Maximum Principle for Quasilinear Stochastic PDEs with Obstacle

We prove a maximum principle for local solutions of quasilinear stochastic PDEs with obstacle (in short OSPDE). The proofs are based on a version of It\^o's formula and estimates for the positive part of a local solution which is non-positive on the lateral boundary.


Introduction
In this paper, we consider an obstacle problem for the following parabolic Stochastic PDE (SPDE in short)                  du t (x) = ∂ i (a i,j (x)∂ j u t (x) + g i (t, x, u t (x), ∇u t (x))) dt + f (t, x, u t (x), ∇u t (x))dt Here, S is the given obstacle, a is a matrix defining a symmetric operator on an open bounded domain O, f, g, h are random coefficients.
In a recent work [9] we have proved existence and uniqueness of the solution of equation (1) under standard Lipschitz hypotheses and L 2 -type integrability conditions on the coefficients. Let us recall that the solution is a couple (u, ν), where u is a process with values * The work of the first and third author is supported by the chair risque de crédit, Fédération bancaire Française † The research of the second author was partially supported by the Chair Financial Risks of the Risk Foundation sponsored by Société Générale, the Chair Derivatives of the Future sponsored by the Fédération Bancaire Française, and the Chair Finance and Sustainable Development sponsored by EDF and Calyon 1 in the first order Sobolev space and ν is a random regular measure forcing u to stay above S and satisfying a minimal Skohorod condition. In order to give a rigorous meaning to the notion of solution, inspired by the works of M. Pierre in the deterministic case (see [18,19]), we introduce the notion of parabolic capacity. The key point is that in [9], we construct a solution which admits a quasi continuous version hence defined outside a polar set and that regular measures which in general are not absolutely continuous w.r.t. the Lebesgue measure, do not charge polar sets. There is a huge literature on parabolic SPDE's without obstacle. The study of the L p −norms w.r.t. the randomness of the space-time uniform norm on the trajectories of a stochastic PDE was started by N. V. Krylov in [13], for a more complete overview of existing works on this subject see [7,8] and the references therein. Concerning the obstacle problem, there are two approaches, a probabilistic one (see [15,12]) based on the Feynmann-Kac's formula via the backward doubly stochastic differential equations and the analytical one (see [10,17,22]) based on the Green function.
To our knowledge, up to now there is no maximum principle result for quasilinear SPDE with obstacle and even very few results in the deterministic case. The aim of this paper is to obtain, under suitable integrability conditions on the coefficients, L p -estimates for the uniform norm (in time and space) of the solution, a maximum principle for local solutions of equation (1) and comparison theorems similar to those obtained in the without obstacle case in [5,7]. This yields for example the following result: Theorem 1.1. Let (M t ) t≥0 be an Itô process satisfying some integrability conditions, p ≥ 2 and u be a local weak solution of the obstacle problem (1). Assume that ∂O is Lipschitz and u ≤ M on ∂O, then for all t ∈ [0, T ]: where C(S, f, g, h, M ) depends only on the barrier S, the initial condition ξ, coefficients f, g, h, the boundary condition M and k is a function which only depends on p and t, · ∞,∞;t is the uniform norm on [0, t] × O.
Let us remark that in order to get such a result, we define the notion of local solutions to the obstacle problem (1) and so introduce what we call local regular measures.
The paper is organized as follows: in section 2 we introduce notations and hypotheses. In section 3, we establish the L p −estimate for uniform norm of the solution with null Dirichlet boundary condition. Section 4 is devoted to the main result: the maximum principle for local solutions whose proof is based on an Itô formula satisfied by the positive part of any local solution with lateral boundary condition, M . The last section is an Appendix in which we give the proofs of several lemmas.
In general, we shall extend the notation where u, v are measurable functions defined on O such that uv ∈ L 1 (O).
The first order Sobolev space of functions vanishing at the boundary will be denoted by H 1 0 (O), its natural scalar product and norm are As usual we shall denote H −1 (O) its dual space. We shall denote by H 1 loc (O) the space of functions which are locally square integrable in O and which admit first order derivatives that are also locally square integrable. For each t > 0 and for all real numbers p, q ≥ 1, we denote by L p,q ([0, t] × O) the space of (classes of) measurable functions u : [0, t] × O −→ R such that is finite. The limiting cases with p or q taking the value ∞ are also considered with the use of the essential sup norm. Now we introduce some other spaces of functions and discuss a certain duality between them. Like in [5] and [7], for self-containeness, we recall the following definitions: Let (p 1 , q 1 ), (p 2 , q 2 ) ∈ [1, ∞] 2 be fixed and set This means that the set of inverse pairs 1 p , 1 q , (p, q) belonging to I, is a segment contained in the square [0, 1] 2 , with the extremities 1 p 1 , 1 q 1 and 1 p 2 , 1 q 2 . We introduce: We know that this space coincides with the intersection of the extreme spaces, and that it is a Banach space with the following norm The other space of interest is the algebraic sum which represents the vector space generated by the same family of spaces. This is a normed vector space with the norm Clearly one has L I;t ⊂ L 1,1 ([0, t] × O) and u 1,1;t ≤ c u I;t , for each u ∈ L I;t , with a certain constant c > 0.
We also remark that if (p, q) ∈ I, then the conjugate pair (p ′ , q ′ ) , with 1 p + 1 p ′ = 1 q + 1 q ′ = 1, belongs to another set, I ′ , of the same type. This set may be described by and it is not difficult to check that for any u ∈ L I;t and v ∈ L I ′ ;t . This inequality shows that the scalar product of L 2 ([0, t] × O) extends to a duality relation for the spaces L I;t and L I ′ ;t . Now let us recall that the Sobolev inequality states that for each u ∈ H 1 0 (O) , where c S > 0 is a constant that depends on the dimension and for each t ≥ 0 and each u ∈ L 2 loc R + ; For d ≥ 3 and some parameter θ ∈ [0, 1[ we set: and by using similar calculations with the convention 2 * Moreover we have the following duality relation: for any u ∈ L θ;t and v ∈ L * θ;t and the following inequality:

Hypotheses
We consider a sequence ((B i (t)) t≥0 ) i∈N * of independent Brownian motions defined on a standard filtered probability space (Ω, F, (F t ) t≥0 , P ) satisfying the usual conditions. Let A be a symmetric second order differential operator defined on the open bounded subset O ⊂ R d , with domain D(A), given by We assume that a = (a i,j ) i,j is a measurable symmetric matrix defined on O which satisfies the uniform ellipticity condition where λ and Λ are positive constants. The energy associated with the matrix a will be denoted by It's defined for functions w, v ∈ H 1 0 (O), or for w ∈ H 1 loc (O) and v ∈ H 1 0 (O) with compact support. We assume that we have predictable random functions We define f (·, ·, ·, 0, 0) := f 0 , g(·, ·, ·, 0, 0) := g 0 = (g 0 1 , ..., g 0 d ) and h(·, ·, ·, 0, 0) : In the sequel, | · | will always denote the underlying Euclidean or l 2 −norm. For example Remark 2.1. Let us note that this general setting of the SPDE (1) we consider, encompasses the case of an SPDE driven by a space-time noise, colored in space and white in time as in [21] for example (see also Example 1 in [9]).
Moreover we introduce some integrability conditions on the coefficients f 0 , g 0 , h 0 and the initial data ξ. Along this article, we fix a terminal time T > 0.

Weak solutions
We now introduce H T , the space of We define H loc = H loc (O) to be the set of H 1 loc (O)-valued predictable processes defined on [0, T ] such that for any compact subset K in O: The space of test functions is the algebraic tensor product Now we recall the definition of the regular measure which has been defined in [9].
C denotes the space of continuous functions with compact support in [0, T [×O and finally: . It is known (see [14]) that W is continuously embedded in C([0, T ]; L 2 (O)), the set of L 2 (O)-valued continuous functions on [0, T ]. So without ambiguity, we will also consider W T = {ϕ ∈ W; ϕ(T ) = 0}, W + = {ϕ ∈ W; ϕ ≥ 0}, W + T = W T ∩ W + . Definition 2.1. An element v ∈ K is said to be a parabolic potential if it satisfies: We denote by P the set of all parabolic potentials.
The next representation property is crucial: Let v ∈ P, then there exists a unique positive Radon measure on [0, T [×O, denoted by ν v , such that: Moreover, v admits a right-continuous (resp. left-continuous) versionv (resp.v) : [0, T ] → L 2 (O) . Such a Radon measure, ν v is called a regular measure and we write: T [×O be compact, v ∈ P is said to be ν−superior than 1 on K, if there exists a sequence v n ∈ P with v n ≥ 1 a.e. on a neighborhood of K converging to v in L 2 ([0, T ]; H 1 0 (O)). We denote: Proposition 2.4. (Proposition 2.1 in [19]) Let K ⊂ [0, T [×O compact, then S K admits a smallest v K ∈ P and the measure ν v K whose support is in K satisfies Definition 2.6. A property is said to hold quasi-everywhere (in short q.e.) if it holds outside a set of null capacity. We say that u admits a quasi-continuous version, if there existsũ quasi-continuous such thatũ = u a.e.
The next proposition, whose proof may be found in [18] or [19] shall play an important role in the sequel:  Proof. Let A be a polar set and consider a sequence We end this part by a convergence lemma which plays an important role in our approach (Lemma 3.8 in [19]): ; if u is a quasi-continuous function and |u| is bounded by a element in P. Then We now give the assumptions on the obstacle that we shall need in the different cases that we shall consider.
is an adapted random field almost surely quasi-continuous, in the sense that for P -almost all ω ∈ Ω, the map (t, x) → S t (ω, x) is quasi-continuous. Moreover, S 0 ≤ ξ P -almost surely and S is controlled by the solution of an SPDE, i.e. ∀t ∈ [0, T ], where S ′ is the solution of the linear SPDE with null boundary Dirichlet conditions.

Assumption (OL):
The obstacle S : [0, T ] × Ω × O → R is an adapted random field, almost surely quasi-continuous, such that S 0 ≤ ξ P -almost surely and controlled by a local solution of an SPDE, i.e. ∀t ∈ [0, T ], where S ′ is a local solution of the linear SPDE Assumption (HO2) It is well-known that under (HO2) S ′ belongs to H T , is unique and satisfies the following estimate: see for example Theorem 8 in [4]. Moreover, as a consequence of Theorem 3 in [9], we know that S ′ admits a quasi-continuous version.
3. the following relation holds almost surely, for all t ∈ [0, T ] and all ϕ ∈ D, 4. u admits a quasi-continuous version,ũ, and we have We denote by R(ξ, f, g, h, S) the solution of the obstacle problem when it exists and is unique.
3. the following relation holds almost surely, for all t ∈ [0, T ] and all ϕ ∈ D, 4. u admits a quasi-continuous version,ũ, and we have We denote by R loc (ξ, f, g, h, S) the set of all the local solutions (u, ν). Finally, in the sequel, we introduce some constants ǫ, δ > 0, we shall denote by C ǫ , C δ some constants depending only on ǫ, δ, typically those appearing in the kind of inequality 3. L p −estimate for the uniform norm of solutions with null Dirichlet boundary condition In this section, we want to study, for some p ≥ 2, the L p − estimate for the uniform norm of the solution of (1). To get such estimate, we need stronger integrability conditions on the coefficients and the initial condition. To this end, we consider the following assumptions: for θ ∈ [0, 1[ and p ≥ 2:

Assumption (HO∞p)
To get the estimates we need, we apply Itô's formula to u − S ′ , in order to take advantage of the fact that S − S ′ is non-positive and that as u is solution of (1) and S ′ satisfies (8), that is why we introduce the following functions: It is obvious thatf ,ḡ andh satisfy the Lipschitz conditions with the same Lipschitz coefficients as f , g and h and ξ − S ′ 0 ∞ ∈ L p (Ω, P ). Nevertheless, we need a supplementary hypothesis: This assumption is fulfilled in the following case: , and assumptions (H) and (HO∞p) hold, then: f satisfies the Lipschitz condition with the same Lipschitz coefficients: f satisfies the integrability condition: And the same forḡ andh, which proves that (HDθp) holds.
We now give the main result of this Section, which is a version of the maximum principle in the case of a solution vanishing on the boundary of O: Let (u, ν) be the solution of OSPDE (1) with null boundary condition, then for all t ∈ [0, T ], where c(p) is a constant which depends on p and k(t) is a constant which depends on the structure constants and t ∈ [0, T ].
As the proof of this theorem is quite long, we split it into several steps.
3.1. The case where ξ,f 0 ,ḡ 0 andh 0 are uniformly bounded In this subsection, we assume that the hypotheses (H), (O), (HI2p), (HO∞p) hold and we add the following stronger ones: Then it is obviously that . Under these hypotheses, we know that the SPDE with obstacle (1) admits a unique weak We start by proving the following L l −estimate: Moreover there exist constants c, c ′ > 0 which only depend on C, α, β and on the quantity such that, for all real l ≥ 2, and and consequently we can apply Itô's formula to (u − S ′ , ν) (See Theorem 5 in [9]). We fix a real l ≥ 2, T > 0 and introduce the sequence (ϕ n ) n∈N * of functions such that for all n ∈ N * : One can easily verify that for fixed n, ϕ n is twice differentiable with bounded second derivative, ϕ ′′ n (x) ≥ 0, and as n → ∞ one has ϕ n ( Moreover, the following relations hold, for all x ∈ R and n ≥ l: Since the support of ν is {u = S}, the last term is equal to and it is negative, because By the uniform ellipticity of the operator A we get Let ǫ > 0 be fixed. Using the Lipschitz condition onf and the properties of the functions (ϕ n ) n we get

Now using Cauchy-Schwarz inequality and the Lipschitz condition onḡ we get
In the same way as before Thus taking the expectation, we deduce On account of the contraction condition, one can choose ǫ > 0 small enough such that and then We obtain by Gronwall's Lemma, that and so it is now easy from (18) to get Finally, letting n → ∞ by Fatou's lemma we deduce (14) and (15). Then with (17), we know that This yields (16) by Fatou's lemma.
With the help of Lemma 3.2, we are able to prove the following Itô formula: Assume the hypotheses of the previous lemma. Let (u, ν) be the solution of the problem (1). Then for l ≥ 2, we get the following Itô's formula, P -almost surely, for Proof. From Itô's formula (see Theorem 5 in [9]), with the same notations as in the previous lemma, we have P -almost surely, and for all t ∈ [0, T ] and all n ∈ N * , Therefore, passing to the limit as n → ∞, the convergences come from the Lemma 3.2 and the dominated convergence theorem.
From now on, we assume the following stronger hypothesis: At this stage, the idea is to adapt the Moser iteration technics to our setting. To this end, in order to control uniformly the L l norms and make l tend to +∞, we introduce for each l ≥ 2, the processes v and v ′ given by where the constants are given by The main difficulty in the stochastic case is to control the martingale part. We start by estimating the bracket of the local martingale in (21) The proof is the same as Lemma 12 in [5] replacing u by u − S ′ and also h byh.
In what follows we will use the notion of domination, which is essential to handle the martingale part. We recall the definition from Revuz and Yor [20].

Definition 3.5. A non-negative, adapted right continuous process X is dominated by an increasing process
for any bounded stopping time, ρ.
One important result related to this notion is the following domination inequality (see Proposition IV.4.7 in Revuz-Yor, p. 163), for any k ∈]0, 1[, where C k is a positive constant and X * t := sup s≤t |X s |. We will also use the fact that if A, A ′ are increasing processes, then the domination of a process X by A is equivalent to the domination of X + A ′ by A + A ′ . Lemma 3.6. The Process τ v is dominated by the process v ′ where In other words, we have where γ, c 1 , c 2 and c 3 are the constants given above.
Proof. Starting from the relation (21): The last term is negative: from the condition of minimality, we have the following relation, Then we can do the same calculus as in the proof of Lemma 14 in [5], replacing u by u − S ′ and f , g, h byf ,ḡ,h respectively.
The proofs of the next 3 lemmas are similar to the proofs of Lemmas 15, 16 and 17 in [5], just replacing u by u − S ′ and replacing f , g and h byf ,ḡ andh respectively.
is dominated by the process where σ = d+2θ d and k : R + → R + is a function independent of l, depending only on the structure constants. Lemma 3.9. There exists a function k 1 : R + × R + → R + which involves only the structure constants of our problem and such that the following estimate holds We now prove Theorem 3.1 in the case where ξ,f 0 ,ḡ 0 andh 0 are uniformly bounded: We set l = pσ n , with some n ∈ N * . By Lemma 3.8 and the domination inequality (25) we deduce, for n ≥ 1, where C σ −n is the constant in the domination inequality. This constant is estimated by (See the exercise IV.4.30 in Revuz -Yor, p. 171). So let us denote by and deduce from the above inequality the following one Iterating this relation n times we get Now we shall let n tend to infinity in this relation. Since in general one has for any function F : which implies Now we estimate Ea 1 by using the fact that δ |u − S ′ | pσ 1 σ θ;t ≤ v t , with p replacing l in the expression of v. So we have Finally one deduces the following estimate by applying Lemma 3.9 with l = p: Moreover (see Theorem 11 [5]), we have Hence, This ends the proof of Theorem 3.1 in this particular case where ξ,f 0 ,ḡ 0 andh 0 are uniformly bounded. We now turn out to the general case.

Proof of Theorem 3.1 in the general case
We now assume that (H), (O), (HI2p), (HO∞p) and (HDθp) hold. We are going to prove Theorem 3.1 in the general case by using an approximation argument. For this, for all n ∈ N * , One can check that for all n,f n ,ḡ n ,h n and ξ n − S ′ 0 satisfy all the assumptions of the Step 1 of the proof, and that Lipschitz constants do not depend on n. And the obstacle S − S ′ is controlled by 0, which obviously satisfies (HO2). For each n ∈ N * , we put (ū n , ν n ) = R(ξ n − S ′ 0 ,f n ,ḡ n ,h n , S − S ′ ) and we know thatū n satisfies the estimate of Step 1. We are now going to prove that (ū n , ν n ) converges to (ū, ν) = R(ξ − S ′ 0 ,f ,ḡ,h, S − S ′ ). Let us fix n ≤ m in N * and putū n,m :=ū n −ū m and ν n,m := ν n − ν m We first note that u n,m satisfies the equation andḡ i,n,m ,h j,n,m have similar expressions. Clearly one has and some similar relations forḡ i,n,m (t, w, x, 0, 0) andh j,n,m (t, w, x, 0, 0) . On the other hand, one can easily verify that By Lemma 5.4 with l = 2 (see Appendix) we deduce that Therefore, (ū n ) has a limitū in H T .
We now study the convergence of (ν n ). Denote by v n the parabolic potential associated to ν n , and z n =ū n − v n , so z n satisfies the following SPDE We define z 1,n to be the solution of the following SPDE with initial value ξ n − S ′ 0 and zero boundary condition: This is a linear SPDE in z 1,n , its solution uniquely exists and belongs to H T . Applying Itô's formula to (z 1,n ) 2 and doing a classical calculation, we get: Then, we define z 2,n to be the solution of the following SPDE with initial value 0 and zero boundary condition: This is still a linear SPDE in z 2,n , its solution uniquely exists and from the proof of Theorem 11 in [5], we know that This yields: E z n − z m 2 T −→ 0, as n, m → ∞. Hence, using (29) and the fact thatū n = z n + v n , we get: T −→ 0, as n, m → ∞. Therefore, (v n ) has a limit v in H T . So, by extracting a subsequence, we can assume that (v n ) converges to v in K almost-surely. Then, it's clear that v ∈ P, and we denote by ν the random regular measure associated to the potential v. Moreover, we have P -a.s., ∀ϕ ∈ W + t : As a consequence of Lemma 5.3 in the Appendix, we know that Therefore, we can apply Proposition 3.3 toū n and pass to the limit and so we obtain that this proposition remains valid in this case. Then, one can end the proof by repeating the first part of Step 1 starting from Proposition 3.3.
We conclude thanks to the uniqueness of the solution of the obstacle problem ensuring thatū is equal to u − S ′ .

Maximum Principle for local solutions
We now introduce the lateral condition on the boundary that we consider:

Itô's formula for the positive part of a local solution
The following proposition represents a key technical result which leads to a generalization of the estimates of the positive part of a local solution. Let (u, ν) ∈ R loc (ξ, f, g, h, S), denote by u + its positive part. For this we need the following notation: Proposition 4.2. Assume that ∂O is Lipschitz and that u + belongs to H T , i.e. u is nonpositive on the boundary of O and that the data satisfy the following integrability conditions for each t ≥ 0. Let ϕ : R → R be a function of class C 2 , which admits a bounded second order derivative and such that ϕ ′ (0) = 0. Then the following relation holds, a.s., for each t ∈ [0, T ], Proof. We consider φ ∈ C ∞ c (O), 0 ≤ φ ≤ 1, and put A direct calculation yields the following relation: Now we prove that φν is a regular measure: We know that: We replace ϕ by φϕ in (32), where φ is the same as before, and we obtain the following relation: note that φ does not depend on t and by a similar calculation as before, we get (− ∂ϕ s ∂s , φv s )ds + E(ϕ s , φv s )ds + (K s , ϕ s )ds − (k s , ∇ϕ s )ds = ϕ(s, x)dφν We denote byz the solution of the following PDE with Dirichlet boundary condition and the initial value 0: dz t + Az t dt = K t dt + divk t dt.
It is easy to verify that (ψ n ) converges uniformly to the function ψ, (ψ ′ n ) converges everywhere to the function (y → ϕ ′ (y + )) and (ψ ′′ n ) converges everywhere to the function (y → 1I {y>0} ϕ ′′ (y + )). Moreover we have the estimates: where C is a constant. Thanks to Itô's formula for the solution of OSPDE (1) (see Theorem 5 in [9]), we have almost surely, for t ∈ [0, T ], Making n tends to +∞ and using the fact that 1I {ws>0} ∂ i w s = ∂ i w + s , we get by the dominated convergence theorem: Then we consider a sequence (φ n ) in C ∞ c (O), 0 ≤ φ n ≤ 1, converging to 1 everywhere on O and such that for any y ∈ H 1 0 (O) the sequence (φ n y) tends to y in H 1 0 (O) and where C is a constant which does not depend on y. Such a sequence (φ n ) exists because ∂O is assumed to be Lipschitz (see Lemma 19 in [8]).
One has to remark that if i ∈ {1, ...d} and y ∈ H 1 0 (O), then (y∂ i φ n ) tends to 0 in L 2 (O). Now, we set w n = φ n u and We have Remarking that for all s ∈ (0, T ], (φ n ϕ ′ (w + n,s )) (resp. (∂ i φ n ϕ ′ (w + n,s ))) tends to ϕ ′ (u + s ) (resp. 0) in H 1 0 (O) (resp. L 2 (O)) we get by the dominated convergence theorem the convergence of all the terms in equality (34) excepted the one involving the measure ν. For this last term, we know that w n is quasi-continuous and from (33) and (34) it is easy to verify Then, by Fatou's lemma, we have Hence, the convergence of the last term comes from the dominated convergence theorem.
From the proof of Proposition 4.2, we know that (φu 1 , φν 1 ) and (φu 2 , φν 2 ) are the solutions of problem (1) with null Dirichlet boundary conditions. We have the Itô formula forŵ, see Theorem 6 in [9]. Then we do the same approximations as in the proof of Proposition 4.2, we can get the desired formula.

Maximum principle
We first consider the case of a solution u such that u ≤ 0 on ∂O.
Let (u, ν) ∈ R loc (ξ, f, g, h, S) be such that u + ∈ H. Then one has where k (t) is constant that depends on the structure constants and t ∈ [0, T ].
So we have: . Therefore, applying the preceding theorem to u − M , we obtain (41). On the other hand, one has the following estimates: This allows us to conclude the proof.
Proof. Beginning from the Itô formula for the difference of solutions of two obstacle problems which has been proved in [9]: we take the same ϕ n as in the proof of Lemma 3.2, Proof. From the Itô formula for the difference of two solutions (see Theorem 6 in [9]), we have P -almost surely for all t ∈ [0, T ] and n ∈ N * O ϕ n (û t (x)) dx + t 0 E ϕ ′ n (û s ),û s ds = O ϕ n (ξ(x))dx Then, passing to the limit as n → ∞, the convergences come from the dominated convergence theorem.
Similar as before, we define the processesv andv ′ bŷ where above and below γ, c 1 , c 2 and c 3 are the constants given by relations (23). We remark first that the last term in (46) is non positive, indeed: Then applying the same proof as the one of Lemma 3.6, we obtain: and this yields that the process τv is dominated byv ′ . Starting from here, we can repeat line by line the proofs of Lemmas 15-17 in [5] and apply the Moser iteration as at the end of Subsection 3.1 to obtain the desired estimations, namely: Lemma 5.3. There exists a function k 2 : R + → R + which involves only the structure constants of our problem and such that the following estimate holds Lemma 5.4. There exists a function k 1 : R + × R + → R + which involves only the structure constants of our problem and such that the following estimate holds