Lewy-Stampacchia’s inequality for a pseudomonotone parabolic problem

where u 7→ −div[a(t, x, u,∇u)] is a pseudomonotone operator under the constraint u ≥ ψ. After the rst results of H. Lewy and G. Stampacchia [14] concerning inequalities in the context of superharmonic problems, many authors have been interested in the so-called Lewy-Stampacchia’s inequality associated with obstacle problems. Without exhaustiveness, let us cite the monograph of J.F. Rodrigues [21] and the papers of A.Mokrane and F.Murat [18] for pseudomonotone elliptic problems, A.Mokrane andG. Vallet [19] in the context of Sobolev spaces with variable exponents, A. Pinamonti and E. Valdinoci [20] in the framework of Heisenberg group, R. Servadei and E. Valdinoci [24] for nonlocal operators or N. Gigli and S. Mosconi [11] concerning an abstract presentation. The literature on Lewy-Stampacchia’s inequality is mainly aimed at elliptic problems, or close to elliptic problems and fewer papers are concerned with other type of problems. Let us cite J. F. Rodrigues [22] for hyperbolic problems, F. Donati [7] for parabolic problems with a monotone operator or L. Mastroeni and M. Matzeu [17] in the case of a double obstacle. There is a large literature on parabolic problems with constraints. To cite some recent ones, consider [6, 13] where the main operator is monotone, associated with a nonlinear, possible graph, reaction term.

Concerning Lewy-Stampacchia's inequality, to the best of the author's knowledge, F. Donati's work [7] has not been extended to pseudomonotone parabolic problems with a Leray Lions operator. In this paper, we propose such a result, with very general assumptions on the Carathéodory function a, by using a method of penalization of the constraint associated with a suitable perturbation of the operator. As proposed e.g. by [12, p.102] and [4] for sub/super solutions to obstacle quasilinear elliptic problems, this perturbation is one of the main new point of the proof. Indeed, without it, one is usually only concerned by Lewy-Stampacchia's inequality in the elliptic case, and one needs to assume, as in [18], some additional, now useless, Höldercontinuity assumptions for a with respect to u and ∇u. Thus, this perturbation allows us on the one hand to prove Lewy-Stampacchia's inequality in the pseudomonotone parabolic case, and on the other hand to reduce signi cantly the list of assumptions. Let us mention also that, with this method, one is to revisit Lewy-Stampacchia's inequality proposed in [18,19] by assuming only basic assumptions. The second essential result is an extension of the formula of time-integration by parts of Mignot-Bamberger [2] & Alt-Luckhaus [1] to non-classical situations. Some information are given too about the time-continuity of an element u when u and ∂ t u are not in spaces in duality relation.
The paper is organized in the following way: after giving the hypotheses and the main result (Theorem 2.2) in Section 2, Section 3 is devoted to the proof of this result. A rst step is devoted to the existence of a solution to the penalized/perturbed problem associated with a parameter ε; then, some a priori estimates and passage to the limit with respect to ε are considered when g − is a regular non-negative element. A rst proof of Lewy-Stampacchia's inequality is given when g − is still regular; nally, the proof of Lewy-Stampacchia's inequality is extended to the general case. A last part, Section 4, presents an annex containing technical results used in the proofs, in particular the time-integration by part and the time-continuity mentioned above. which acts from W ,p (Ω) into W − ,p (Ω) where H , a : (t, x, u, ξ ) ∈ Q × R × R d → a(t, x, u, ξ ) ∈ R d is a Carathéodory function on Q × R d+ , H , a is strictly monotone with respect to its last argument:

Notation, hypotheses and main result
∀(t, x) ∈ Q a.e., ∀u ∈ R, ∀ ξ , η ∈ R d , H , a is coercive and bounded: there exist constantsᾱ > ,β > andγ ≥ , a functionh in L (Q) and a functionk in L p (Q) and two exponents q, r < p such that, for a.e. (t, x) ∈ Q, for all u ∈ R and for all ξ ∈ R d , H : assume that the obstacle ψ belongs to L p ( , T; W ,p (Ω)) ∩ L p ( , T; L (Ω)); that ∂ t ψ belongs to L p ( , T; V ) and ψ ≤ on ∂Ω (See Section 4.4 for some comments on the time regularity of such elements).
H : the right hand side f , which is assumed to be such that belongs to the order dual where (L p ( , T; V )) + denotes the non-negative elements of L p ( , T; V ). H : u ∈ L (Ω) satis es the constraint, i.e. u ≥ ψ( ).
As usual concerning obstacle problems one denotes by is an admissible test-function and one has that Then, Corollary 4.5 with β = and α = yields for any t ∈ ( , T) As a consequence, v * ≥ ψ and v * ∈ K(ψ).
Our aim is to prove the following result.

Theorem 2.2. Under the above assumptions (H )-(H )
, there exists at least u in K(ψ) with u(t = ) = u and such that, for any v ∈ L p ( , T; V), v ≥ ψ implies that Moreover, the following Lewy-Stampacchia's inequality holds

Proof of Theorem 2.2
Theorem 2.2 will be proved in four steps. In a rst part, we establish the existence of a solution to a problem where the constraint u ≥ ψ is penalized. Moreover, the crucial point in the method developed in the present paper is to replace a(·, ·, u, ξ ) by a(·, ·, max(u, ψ), ξ ). The aim of this additional perturbation is to ensure, formally, a monotone behavior of the operator when u violates the constraint. This is the aim of Theorem 3.2.
For technical reasons, some a priori estimates and the passage to the limit will be obtained rstly by assuming that g − is regular. This is the object of Lemmas 3.3, 3.4, 3.5 and Theorem 3.7. Then a proof of Lewy-Stampacchia's inequality, still with a regular g − , will be presented in Lemma 3.9.
Finally, one will be able to prove Lewy-Stampacchia's inequality in the general case.

. Penalization
Denote byq = min(p, ) and let us de ne the function Θ and the perturbed operator Remark 3.1. We wish to draw the reader's attention to the fact that with the proposed perturbation: a(t, x, u, ξ ) = a(t, x, max(u, ψ), ξ ), the idea is to make formally the operator monotone and not pseudomonotone any more on the free-set where the constraint is violated.
. The regular case: g − ∈ Lq (Q) → L p (Q) Following Assumption H let us recall that f − ∂ t ψ − Aψ = g = g + − g − belongs to the order dual L p ( , T; V) * . In this subsection we impose an additional regularity on g − , namely ≤ g − ∈ Lq (Q) → L p (Q).

. . A priori estimates with respect to ε
Let us test the penalized problem (6) with uε − v * , Thus, by using (1), for any positive δ , there exists C δ depending on δ and Ω such that For the third term, Θ ≤ and v * ≥ ψ yield By using (2), for any positive δ , there exists C δ depending on δ and Ω such that Finally, for any positive δ , there exists C δ depending on δ and Ω such that In conclusion we have Then, using Young's inequality and a convenient choice of the parameters δ , δ , δ yield that for any positive δ there exists C depending on the listed parameters such that Proof. If p ≥ , W ,p (Ω) = V so that Lemma 3.3 is a straightforward consequence of (7). If p < , it is enough to remark that It is worth noting that Lemma 3.3 gives that ε Q ((uε − ψ) − )q dxdt is bounded (with respect to ε) so that we cannot expect to have a bound of the penalized term ε Θ(uε − ψ) in L p (Q) nor in L p ( , T; V ). Using the additional regularity g − ∈ Lq (Q) we prove in the following lemma more precise estimates on (uε − ψ) − .
Gathering Lemmas 3.3 and 3.4 we prove the following estimates Lemma 3.5. There exists a constant C depending on C , C and g − L p (Q) such that for any ε > Proof. The growth condition (5) onã and Lemma 3.3 imply that . We distinguish the two cases p ≥ and < p < .
which concludes the proof of Lemma 3.5.

. . At the limit when ε → .
The sequence (uε) is bounded in W( , T), therefore, up to a subsequence denoted the same, there exists u ∈ W( , T) such that uε converges weakly to u in W( , T). In particular, one gets that u(t = ) = u . Then, by classical compactness arguments of type Aubin-Lions-Simon [26], the convergence is strong in L p (Q), and a.e. in Q † .
By (2), the following estimate holds for any v ∈ L p ( , T; V), is a continuous function, the theory of Nemytskii operators gives that and Testing the penalized equation (6) Using (14) we obtain The monotone character of the operatorã(x, t, u, ξ ) with respect to ξ (see Assumption H , and (3) It follows that uε(t) → u(t) in L (Ω) for any t (16) and in view of (14) Set using (15) and information similar to (14) allow one to pass to the limit and to conclude that By the classical Minty's trick, considering v = ∇u + λ w, w ∈ L p (Q) d and λ ∈ R, we have necessarily Thus, a classical property of radial continuity coming from the assumptions on a yields, for any w ∈ L p (Q) d , i.e. ξ =ã(t, x, u, ∇u) = a(t, x, u, ∇u), since u ≥ ψ.
Remark 3.6. Note that, following [3, Proof of Lemma 1] , (15) yields the convergence in measure, then the a.e. convergence of ∇uε to ∇u (up to a subsequence if needed), so that this is also a way to identify ξ has being a(t, x, u, ∇u).
We are now in a position to pass to the limit in the penalized problem and to conclude the existence of a solution to the obstacle problem under the additional regularity on g − . Let us consider v ∈ L p ( , T; V), v ≥ ψ as a test function in the penalized problem (6), In view of (16) we have From (17) and the identi cation ξ =ã(t, x, u, ∇u) = a(t, x, u, ∇u) it follows that The weak convergence of uε to u in L p ( , T; V) yields that At last splitting the penalized term in the following way

Lq(Q)
→ thanks to (11) allows one to pass to the limit in (18). One concludes that a solution exists, i.e. Using an idea from A. Mokrane and F. Murat [18], denote by zε Observing that ∂ t uε + A(uε) − f = −zε + g − as in [18] in the elliptic case and under more restrictive assumptions on the operator a, proving that z − ε converges to in an appropriate space leads to the Lewy-Stampacchia's inequality. Due to the time variable and the weak assumption on a we have to face to additional di culties. For technical reasons, we will assume in this section only that, on top of g − ∈ L p (Q) ∩ L p ( , T; V), g − ≥ , that ∂ t g − ∈ Lq (Q). Roughly speaking it allows one to use a test function depending on g − and together with Lemma 4.3 to perform an integration by part formula and then the convergence analysis of z − ε . The general case will be obtained in the next section by a regularization argument based on Lemma 4.1 of the Annex.
Our aim is now to show the convergence of z − ε to in L (Q) to prove the following lemma.
Note that Ψ k (t, x, ) = and ∂ t Ψ k (t, x, λ) = ∂ t g − {g − − ε η k (λ − )< } so that, since Ψ k (t, x, u) is a test-function, by Lemma 4.3, for any t, We now pass to the limit rst as k → ∞ and then as ε → . Since g − ≥ , one has that Ψ k (t, x, λ) = if λ ≥ and as uε( ) = u ≥ ψ( ), one gets that Note that (Ψ k (t, x, λ)) k is a non-increasing sequence of functions with non-positive values so that by monotone convergence theorem since the integration holds on the set of negative values of uε(t) − ψ(t). Due to the de nition of zε we have from which it follows using again the monotone convergence theorem As far as the rst term of (19) is concerned we obtain For the fourth term of (19) we have the following equality since in this situation, the integration holds in the set where uε ≤ ψ. Thus, We now claim that estimate (10) of Lemma 3.4 which gives and Assumptions H , to H , imply that, up to a subsequence (still denoted by ε), ∇(uε − ψ) − converges to a.e. in Q.
Consider (t, x) such that the above limits hold. Since, and x). Else, at the limit, one has that u(t, x) = ψ(t, x). If one assumes that ∇(uε −ψ) − (t, x) is not converging to , then there exists a subsequence ε (depending on (t, x)) and a positive δ such that ∇(u ε −ψ) − (t, x) ≥ δ > . Then, x) and, since it is a bounded sequence in R d , there exists ξ ∈ R d and a new subsequence still labeled ε such that ∇u ε (t, x) converges to ξ , with the additional information: But, this is in contradiction with the convergence of the same sequence to and the result holds.

. Proof of the main result
In this section, H is assumed and g = f −∂ t ψ−A(ψ) = g + −g − where g + , g − ∈ (L p ( , T; V )) + are non-negative elements of L p ( , T; V ). Thanks to Lemma 4.1, there exists positives (g − n ) ⊂ L p (Q) such that g − n → g − in L p ( , T; V ). Then, by a regularization procedure, one can assume that g − n ∈ L p (Q) ∩ L p ( , T; V), g − n ≥ with ∂ t g − n ∈ Lq (Q). Then, the corresponding sequence fn converges to f in L p ( , T; V ). Remark 3.11. In fact, since D(Q) + is dense in L p (Q) + , one can consider g − n as regular as needed.
Since this solution comes from the above penalization method, and as C of Lemma 3.3 can be chosen independent of n, one gets that sup t un L (Ω) (t) + un p L p ( ,T;V) ≤ C .
Thus, a(·, un , ∇un) is bounded in L p (Q) d and, thanks to the above Lewy-Stampacchia's inequality, ∂ t un is bounded in L p ( , T; V ).
Up to a subsequence denoted similarly, un converges weakly to an element u ∈ K(ψ) in W( , T) and strongly in L p (Q); and a(·, un , ∇un) converges to an element ξ in L p (Q) d . Finally, the embedding of W( , T) in C([ , T], L (Ω)) yields the weak convergence of un(t) to u(t) in L (Ω), for any t.
Since u ∈ K(ψ), Therefore, passing to the limit with respect to n in Thus, (2) and the continuity property of Nemytskii's operator ensure the following limit argument: Proof. This result is given in [7,Lemma p.593]. We propose here a sketch of a proof following the idea of [18]. For any ε > and n ∈ N, denote by v n ε the solution to Bv n ε + ε where Tn is the truncation at the height n.
Then, up to a subsequence, vε − v → in L (Q), vε v in L p ( , T; V) and Finally, as lim sup ε J(vε) ≤ J(v), an argument of uniform convexity yields the convergence of vε to v in . Compactness when p < .
Concerning the compactness argument in L p (Q) when p < : note that there exists an integer k ≥ such that  , one has that the sequence (un) is bounded in L p (Ω) and that, up to a subsequence if needed, it converges weakly to a given u in L p (Ω). Thus, for any v ∈ L p (Ω), Since this element u is unique in its way, the identi cation holds.
Then, since the embedding of V is compact in L p (Ω), by Aubin-Lions-Simon [26] compactness theorems, if a sequence is bounded in W( , T), it is also bounded in {u ∈ L p ( , T; V), ∂ t u ∈ L p ( , T; W −k,p (Ω))} and relatively compact in L p (Q).

. On Mignot-Bamberger -Alt-Luckhaus integration by part formula
We propose in next Lemma a time integration by part formula adapted to our situation. Its proof has been inspired by [9].

. Strong continuity in L (Ω)
We consider the following notations in the sequel: V(Ω) = W ,p (Ω) ∩ L (Ω), V (Ω) = W ,p (Ω) ∩ L (Ω) and V (Ω) = W − ,p (Ω) + L (Ω). Let us prove in this section the following result of continuity.  Proof. This result is based on a classical method: rst in R N , then in the half-space R N + and nally in Ω thanks to an atlas of charts.
If Ω = R N +/resp.− = {(x , x N ) ∈ R N ; x N > /resp. < }, the method is based on a suitable extension of u to R N . Following a recommendation of F. Murat, we consider the following extension, proposed in [16, (12.21-22) p.83] and revisited in [8, p.2]: Note thatũ ∈ L p ( , T; V(R N )) and, thanks to a change of variables, that for any φ ∈ C x, x N )dxdt.