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 , which was followed by many other works, including the extension to pseudo-monotone operators by Brézis in 1968  (see also [23, 33]). Quasi-variational inequalities were introduced later by Bensoussan and Lions in 1973 to describe impulse control problems  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  and , 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 , including other applications in elastoplasticity and in electrostatics, and, in , 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 , always in the scalar cases.
In this work, we consider weak solutions to a class of quasi-variational inequalities associated with evolution equations or systems of the type
formally in the unsaturated region of the scalar constraint
i.e. in the domain , with a nonlocal positive and compact operator G, where denotes the partial time derivative, L is a linear partial differential operator in x with bounded coefficients and is its formal dual. Here the monotone vector fields and are of power-type growth, and the boundary value problems may be coercive or not. However, in the region 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 and a second one for a coercive regularization, as in . 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 .
When the constraint G, which may depend on time and space, is independent of the solution, i.e. , 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 and 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 function. Here our method is adapted to gradient-type constraints and it develops and extends the pioneer work of , which was continued in , extended to a p-curl system in  and to thick flows by  (see also ). 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 , 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 , using the pseudo-monotonicity and the -semigroup approach of Brézis-Lions, and in , using contractive iteration arguments that yield uniqueness results and numerical approximation schemes in interesting but special situations. Although the elegant and abstract approach of  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 , 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 with a Lipschitz boundary, and for denote . For a real vector function , , and a multi-index , with and , we denote
the partial derivatives of . Given real numbers , we set and .
We introduce now several assumptions which will be important to set the functional framework of our problem.
For , let L be a linear differential operator of order , given by
is endowed with the graph norm, .
In general, the operator L can have the form
where , is a multi-index and each is in , 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:
(gradient of u, , ).
(Laplacian of u, ).
(curl of , ).
(symmetric part of the Jacobian of , , ).
, where , where are appropriate scalar real functions (subelliptic gradient of u, , , ).
Let and be Carathéodory functions, i.e. they are measurable functions in the variables for all and , respectively, and they are continuous in the variables and for a.e. . Suppose, additionally, that and satisfy the following structural conditions: for all and and a.e. ,
where and are positive constants and .
For a given , we work with a closed subspace of such that and is a norm in equivalent to the norm induced from .
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.
There exists a Hilbert subspace of such that is a Gelfand triple and the inclusion of into is compact for the given p, .
From now on, we set
and we observe that , with for .
By well-known embedding theorems on Sobolev–Bochner spaces (see, for instance, [33, Chapter 7]), we have
and Assumption 2.5 implies, by the Aubin–Lions lemma, that the embedding is also compact for .
We consider a nonlinear continuous functional such that its restriction to is compact with values in , i.e. is compact. In addition, we assume
for given constants and .
Since G is compact in , in particular for any sequence weakly convergent to in , there exists a subsequence, still denoted by , such that converges uniformly to in .
For and a.e. , we define the nonempty convex set for :
where is the Euclidean norm in and we denote if and only if for a.e. .
We note that by Assumption 2.6 the solutions have bounded but, in general, this may not imply that 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 and the special case of the convex set (2.3) with
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.
The variational inequality (2.4) with a fixed convex , as in (2.5), for a given strictly positive function , and , has at most one solution provided and one of the monotonicity conditions is strict, i.e.
and Assumption 2.3 holds.
We can now introduce the strong formulation of the corresponding variational inequality. Find
satisfying, for all ,
we immediately conclude that also satisfies (2.4).
We consider also a stronger non-coercive framework with a potential vector field and a lower-order term with linear growth, by replacing Assumption 2.2 by the following.
Let and be Carathéodory functions, i.e. they are measurable functions in the variables for all and , respectively, and they are continuous in the variables and , respectively, for a.e. . Suppose, additionally, that there exists such that, for all and a.e. , the function A is differentiable in t and in , and
and satisfies the following structural conditions: for all and a.e. ,
where , , and are positive constants, .
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 , the continuous dependence result in the coercive case also holds in the space .
Then the variational inequality (2.6) has a unique solution .
If, in addition, satisfies
where is a positive constant depending on p, , then there exists such that
For strong solutions , the variational inequality (2.6) is, for a.e. , equivalent to
provided we assume , . Indeed, for arbitrary , , for fixed , we set
and we may define
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 , extending [18, Theorem 3.8].
3 Applications with particular G and L
3.1 Nonlocal compact operators
Here we are interested in two examples of compact operators G given in the form
where is a positive function, continuous in and in , and is a completely continuous mapping.
3.1.1 Regularization by integration in time
We define the compact operator by
where is a given kernel satisfying
For simplicity, we assume here the existence of a constant and a real bounded function such that
We also assume that the embedding
which, by the Rellich–Kondratchov theorem, is satisfied if with .
Suppose , , and observe that, by assumption (3.2), not only but also , i.e.
Hence, by [4, Lemma 2.2], for instance the image by of a bounded subset of , being bounded in , by (3.3) is relatively compact in . So 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
It is well known that for each and , the weak solution
to (3.4), (3.5) and (2.6) depends continuously, in these spaces, on the variation of φ in the weak topology of . Moreover, if is Hölder continuous for some and for , the following estimate holds (see [22, p. 419]):
for some λ, , where is a constant independent of the data φ.
for some fixed . Hence, by (3.6) and the Ascoli theorem, the mapping is completely continuous from into . Indeed, if in , is bounded in
and, for some subsequence, weakly in and uniformly in , for a
We observe that, if , , in (3.4) we can also choose
with , provided , and even more general terms involving linear combinations of the gradients of the , , provided .
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
and we assume that the operator G satisfies Assumption 2.6.
3.2.1 A problem with gradient constraint
Let Ω be a bounded open subset of with a Lipschitz boundary, let and let . Let further and . Then the following quasi-variational inequality has a weak solution:
The degenerate case corresponds to the variational model of sandpile growth where G models the slope of the pile (see ). In , 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
Let Ω be a bounded open subset of with a boundary, and let with . Let and . Then the following quasi-variational inequality has a weak solution:
Here we choose
i.e. the operator L is the Laplacian. The subspace is endowed with the norm
which is equivalent to the usual norm of because Δ is an isomorphism between and . Besides, is a Gelfand triple and the inclusion is compact because .
3.2.3 A problem with curl constraint
Let Ω be a bounded open subset of with a Lipschitz boundary, and let and . Define
If , the following quasi-variational inequality has a weak solution:
Here and . In both choices of , corresponding to different boundary conditions, it is well known that is a closed subspace of and that the semi-norm is a norm equivalent to the one induced in by the usual norm in (for details see ). Here is compactly embedded in .
This model is related to the Bean-type superconductivity variational inequality, which was solved in , 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 , 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
Let and observe that is compactly embedded in
if , 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.
Let Ω be a bounded open subset of with a Lipschitz boundary, and let , , and . Then the quasi-variational inequality
has a weak solution.
3.2.5 A problem with first-order vector fields constraint
Let , , be a connected bounded open set and let be a family of Lipschitz vector fields on that connect the space. We shall assume that the regularity of and the structure of L support the following Sobolev–Poincaré compact embedding for :
This is the case of an Hörmander operator with
with such that the Lie algebra generated by these vector fields has dimension d, when the set is the closure of in
holds, and so 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.
Suppose that Ω is a bounded open subset of with a smooth boundary. Under assumption (3.7), if , and , the quasi-variational inequality
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 , and . With this regularization and penalization of the quasi-variational inequality (2.4) with convex sets , , we apply a fixed point argument. Consider the following increasing continuous function :
Observe that the function approximates the maximal monotone graph
We start with an auxiliary lemma.
Let ψ be a scalar real function defined in . Then, for , the operator
To simplify, we omit the argument and we denote simply by . We may assume, without loss of generality, that . Because is monotone and is a nonnegative increasing function, we have
the problem that consists of finding such that
has a unique solution with , i.e. .
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]). ∎
where C and are positive constants independent of and of δ, and C is also independent of ε.
Using as a test function in (4.4), we get, for a.e. ,
Set . Integrating the last equality between 0 and t, recalling the monotonicity of , and of defined in (4.2), and applying the Hölder and Young inequalities to the right-hand side of the above equation, we obtain the inequality
By the Gronwall inequality we conclude that
Next we prove that, given ,
with C being a positive constant. We notice that, by Assumption 2.3, . So there exists a positive constant C such that, for all , we have
We split the proof in two cases.
(i) . Then
(ii) . Then
From the first equation of (4.4) we conclude that
Using again Assumption 2.3, we obtain
concluding now easily that
Let us consider a sequence converging to in . Setting and , we need to prove that
The argument is standard, but we present it here for the sake of completeness. Both functions and solve (4.4), so, for any ,
Replacing by in the last expression and integrating it over , we get
Using the monotonicity of , and the operator defined in (4.2), we can neglect the second, third and fifth terms of the inequality above.
In the case , we obtain, applying the Hölder and Young inequalities and denoting by the constant related to the strongly monotone term in δ (see (2.12)),
and therefore we get
Consider now the case . From (4.9) we get again
and, using also the coercive condition on δ (see (2.12)) and the Hölder inverse inequality, we obtain
Recalling, by (4.6),
and applying the Hölder and Young inequalities to the right-hand side, we obtain
Observe now that we have
a.e. in , and is a continuous function. By recalling that in , Assumption 2.6 implies that
in . Hence, at least for a subsequence,
and, by the dominated convergence theorem,
Applying the Hölder inequality, we conclude that
and, arguing as before, we conclude the proof. ∎
Suppose that Assumptions 2.1–2.6 and (4.3) are verified. Let i be the inclusion of into and let be the function defined in Proposition 4.4. Then the function has a fixed point in . This fixed point solves the problem that consists of finding such that
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 is compact for , and so is completely continuous as a map of into itself. By Proposition 4.3, given , we have
where C is a constant independent of and δ (it may depend on ε). Because i is continuous, there exists such that , and we get
Then the image of is bounded, so we may apply the Schauder fixed point theorem, obtaining immediately the conclusion of the existence of a in . ∎
5 Weak solutions of the quasi-variational inequality
Firstly we collect the a priori estimates for the solution of problem (4.12) which are independent of ε.
Given , we have
From (4.8) we have
But is uniformly bounded by (5.1), and
Choosing as test function in (4.12), and integrating between 0 and t, we get
Therefore, because and are monotone, , and by using the Gronwall inequality, we obtain
where is a constant independent of ε and δ. As and in , we obtain
Using (5.6), we obtain
concluding the proof of (5.5). ∎
Let be a solution of the approximated problem (4.12). If is the weak limit of a subsequence of when , then .
To prove that belongs to the convex set , we use arguments as in , which we adapt to our problem. We split in three sets:
We recall that, by Assumption 2.6, the operator G is compact. So, as a subsequence of (still denoted by ) converges weakly to in , we obtain that converges to strongly in and
We observe that
which means that
Let be such that , and let . Then there exists a regularizing sequence and a sequence of scalar functions with the following properties:
strongly in .
, where and in .
Let be the unique solution of the ordinary differential equation with . The function has the following expression:
where denotes the extension of by for , we have
and, by the uniform continuity of in with values in , we also have the uniform convergence of in , concluding (iv). ∎
Proof of Theorem 2.7.
The boundedness of in implies that there exists a subsequence, still denoted by , converging weakly- to a function when , in . So, by Lemma 5.2, and we may extract a subsequence of converging weakly- to some in when .
Observe that, for any measurable set ,
using Assumption 2.6. Consequently,
and so .
Define the operator by
Given belonging to the space defined in (2.2), we have
From now on, we denote simply by , with no risk of confusion.
Using as test function in (4.12) and integrating between 0 and T, we obtain
Hence, for all ,
Let be the regularizing sequence of defined in the previous lemma. Using as test function in (5.10), we get
In fact, the term is less than or equal to zero because when , then .
So, noticing that in and recalling (5), we obtain
Step 2: The limit when . Because there exists a positive constant C independent of ε and δ such that
we obtain, for a subsequence,
Since in , we obtain
Letting in the above inequality, using that and in , we conclude that
If is defined by
Step 3: Conclusion. Let . As , given , there exists such that . But, because
we can find for which . So,
From now on we set and .
As the operator is bounded, monotone and hemicontinuous, it is pseudo-monotone. Therefore, as , we obtain
Finally, we conclude, going back to (5.9), that if , then
because in . Hence,
concluding the proof since we already know that . ∎
Proof of Theorem 2.8.
we have and it may be chosen as test function in (2.4) for and . We obtain, by addition,
and integrating in time, since , we have
Therefore, taking the limit in (5.14), since , we obtain
and the conclusion follows by the strict monotonicity of or 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 is monotone or strongly monotone.
Proof of Theorem 2.10.
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 instead of .
Using Galerkin’s approximation, we can argue formally with as a test function on (6.1) and we get
Set and observe that
Integrating (6.5) between 0 and T, we obtain
since A satisfies (2.7). But
and, using assumption (2.9), the Hölder and Young inequalities,
Then, recalling that Assumption 2.5 implies, by the Aubin–Lions lemma, the compactness of , there exists a subsequence that we still denote by such that, for every ,
Recalling Lemma 4.1 and observing that if , we have, for any ,
Passing to the limit when ε tends to zero, we get
Arguing as in Lemma 5.2, we also prove that .
The next step is to let . From (6.7) we have
Using the sets defined in (5.7), we get
because, for , , and by (5.6),
so is also uniformly bounded in . Since , we have bounded in independently of δ. Then, for a subsequence, we have
We can pass to the limit when in (6.9), writing
Because in -weak yields
for each , we recover in the limit that satisfies
The uniqueness of the solution is immediate since if and are two solutions of (2.6), then
and, by monotonicity of and , we get
concluding that . ∎
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 , and we are able to give a result even in the very degenerate case and .
Proof of Theorem 2.11.
we observe that
Considering, for , , the solution of the variational inequality (2.6) associated to the constraint , and using as test function, we have, for ,
Adding the inequalities we obtained in the former expression to and , and setting , we get
Applying the Gronwall inequality, we obtain
and so we conclude that there exists another positive constant C depending on T such that
obtaining (2.13) when .
Applying the reverse Hölder inequality, we obtain
Since and a.e. in , , we have
where C is a positive constant depending only on and . From (6.14) we obtain
and so, using the Hölder and Young inequalities, we get the inequality
Proof of Theorem 2.13.
Consider a sequence of solutions given by Theorem 2.10 for a sequence of such that
First we show that is relatively compact in . For arbitrary , there exists such that
for all n sufficiently large and all . Setting
we obtain for all , and can be chosen as test function in (2.14) for at , obtaining
Since and the solutions are uniformly bounded in , from (6.16) for fixed s we can integrate in τ on obtaining
and all n sufficiently large. Hence is equicontinuous on with values in . Therefore, we can take for a subsequence
for some which is such that and .
We conclude that is a weak solution to (2.4) with . Using Minty’s lemma and taking for arbitrary in
we observe that in . By the uniqueness, the whole sequence converges to . ∎
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About the article
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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