ0, given η>0, there exists δη>0 such that aδη<η2. But, because

lim¯ε0𝒜𝒖εδη,𝒖εδη-𝒖=aδη,

we can find εη=ε(δn)>0 for which 𝒜𝒖εηδη,𝒖εηδη-𝒖<η. So,

lim¯η0𝒜𝒖εηδη,𝒖εηδη-𝒖0.

From now on we set 𝒖η=𝒖εηδη and kη=kεη.

As the operator 𝒜 is bounded, monotone and hemicontinuous, it is pseudo-monotone. Therefore, as lim¯η0𝒜𝒖η,𝒖η-𝒖0, we obtain

𝒜𝒖,𝒖-𝒗lim¯η0𝒜𝒖η,𝒖η-𝒗for all 𝒗𝕂G[𝒖].

Finally, we conclude, going back to (5.9), that if 𝒗𝕂G[𝒖], then

𝒜𝒖,𝒖-𝒗lim¯η0(12Ω|𝒖0-𝒗(0)|2+0Tt𝒗,𝒗-𝒖ηp+QTkη|L𝒖η|p-2L𝒖ηL(𝒗-𝒖η)-QT𝒇(𝒗-𝒖η)).

But, as 𝒗𝕂G[𝒖], as in (5.12) and (5.13),

QTkη|L𝒖η|p-2L𝒖ηL(𝒗-𝒖η)QTkη|L𝒖η|p-1(|L𝒗|-|L𝒖η|)QTkη|L𝒖η|p-1(G[𝒖η]-|L𝒖η|)+QTkη|L𝒖η|p-1(G[𝒖]-G[𝒖η])CG[𝒖]-G[𝒖η]𝒞([0,T];L(Ω)).

Then

lim¯η0QTkη|L𝒖η|p-2L𝒖ηL(𝒗-𝒖η)0

because limη0G[𝒖η]=G[𝒖] in 𝒞([0,T];L(Ω)). Hence,

𝒜𝒖,𝒖-𝒗12Ω|𝒖0-𝒗(0)|2+0Tt𝒗,𝒗-𝒖-QT𝒇(𝒗-𝒖),

concluding the proof since we already know that 𝒖𝕂G[𝒖]. ∎

Proof of Theorem 2.8.

Let 𝒖1,𝒖2𝕂g be two solutions of (2.4) and denote by {𝒘n}n and by {gn}n the regularizing sequences of Lemma 5.3 of 𝒘=𝒖1+𝒖22𝕂g and g, respectively, with 𝒛=𝒖0. Considering

εn=gn-g𝒞([0,T];L(Ω))andρn=g*g*+εn𝑛1,

we have 𝒘^n=ρn𝒘n𝕂g𝒴p and it may be chosen as test function in (2.4) for 𝒖1 and 𝒖2. We obtain, by addition,

20Tt𝒘^n,𝒘-𝒘^np+𝒜𝒖1,𝒖1-𝒘^n+𝒜𝒖2,𝒖2-𝒘^n2QT𝒇(𝒘-𝒘^n)+(ρn-1)2Ω|𝒖0|2.(5.14)

Observing that

t𝒘^n,𝒘^n-𝒘p=ρnt𝒘n,𝒘n-𝒘p+ρn(ρn-1)t𝒘n,𝒘np

and integrating in time, since 𝒘nW1,p(0,T;𝕏p)𝒞(0,T;L2(Ω)m), we have

0Tt𝒘^n,𝒘^n-𝒘p=ρn0Tt𝒘n,𝒘n-𝒘p+12ρn(1-ρn)(𝒖0L2(Ω)m2-𝒘n(T)L2(Ω)m2)12ρn(1-ρn)𝒖0L2(Ω)m2𝑛0.

Therefore, taking the limit in (5.14), since 𝒘^n𝑛𝒖1+𝒖22, we obtain

QT(𝒂(L𝒖1)-𝒂(L𝒖2))(L𝒖1-L𝒖2)+QT(𝒃(𝒖1)-𝒃(𝒖2))(𝒖1-𝒖2)=𝒜𝒖1-𝒜𝒖2,𝒖1-𝒖20,

and the conclusion 𝒖1=𝒖2 follows by the strict monotonicity of 𝒃 or 𝒂 with Assumption 2.2. ∎

6 Solution of the variational inequality

We study now the variational inequality case as well as the continuous dependence of its solution on the given data. We obtain different stability results whether we consider the case where the operator 𝒂 is monotone or strongly monotone.

Proof of Theorem 2.10.

We penalize the variational inequality using the function kε defined in (4.1), as we have done in Section 4. For ε(0,1) and δ>0, let us consider the problem of finding

𝒘εδ𝒱pH1(0,T;L2(Ω))

such that

{Ωt𝒘εδ(t)𝝍+Ω𝒂(t,L𝒘εδ(t))L𝝍+Ω𝒃(t,𝒘εδ(t))L𝝍+Ω(δ+kε(|L𝒘εδ(t)|p-g(t)p))|L𝒘εδ(t)|p-2L𝒘εδ(t)L𝝍=Ω𝒇(t)𝝍for all 𝝍𝕏p for a.e. t(0,T),𝒘εδ(0)=𝒘0.(6.1)

The proof of the existence of a solution for this problem is similar to the proof of Proposition 4.2 and can be done with the Galerkin method (see, for instance, [33, p. 240]). We observe that we consider here the function kε(|L𝒘εδ|p-gp) instead of kε(|L𝒘εδ|-G[𝒘εδ]).

As in estimates (4.5), (4.6) and (5.5), we obtain, with a constant C>0 independent of ε and δ,

𝒘εδL(0,T;L2(Ω)m)C,(6.2)L𝒘εδLp(QT)Cδ1/p,(6.3)kε(|L𝒘εδ|p-gp)L1(QT)C.(6.4)

Using Galerkin’s approximation, we can argue formally with t𝒘εδ as a test function on (6.1) and we get

Ω|t𝒘εδ(t)|2+Ω𝒂(t,L𝒘εδ(t))tL𝒘εδ(t)+Ω𝒃(t,𝒘εδ(t))t𝒘εδ(t)   +Ω(δ+kε(|L𝒘εδ|p-g(t)p))|L𝒘εδ(t)|p-2L𝒘εδ(t)tL𝒘εδ(t)=Ω𝒇(t)t𝒘εδ(t).(6.5)

Set ϕε(s)=0skε(τ)𝑑τ and observe that

kε(|L𝒘εδ(t)|p-gp(t))|L𝒘εδ(t)|p-2L𝒘εδ(t)tL𝒘εδ(t)=1pkε(|L𝒘εδ(t)|p-gp(t))t(|L𝒘εδ(t)|p-gp(t))+kε(|L𝒘εδ(t)|p-gp(t))gp-1(t)tg(t)=1pt(ϕε(|L𝒘εδ(t)|p-gp(t))+kε(|L𝒘εδ(t)|p-gp(t))gp-1(t)tg(t).

Integrating (6.5) between 0 and T, we obtain

QT|t𝒘εδ|2+ΩA(T,L𝒘εδ(T))-ΩA(0,L𝒘0)-QT(tA)(L𝒘εδ)+δpΩ|L𝒘εδ(T)|p-δpΩ|L𝒘0|p   +1pΩϕε(|L𝒘εδ(T)|p-gp(T))-1pΩϕε(|L𝒘0|p-gp(0))+QTkε(|L𝒘εδ|p-gp)gp-1tg=QT(𝒇-𝒃(𝒘εδ))t𝒘εδ(6.6)

since A satisfies (2.7). But

ϕε(|L𝒘εδ(T)|p-gp(T))0,ϕε(|L𝒘0|p-gp(0))=0because |L𝒘0|g(0),

and, using assumption (2.9), the Hölder and Young inequalities,

QT(𝒇-𝒃(𝒘εδ))t𝒘εδQT(|𝒇|+b*|𝒘εδ|)|t𝒘εδ|C(𝒇L2(QT)m2+b*𝒘εδL2(QT)m2)+12t𝒘εδL2(QT)m2

and (2.8), from (6.6) we have

12t𝒘εδL2(Ω)m2C(𝒇L2(QT)m2+b*𝒘εδL2(QT)m2)+2a*L𝒘0Lp(Ω)p+A1|QT|+A2L𝒘εδLp(QT)p+δpL𝒘0Lp(Ω)p+kε(|L𝒘εδ|p-gp)L1(QT)gLp(QT)p-1tgL(QT).

From (6.2)–(6.4) we obtain, with a constant C>0 independent of ε and δ,

t𝒘εδL2(QT)mC(1+L𝒘εδLp(QT)p).(6.7)

Then, recalling that Assumption 2.5 implies, by the Aubin–Lions lemma, the compactness of 𝒴p, there exists a subsequence that we still denote by {𝒘εδ}ε such that, for every t(0,T],

𝒘εδε0𝒘δin L2(QT)m,L𝒘εδε0L𝒘δin Lp(QT) weak,t𝒘εδε0t𝒘δin L2(QT)m weak.

Recalling Lemma 4.1 and observing that kε(|L𝒗|p-gp)=0 if 𝒗𝕂g, we have, for any t(0,T],

Qtkε(|L𝒘εδ|p-gp)|L𝒘εδ|p-2L𝒘εδL(𝒗-𝒘εδ)=Qt(kε(|L𝒘εδ|p-gp)|L𝒘εδ|p-2L𝒘εδ-kε(|L𝒗|p-gp)|L𝒗|p-2L𝒗)L(𝒗-𝒘εδ)   +Qtkε(|L𝒗|p-gp)|L𝒗|p-2L𝒗L(𝒗-𝒘εδ)0for all 𝒗𝕂g.(6.8)

Using 𝒗-𝒘εδ as test function in (6.1) and integrating over (0,t), by (6.8) and by the monotonicity of the operators 𝒂, 𝒃 and 𝝃|L𝝃|p-2L𝝃, we obtain

Qtt𝒘εδ(𝒗-𝒘εδ)+Qt𝒂(L𝒗)L(𝒗-𝒘εδ)+Qt𝒃(𝒗)(𝒗-𝒘εδ)+δQt|L𝒗|p-2L𝒗L(𝒗-𝒘εδ)Qt𝒇(𝒗-𝒘εδ).

Passing to the limit when ε tends to zero, we get

Qtt𝒘δ(𝒗-𝒘δ)+Qt𝒂(L𝒗)L(𝒗-𝒘δ)+Qt𝒃(𝒗)(𝒗-𝒘δ)+δQt|L𝒗|p-2L𝒗L(𝒗-𝒘δ)Qt𝒇(𝒗-𝒘δ).(6.9)

Arguing as in Lemma 5.2, we also prove that 𝒘δ𝕂g.

The next step is to let δ0. From (6.7) we have

t𝒘δL2(QT)mlim¯ε0t𝒘εδL2(QT)mC(1+lim¯ε0QT|L(𝒘εδ|p).

Using the sets defined in (5.7), we get

QT|L𝒘εδ|p=Aεδ|L𝒘εδ|p+Bεδ(|L𝒘εδ|p-gp+gp)+Cεδ|L𝒘εδ|p2gLp(QT)p+|QT|+Bεδεkε(|L𝒘εδ|p-gp)+C1e1/ε2-1C

because, for s(0,1ε), sεkε(s), and by (5.6),

Cεδ(e1/ε2-1)|L𝒘εδ|p=Cεδkε(|L𝒘εδ|p-gp)|L𝒘εδ|pC1,

so {t𝒘δ}δ is also uniformly bounded in L2(QT)m. Since 𝒘δ𝕂g, we have |L𝒘δ| bounded in L(QT) independently of δ. Then, for a subsequence, we have

𝒘δδ0𝒘in H1(0,T;L2(QT)m) weak,L𝒘δδ0L𝒘in L(QT) weak-*,𝒘δ(t)δ0𝒘(t)in L2(Ω)m weak for all 0<tT.

We can pass to the limit when δ0 in (6.9), writing

Qtt𝒘δ𝒗+Qt𝒂(L𝒗)L(𝒗-𝒘δ)+Qt𝒃(𝒗)(𝒗-𝒘δ)+δQt|L𝒗|p-2L𝒗L(𝒗-𝒘δ)Qt𝒇(𝒗-𝒘δ)+12Ω|𝒘δ(t)|2-12Ω|𝒘0|2.

Because 𝒘δ(t)𝒘(t) in L2(Ω)m-weak yields

lim¯δ0Ω|𝒘δ(t)|2Ω|𝒘(t)|2

for each 0<tT, we recover in the limit that 𝒘 satisfies

Qtt𝒘(𝒗-𝒘)+Qt𝒂(L𝒗)L(𝒗-𝒘)+Qt𝒃(𝒗)(𝒗-𝒘)Qt𝒇(𝒗-𝒘)for all 𝒗𝕂g.

Finally, as in the proof of Theorem 2.7, 𝒘 also belongs to 𝕂g. We may apply Minty’s lemma and conclude that it solves (2.6).

The uniqueness of the solution is immediate since if 𝒘1 and 𝒘2 are two solutions of (2.6), then

Qtt(𝒘1-𝒘2)(𝒘1-𝒘2)+Qt(𝒂(L𝒘1)-𝒂(L𝒘2))L(𝒘1-𝒘2)+Qt(𝒃(𝒘1)-𝒃(𝒘2))(𝒘1-𝒘2)0

and, by monotonicity of 𝒂 and 𝒃, we get

Ω|𝒘1(t)-𝒘2(t)|20 for all t(0,T),

concluding that 𝒘1=𝒘2. ∎

Next we prove the stability of the solutions of the variational inequality (2.6) with respect to the given data. The results we obtain depend on the assumptions on 𝒂, and we are able to give a result even in the very degenerate case 𝒂𝟎 and 𝒃𝟎.

Proof of Theorem 2.11.

Considering the threshold functions g1 and g2 satisfying (2.10), let 𝕂g1(t) and 𝕂g2(t) be, respectively, the corresponding convex sets defined in (2.5). For β(t)=gi(t)-gj(t)L(Ω) for i,j{1,2}, ij, and given 𝒘i such that 𝒘i(t)𝕂gi(t) for a.e. t(0,T), we define the functions

𝒘ij(t)=g*𝒘i(t)g*+β(t)𝕂gj(t).

Choosing

Cmax{1g*𝒘iL2(QT)m,1g*L𝒘iLp(QT)},

we observe that

𝒘i(t)-𝒘ij(t)L2(Ω)mCβ(t)andL(𝒘i(t)-𝒘ij(t))Lp(Ω)Cβ(t).(6.10)

Considering, for i,j{1,2}, ij, the solution 𝒘i of the variational inequality (2.6) associated to the constraint gi, and using 𝒘ji as test function, we have, for t(0,T],

Qtt𝒘i(𝒘i-𝒘ji)+Qt𝒂(L𝒘i)L(𝒘i-𝒘ji)+Qt𝒃(𝒘i)(𝒘i-𝒘ji)Qt𝒇i(𝒘i-𝒘ji),

and so

Qtt𝒘i(𝒘i-𝒘j)+Qt𝒂(L𝒘i)L(𝒘i-𝒘j)+Qt𝒃(𝒘i)(𝒘i-𝒘j)Qt𝒇i(𝒘i-𝒘j)+Qtt𝒘i(𝒘ji-𝒘j)+Qt𝒂(L𝒘i)L(𝒘ji-𝒘j)   +Qt𝒃(𝒘i)(𝒘ji-𝒘j)+Qt𝒇i(𝒘j-𝒘ji).

Adding the inequalities we obtained in the former expression to (i,j)=(1,2) and (i,j)=(2,1), and setting 𝒘=𝒘1-𝒘2, we get

Qtt𝒘𝒘+Qt(𝒂(L𝒘1)-𝒂(L𝒘2))L𝒘+Qt(𝒃(𝒘1)-𝒃(𝒘2))𝒘Qt(𝒇1-𝒇2)𝒘+Θ(t),(6.11)

where

Θ(t)=Qtt𝒘1(𝒘21-𝒘2)+Qtt𝒘2(𝒘12-𝒘1)+Qt𝒂(L𝒘1)L(𝒘21-𝒘2)+Qt𝒂(L𝒘2)L(𝒘12-𝒘1)+Qt𝒃(𝒘1)(𝒘21-𝒘2)+Qt𝒃(𝒘2)(𝒘12-𝒘1)+Qt𝒇1(𝒘2-𝒘21)+Qt𝒇2(𝒘1-𝒘12).

Estimates (6.2), (6.3), (6.7) and (6.10) allow us to conclude that there exists a positive constant C such that, for any t(0,T),

Θ(t)C0tg1(τ)-g2(τ)L(Ω)𝑑τ.(6.12)

From (6.11) and (6.12), using the monotonicity of 𝒂 and 𝒃, we obtain

ddt𝒘(t)L2(Ω)m2𝒘(t)L2(Ω)m2+𝒇1(t)-𝒇2(t)L2(Ω)m2+2Cg1(t)-g2(t)L(Ω).

Applying the Gronwall inequality, we obtain

𝒘1(t)-𝒘2(t)L2(Ω)m2eT(0T𝒇1(t)-𝒇2(t)L2(Ω)m2𝑑t+𝒘10-𝒘20L2(Ω)m+0Tg1(t)-g2(t)L(Ω)𝑑t),

concluding (2.11).

Consider now the case where 𝒂 is strongly monotone. When p2, after integration, from (6.11) and (6.12) we obtain

12𝒘(t)L2(Ω)m2+a*0tL𝒘(τ)Lp(Ω)p𝑑τ0t𝒇1(τ)-𝒇2(τ)L2(Ω)m𝒘(τ)L2(Ω)m𝑑τ+12𝒘(0)L2(Ω)m2+Cg1-g2L1(0,T;L(Ω)),

and so we conclude that there exists another positive constant C depending on T such that

𝒘1-𝒘2L(0,T;L2(Ω)m)2+𝒘1-𝒘2𝒱p(QT)pC(𝒇1-𝒇2L2(QT)m2+𝒘10-𝒘20L2(Ω)m2+g1-g2L1(0,T;L(Ω))),(6.13)

obtaining (2.13) when p2.

If 1<p<2, using the strong monotonicity (2.12) of the operator 𝒂, from (6.11) we get

Qtt𝒘𝒘+a*Qt|L𝒘|2(|L𝒘1|+|L𝒘2|)p-2Qt(𝒇1-𝒇2)𝒘+Θ(t).

Applying the reverse Hölder inequality, we obtain

Qtt𝒘𝒘+a*(Qt|L𝒘|p)2p(Qt(|L𝒘1|+|L𝒘2|)p)p-22Qt(𝒇1-𝒇2)𝒘+Θ(t).(6.14)

Since p<2 and |L𝒘i|g* a.e. in QT, i=1,2, we have

(Qt(|L𝒘1|+|L𝒘2|)p)p-22C,

where C is a positive constant depending only on 𝒇iLp(QT)m and 𝒘i0L2(Ω)m. From (6.14) we obtain

12𝒘(t)L2(Ω)m2+Ca*(Qt|L𝒘|p)2p0t𝒇1-𝒇2L2(Ω)m𝒘L2(Ω)m+12𝒘(0)L2(Ω)m2+0tΘ,

and so, using the Hölder and Young inequalities, we get the inequality

𝒘L(0,T;L2(Ω)m)2+𝒘𝒱p2C(𝒇1-𝒇2L2(QT)m2+𝒘10-𝒘20L2(Ω)m2+g1-g2L1(0,T;L(Ω))).(6.15)

From (6.13) and (6.15) the conclusion follows. ∎

Proof of Theorem 2.13.

Consider a sequence of solutions 𝒘n given by Theorem 2.10 for a sequence of gnW1,(0,T;L(Ω)) such that

gn𝑛g in 𝒞([0,T];L(Ω)).

First we show that {𝒘n}n is relatively compact in 𝒞([0,T];L2(Ω)). For arbitrary ε>0, there exists δ>0 such that

εn=sup|τ-s|<δgn(τ)-gn(s)L(Ω)<ε

for all n sufficiently large and all τ,s(0,T). Setting

ρn=g*g*+εn,

we obtain ρn𝒘n(s)𝕂gn(τ) for all τ(s-δ,s+δ), and ρn𝒘n(s) can be chosen as test function in (2.14) for 𝒘n at t=τ, obtaining

Ωt𝒘n(τ)(ρn𝒘n(s)-𝒘n(τ))+Ω𝒂(τ,𝒘n(τ))L(ρn𝒘n(s)-𝒘n(τ))+Ω𝒃(τ,𝒘n(τ))(ρn𝒘n(s)-𝒘n(τ))Ω𝒇(τ)(ρn𝒘n(s)-𝒘n(τ)).(6.16)

Since 0<ρn1 and the solutions 𝒘n𝕂gn are uniformly bounded in L(0,T;𝒱pL2(Ω)m), from (6.16) for fixed s we can integrate in τ on [s,t] obtaining

12Ω|𝒘n(t)-𝒘n(s)|2=12stddτΩ|𝒘n(τ)-𝒘n(s)|2(ρn-1)stΩτ𝒘n(τ)𝒘n(s)+C1|t-s|+Cst𝒇(τ)L2(Ω)m(ρn-1)Ω(𝒘n(t)-𝒘n(s))𝒘n(s)+C|t-s|12(T12+𝒇L2(QT)m)C′′(ε+|t-s|12) for all |t-s|<δ

and all n sufficiently large. Hence {𝒘n}n is equicontinuous on [0,T] with values in L2(Ω)m. Therefore, we can take for a subsequence

𝒘n𝑛𝒘 in 𝒞([0,T];L2(Ω)m)  and  𝒘n𝑛𝒘 in L(0,T;𝑽p) weak-*

for some 𝒘 which is such that 𝒘𝕂g𝒱p𝒞([0,T];L2(Ω)m) and 𝒘(0)=𝒘0.

We conclude that 𝒘 is a weak solution to (2.4) with 𝕂g. Using Minty’s lemma and taking 𝒗n=ρn𝒗𝕂gn for arbitrary 𝒗𝕂g in

0Tt𝒗n,𝒗n-𝒘np+QT𝒂(L𝒗n)L(𝒗n-𝒘n)+QT𝒃(𝒗n)(𝒗n-𝒘n)QT𝒇(𝒗n-𝒘n)-12Ω|𝒘0-𝒗n(0)|2,

we observe that 𝒗n𝑛𝒗 in 𝒴p. By the uniqueness, the whole sequence {𝒘n}n converges to 𝒘. ∎

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About the article

Received: 2018-05-15

Revised: 2018-07-17

Accepted: 2018-09-02

Published Online: 2018-10-09

Published in Print: 2019-03-01


Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 250–277, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0113.

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