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

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 penalisation/regularisation 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.


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, [3,25]), 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.
We introduce now several assumptions which will be important to set the functional framework of our problem.
In general, the operator L can have the form Assumption 2.2. Let a : Q T × ℝ ℓ → ℝ ℓ and b : Q T × ℝ m → ℝ m be Carathéodory functions, i.e. they are measurable functions in the variables (x, t) for all ξ ∈ ℝ ℓ and η ∈ ℝ m , respectively, and they are continuous in the variables ξ ∈ ℝ ℓ and η ∈ ℝ m for a.e. (x, t) ∈ Q T . Suppose, additionally, that a and b satisfy the following structural conditions: for all ξ , ξ ∈ ℝ ℓ and η, η ∈ ℝ m and a.e. (x, t) ∈ Q T , |a(x, t, ξ )| ≤ a * |ξ | p−1 , where a * and b * are positive constants and 1 < p < ∞. Assumption 2.3. For a given p ∈ (1, ∞), we work with a closed subspace p of V p such that p ⊂ L 2 (Ω) m and ‖v‖ p := ‖Lv‖ L p (Ω) ℓ is a norm in p equivalent to the norm induced from V p .

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 ℍ of L 2 (Ω) m such that ( p , ℍ, p ) is a Gelfand triple and the inclusion of p into ℍ is compact for the given p, 1 < p < ∞.
From now on, we set V p = L p (0, T; p ), H = L p (0, T; ℍ), 1 ≤ p ≤ ∞, and we observe that L p (0, T; p ) = V p , with p = p p−1 for 1 < p < ∞. 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 Y p → H is also compact for 1 < p < ∞.
Assumption 2.6. We consider a nonlinear continuous functional G : and a.e. x ∈ Ω, for given constants g * and g * .
Since G is compact in V p , in particular for any sequence {v n } n weakly convergent to v in V p , there exists a subsequence, still denoted by {v n } n , such that For v ∈ V p and a.e. t ∈ (0, T), we define the nonempty convex set for For 1 ≤ p < ∞, we denote the duality pairing between p and p by ⟨ ⋅ , ⋅ ⟩ p and we consider the quasivariational inequality associated with (1.1) and (2.3). Find u ∈ V p satisfying 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 : Q T → ℝ + 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.
Theorem 2.8. The variational inequality (2.4) with a fixed convex g , as in (2.5), for a given strictly positive function g = g(x, t) ∈ C ([0, T]; L ∞ (Ω)), u 0 ∈ g(0) and f ∈ L 2 (Q T ) m , has at most one solution provided p ⊂ ℍ 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 t ∈ (0, T], We observe that if w is a strong solution to (2.6), it is also a weak solution to (2.4).
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 : Q T × ℝ ℓ → ℝ ℓ and b : Q T × ℝ m → ℝ m be Carathéodory functions, i.e. they are measurable functions in the variables (x, t) for all ξ ∈ ℝ ℓ and η ∈ ℝ m , respectively, and they are continuous in the variables ξ ∈ ℝ ℓ and η ∈ ℝ m , respectively, for a.e. (x, t) ∈ Q T . Suppose, additionally, that there exists A : Q T × ℝ ℓ → ℝ such that, for all ξ ∈ ℝ ℓ and a.e. (x, t) ∈ Q T , the function A is differentiable in t and in ξ , and and b satisfies the following structural conditions: for all η, η ∈ ℝ m and a.e. (x, t) ∈ Q T , where a * , A 1 , A 2 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−2 ξ , the continuous dependence result in the coercive case also holds in the space V p .
Theorem 2.11. Suppose that the assumptions of Theorem 2.10 hold and for i = 1, 2 let w i be the solution to the variational inequality (2.6) with data f i , w i 0 , g i satisfying (2.10). Then there exists a positive constant C = C(T) such that where a * is a positive constant depending on p, 1 < p < ∞, then there exists C * = C(a * , p, T) > 0 such that (2.13) Remark 2.12. For strong solutions w ∈ g ∩ H 1 (0, T; L 2 (Ω) m ), the variational inequality (2.6) is, for a.e. t ∈ (0, T), equivalent to and we may define which is such that v ∈ V p ∩ g whenever z ∈ g(t) . Hence we can choose this v as test function in (2.6) with t = T, divide by 2δ and let δ → 0 obtaining, by Lebesgue's theorem, inequality (2.14) for a.e. t ∈ (0, T).

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.

Nonlocal compact operators
Here we are interested in two examples of compact operators G given in the form where g = g(x, t, ζ ) : Q T × ℝ m → ℝ is a positive function, continuous in (x, t) ∈ Q T and in ζ ∈ ℝ m , and ζ : V p → C (Q T ) m is a completely continuous mapping.

Regularization by integration in time
We define the compact operator by 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) ∈ Q T and for all ξ : |ξ | ≤ M.
We also assume that the embedding Hence, by [4, Lemma 2.2], for instance the image by ζ of a bounded subset of V p , being bounded in is a completely continuous mapping, and therefore G defined in (3.1) satisfies Assumption 2.6.

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 where φ v = φ(x, t) depends on v ∈ V p , and the vector field a satisfies (2.1) and (2.12) with p = 2 and ℓ = d.
It is well known that for each φ ∈ L 2 (Q T ) and ζ 0 ∈ L 2 (Ω), 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 L 2 (Q T ). Moreover, if ζ 0 ∈ C γ (Ω) is Hölder continuous for some 0 < γ < 1 and φ ∈ L q (Q T ) for q > d+2 2 , the following estimate holds (see [22, p. 419]): as being the solution of (3.4)-(3.5), with a given ζ 0 ∈ C γ (Ω) and for some fixed φ 0 ∈ L p (Q T ). Hence, by (3.6) and the Ascoli theorem, the and, for some subsequence, ζ(v n ) ⇀ ζ weakly in L 2 (0, T; H 1 0 (Ω)) and uniformly in Q T , for a 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 and even more general terms involving linear combinations of the gradients of the

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.

A problem with gradient constraint
. Then the following quasi-variational inequality has a weak solution: Actually, with Lu = ∇u and V p = W 1,p (Ω), Assumptions 2.1-2.6 are satisfied because the inclusion of The degenerate case α ≡ 0 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.

A problem with Laplacian constraint
. Then the following quasi-variational inequality has a weak solution: Here we choose i.e. the operator L is the Laplacian. The subspace p = W 2,p 0 (Ω) is endowed with the norm which is equivalent to the usual norm of W 2,p (Ω) because ∆ is an isomorphism between p and L p (Ω).

A problem with curl constraint
Let Ω be a bounded open subset of ℝ 3 with a Lipschitz boundary, and let p > 6 5 and f ∈ L 2 (Q T ). Define If u 0 ∈ G[u 0 ] , the following quasi-variational inequality has a weak solution: Here Lv = ∇ × v and V p = {v ∈ L p (Ω) 3 : ∇ × v ∈ L p (Ω) 3 }. In both choices of p , corresponding to different boundary conditions, it is well known that p is a closed subspace of W 1,p (Ω) 3 and that the semi-norm ‖∇ × ⋅ ‖ L p (Ω) 3 is a norm equivalent to the one induced in p by the usual norm in W 1,p (Ω) 3 (for details see [1]).
Here p is compactly embedded in ℍ = {v ∈ L 2 (Ω) 3 : 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 > 5 2 , we obtain the existence of a weak solution to the corresponding thermal and electromagnetic coupled problem.

Non-Newtonian thick fluids -a problem with a constraint on D
Let V p = L p (0, T; V p ) and observe that p is compactly embedded in , by the Sobolev and Korn inequalities. Hence, using the results of [26,30] 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 ℝ d with a Lipschitz boundary, and let d
has a weak solution.

A problem with first-order vector fields constraint
Let Ω ⊂ ℝ d , d ≥ 2, be a connected bounded open set and let L = (X 1 , . . . , X ℓ ) be a family of Lipschitz vector fields on ℝ 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 p ≥ 2: This is the case of an Hörmander operator with with α ij ∈ C ∞ (Ω) such that the Lie algebra generated by these ℓ vector fields has dimension d, when the set p is the closure of D(Ω) in with the graph norm and ∂Ω ∈ C ∞ . Indeed, in this case, it is known (see [9,10,12]) that the extension of the Rellich-Kondratchov theorem, holds, and so ( p , L 2 (Ω), p ) 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, [8,12]. 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 ℝ d with a smooth boundary. Under assumption
has a weak solution.

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 ∈ H , δ ∈ (0, 1) and ε ∈ (0, 1). With this regularization and penalization of the quasi-variational inequality (2.4) with convex sets G[φ](t) , t ∈ [0, T], we apply a fixed point argument. Consider the following increasing continuous function k ε : ℝ → ℝ + 0 : Observe that the function k εδ = δ + k ε approximates the maximal monotone graph We start with an auxiliary lemma.
is monotone.
Proof. To simplify, we omit the argument (x, t) and we denote k ε (|ξ | − ψ) simply by k ε (|ξ |). We may assume, without loss of generality, that |ξ | ≥ |ξ |. Because S(ξ ) = |ξ | p−2 ξ is monotone and k ε is a nonnegative increasing function, we have the problem that consists of finding u εδ,φ such that ‖∂ t u εδ,φ ‖ (V p ) ≤ C ε δ 1/p , where C and C ε are positive constants independent of φ and of δ, and C is also independent of ε.
Proof. Using w = u εδ,φ as a test function in (4.4), we get, for a.e. t ∈ (0, T), Set Q t = Ω × (0, t). Integrating the last equality between 0 and t, recalling the monotonicity of a, b and of T ε 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 and so we proved (4.5). From (4.7) we immediately obtain (4.6).
Next we prove that, given ψ ∈ p , with C being a positive constant. We notice that, by Assumption 2.3, p ⊂ L 2 (Ω) m . So there exists a positive constant C such that, for all v ∈ p , we have We split the proof in two cases.
From the first equation of (4.4) we conclude that Using again Assumption 2.3, we obtain Proof. Let us consider a sequence {φ n } n converging to φ in H . Setting w n = u εδ,φ n and w = u εδ,φ , we need to prove that w n n → w in V p and ∂ t w n n → ∂ t w in V p .
The argument is standard, but we present it here for the sake of completeness. Both functions w n and w solve (4.4), so, for any ψ ∈ p , , w n (t)) − b(t, w(t) Replacing ψ by w n (t) − w(t) in the last expression and integrating it over (0, t), we get Using the monotonicity of a, b and the operator T ε defined in (4.2), we can neglect the second, third and fifth terms of the inequality above.
In the case p ≥ 2, we obtain, applying the Hölder and Young inequalities and denoting by D p the constant related to the strongly monotone term in δ (see (2.12)), and therefore we get Consider now the case 1 < p < 2. 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), C p δ and applying the Hölder and Young inequalities to the right-hand side, we obtain and so Observe now that we have a.e. in Q T , and k ε is a continuous function. By recalling that φ n n → φ in H , Assumption 2.6 implies that in L 1 (Q T ). Hence, at least for a subsequence, a.e. in Q T and, by the dominated convergence theorem, Therefore, the right-hand sides of (4.10) and (4.11) converge to zero a.e. when n → ∞. By definition, Applying the Hölder inequality, we conclude that and, arguing as before, we conclude the proof.

Theorem 4.5. Suppose that Assumptions 2.1-2.6 and (4.3) are verified. Let i be the inclusion of Y p into H and let S : H → Y p be the function defined in Proposition 4.4. Then the function i ∘ S has a fixed point in H . This fixed point solves the problem that consists of finding
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 Y p → H is compact for 1 < p < ∞, and so i ∘ S is completely continuous as a map of H into itself. By Proposition 4.3, given φ ∈ H , we have where C is a constant independent of φ and δ (it may depend on ε). Because i is continuous, there exists C 1 such that ‖v‖ H ≤ C 1 ‖v‖ Y p , and we get Then the image of i ∘ S is bounded, so we may apply the Schauder fixed point theorem, obtaining immediately the conclusion of the existence of a u εδ = i ∘ S(u εδ ) in Y p .

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 ε → 0 and then as δ → 0.
Firstly we collect the a priori estimates for the solution u εδ of problem (4.12) which are independent of ε. Let u εδ be a solution of the approximated problem (4.12). Then there exists a positive constant C independent of ε and δ such that Proof. The first two estimates are direct consequences of inequalities (4.5) and (4.6), respectively, taking But ‖w‖ L ∞ (0,T;L 2 (Ω) m ) is uniformly bounded by (5.1), and Choosing u εδ as test function in (4.12), and integrating between 0 and t, we get Therefore, because a and b are monotone, a(0) = 0, b(0) = 0 and by using the Gronwall inequality, we obtain where C 1 is a constant independent of ε and δ. As Using (5.6), we obtain concluding the proof of (5.5). (4.12). If u δ is the weak limit of a subsequence of {u εδ } ε when ε → 0, then u δ ∈ G[u δ ] .

Lemma 5.2. Let u εδ be a solution of the approximated problem
Proof. To prove that u δ belongs to the convex set G[u δ ] , we use arguments as in [27], which we adapt to our problem. We split Q T in three sets: We recall that, by Assumption 2.6, the operator G is compact. So, as a subsequence of {u εδ } ε (still denoted by {u εδ } ε ) converges weakly to u δ in V p , we obtain that G[u εδ ] converges to G[u δ ] strongly in C ([0, T]; L ∞ (Ω)) and We observe that (0) . Then there exists a regularizing sequence {v n } n and a sequence of scalar functions {g n } n with the following properties: Proof. Let v n be the unique solution of the ordinary differential equation v n + 1 n ∂ t v n = v with v n (0) = z. The function v n has the following expression: and it is well known that it satisfies (i), (ii) and (iii) (see [23, p. 274] or [33, p. 206]). Therefore, it follows Observe that, for any measurable set ω ⊂ Q T , using Assumption 2.6. Consequently, a.e. in Q T , and so u ∈ G[u] .
Step 1: The limit when ε → 0. From estimates (5.3), (5.4) and (5.2) in Proposition 5.1 there exist χ δ ∈ V p , Υ δ ∈ V p and Λ δ ∈ L p (Q T ) ℓ such that for subsequences, Given v belonging to the space Y p defined in (2.2), we have From now on, we denote k ε (|Lu εδ | − G[u εδ ]) simply by k ε , with no risk of confusion.
Using u εδ − v as test function in (4.12) and integrating between 0 and T, we obtain Hence, for all v ∈ Y p , Let u n be the regularizing sequence of u defined in the previous lemma. Using u n as test function in (5.10), we get In fact, the term k ε |Lu εδ | p−1 (G[u εδ ] − |Lu εδ |) is less than or equal to zero because when |Lu εδ | < G[u εδ ], then k ε (|Lu εδ | − G[u εδ ]) = 0. Then, recalling (5.5) and (5.6), we conclude that Going back to (5.11) and using (5.12) and the estimate above, we get ; L ∞ (Ω)) and recalling (5.8), we obtain Step 2: The limit when δ → 0. Because there exists a positive constant C independent of ε and δ such that we obtain, for a subsequence, Letting n → ∞ in the above inequality, using that ∫ Step 3: Conclusion. Let a δ = lim ε→0 ⟨Au εδ , u εδ − u⟩. As lim δ→0 a δ ≤ 0, given η > 0, there exists δ η > 0 such that a δ η < η 2 . But, because lim From now on we set u η = u ε η δ η and k η = k ε η . As the operator A is bounded, monotone and hemicontinuous, it is pseudo-monotone. Therefore, as lim η→0 ⟨Au η , u η − u⟩ ≤ 0, we obtain Finally, we conclude, going back to (5.9), that if v ∈ G [u] , then But, as v ∈ G [u] , as in (5.12) and (5.13), Then lim concluding the proof since we already know that u ∈ G [u] .
Proof of Theorem 2.8. Let u 1 , u 2 ∈ g be two solutions of (2.4) and denote by {w n } n and by {g n } n the regularizing sequences of Lemma 5.3 of w = u 1 +u 2 2 ∈ g and g, respectively, with z = u 0 . Considering ε n = ‖g n − g‖ C ([0,T];L ∞ (Ω)) and ρ n = g * g * + ε n n → 1, we haveŵ n = ρ n w n ∈ g ∩ Y p and it may be chosen as test function in (2.4) for u 1 and u 2 . We obtain, by addition, (5.14) Observing that ⟨∂ tŵn ,ŵ n − w⟩ p = ρ n ⟨∂ t w n , w n − w⟩ p + ρ n (ρ n − 1)⟨∂ t w n , w n ⟩ p and integrating in time, since w n ∈ W 1,p (0, T; p ) ⊂ C (0, T; L 2 (Ω) m ), we have Therefore, taking the limit in (5.14), sinceŵ n n → u 1 +u 2 2 , we obtain and the conclusion u 1 = u 2 follows by the strict monotonicity of b or a with Assumption 2.2.

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 k ε 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 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 As in estimates (4.5), (4.6) and (5.5), we obtain, with a constant C > 0 independent of ε and δ, Using Galerkin's approximation, we can argue formally with ∂ t w εδ as a test function on (6.1) and we get Set ϕ ε (s) = ∫ s 0 k ε (τ) dτ and observe that Integrating (6.5) between 0 and T, we obtain since A satisfies (2.7). But ϕ ε (|Lw εδ (T)| p − g p (T)) ≥ 0, ϕ ε (|Lw 0 | p − g p (0)) = 0 because |Lw 0 | ≤ g(0), and, using assumption (2.9), the Hölder and Young inequalities, and (2.8), from (6.6) we have From (6.2)-(6.4) we obtain, with a constant C > 0 independent of ε and δ, Then, recalling that Assumption 2.5 implies, by the Aubin-Lions lemma, the compactness of Y p → H , there exists a subsequence that we still denote by {w εδ } ε such that, for every t ∈ (0, T], Recalling Lemma 4.1 and observing that k ε (|Lv| p − g p ) = 0 if v ∈ g , we have, for any t ∈ (0, T], Using v − w εδ 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−2 Lξ , we obtain Passing to the limit when ε tends to zero, we get Arguing as in Lemma 5.2, we also prove that w δ ∈ g . The next step is to let δ → 0. From (6.7) we have Using the sets defined in (5.7), we get because, for s ∈ (0, 1 ε ), s ε ≤ k ε (s), and by (5.6), so {∂ t w δ } δ is also uniformly bounded in L 2 (Q T ) m . Since w δ ∈ g , we have |Lw δ | bounded in L ∞ (Q T ) independently of δ. Then, for a subsequence, we have We can pass to the limit when δ → 0 in (6.9), writing for each 0 < t ≤ T, we recover in the limit that w satisfies Finally, as in the proof of Theorem 2.7, w also belongs to g . We may apply Minty's lemma and conclude that it solves (2.6). The uniqueness of the solution is immediate since if w 1 and w 2 are two solutions of (2.6), then and, by monotonicity of a and b, we get concluding that w 1 = w 2 .
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 a ≡ 0 and b ≡ 0.
Proof of Theorem 2.13. Consider a sequence of solutions w n given by Theorem 2.10 for a sequence of g n ∈ W 1,∞ (0, T; L ∞ (Ω)) such that g n n → g in C ([0, T]; L ∞ (Ω)).