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Parabolic problems with data measure
Boundary Value Problems volume 2011, Article number: 46 (2011)
Abstract
The article deals with the existence of solutions of some unilateral problems in the Orlicz-Sobolev spaces framework when the right-hand side is a Radon measure.
Mathematics Subject Classification: 35K86.
1 Introduction
We deal with boundary value problems
where
T > 0 and Ω is a bounded domain of RN, with the segment property. a : Ω × R × RN→ RNis a Carathéodory function (that is, measurable with respect to x in Ω for every (t, s, ξ) in R × R × RN, and continuous with respect to (s, ξ) in R × RNfor almost every x in Ω) such that for all ξ, ξ* ∈ RN, ξ ≠ ξ*,
where c (x,t) belongs to , P is an N-function such that P ≪ M and k i (i = 1,2,3,4) belongs to R+ and α to .
There have obviously been many previous studies on nonlinear differential equations with nonsmooth coefficients and measures as data. The special case was cited in the references (see [1, 2]).
It is noteworthy that the articles mentioned above differ in significant way, in the terms of the structure of the equations and data. In [1], when f ∈ L1(0,T;L1(Ω)) and u0 ∈ L1(Ω). The authors have shown the existence of solutions u of the corresponding equation of the problem for every q such that which is more restrictive than the one given in the elliptic case .
In this article, we are interested with an obstacle parabolic problem with measure as data. We give an improved regularity result of the study of Boccardo et al. [1].
In [1], the authors have shown the existence of a weak solutions for the corresponding equation, the function a(x, t, s, ξ) was assumed to satisfy a polynomial growth condition with respect to u and ∇u. When trying to relax this restriction on the function a(., s, ξ), we are led to replace the space Lp(0, T; W1,p(Ω)) by an inhomogeneous Sobolev space W1,xL M built from an Orlicz space L M instead of Lp, where the N-function M which defines L M is related to the actual growth of the Carathéodory's function.
For simplicity, one can suppose that there exist α > 0, β > 0 such that
2 Preliminaries
Let M : R+ → R+ be an N-function, i.e. M is continuous, convex, with M(t) > 0 for as t → 0 and as t → ∞. Equivalently, M admits the representation: where a : R+ → R+ is non-decreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞. The N-function conjugate to M is defined by , where is given by (see [3, 4]).
The N-function M is said to satisfy the Δ2 condition if, for some k > 0:
when this inequality holds only for t ≥ t0 > 0, M is said to satisfy the Δ2 condition near infinity.
Let P and Q be two N-functions. P ≪ Q means that P grows essentially less rapidly than Q; i.e., for each ε > 0,
Let Ω be an open subset of RN. The Orlicz class (resp. the Orlicz space L m (Ω)) is defined as the set of (equivalence classes of) real-valued measurable functions u on Ω such that
Note that L M (Ω) is a Banach space under the norm and is a convex subset of L M (Ω). The closure in L M (Ω) of the set of bounded measurable functions with compact support in is denoted by E M (Ω). The equality E M (Ω) = L M (Ω) holds if and only if M satisfies the Δ2 condition, for all t or for t large according to whether Ω has infinite measure or not.
The dual of E M (Ω) can be identified with by means of the pairing , and the dual norm on is equivalent to . The space L M (Ω) is reflexive if and only if M and satisfy the Δ2 condition, for all t or for t large, according to whether Ω has infinite measure or not.
We now turn to the Orlicz-Sobolev space. W1L M (Ω) (resp. W1E M (Ω)) is the space of all functions u such that u and its distributional derivatives up to order 1 lie in L M (Ω) (resp. E M (Ω)). This is a Banach space under the norm . Thus, W1L M (Ω) and W1E M (Ω) can be identified with subspaces of the product of N + 1 copies of L M (Ω). Denoting this product by ΠL M , we will use the weak topologies and . The space is defined as the (norm) closure of the Schwartz space in W1E M (Ω) and the space as the closure of in W1L M (Ω). We say that u n converges to u for the modular convergence in W1L M (Ω) if for some for all |α| ≤ 1. This implies convergence for . If M satisfies the Δ2 condition on R+(near infinity only when Ω has finite measure), then modular convergence coincides with norm convergence.
Let (resp. ) denote the space of distributions on Ω which can be written as sums of derivatives of order ≤ 1 of functions in (resp. ). It is a Banach space under the usual quotient norm.
If the open set Ω has the segment property, then the space is dense in for the modular convergence and for the topology (cf. [5, 6]). Consequently, the action of a distribution in on an element of is well defined.
For k > 0, s ∈ R, we define the truncation at height k,T k (s) = [k - (k - |s|)+]sign(s).
The following abstract lemmas will be applied to the truncation operators.
Lemma 2.1 [7] Let F : R → R be uniformly lipschitzian, with F(0) = 0. Let M be an N-function and let (resp.).
Then (resp. ). Moreover, if the set of discontinuity points of F' is finite, then
Let Ω be a bounded open subset of RN, T > 0 and set Q = Ω × ]0, T[. Let m ≥ 1 be an integer and let M be an N-function. For each α ∈ INN, denote by the distributional derivative on Q of order α with respect to the variable x ∈ RN. The inhomogeneous Orlicz-Sobolev spaces are defined as follows .
The last space is a subspace of the first one, and both are Banach spaces under the norm . We can easily show that they form a complementary system when Ω satisfies the segment property. These spaces are considered as subspaces of the product space ΠL m (Q) which have as many copies as there are α-order derivatives, |α| ≤ m. We shall also consider the weak topologies and . If u ∈ Wm, xL M (Q), then the function : t ↦ u(t) = u(t,.) is defined on [0, T] with values in WmL M (Ω). If, further, u ∈ Wm,xE M (Q) then the concerned function is a WmE M (Ω)-valued and is strongly measurable. Furthermore, the following imbedding holds: Wm,xE M (Q) ⊂ L1(0,T; WmE M (Ω)). The space Wm,xL M (Q) is not in general separable, if u ∈ Wm,xL M (Q), we cannot conclude that the function u(t) is measurable on [0,T]. However, the scalar function t ↦ ||u(t)||M,Ω, is in L1(0,T). The space is defined as the (norm) closure in Wm,xE M (Q) of . We can easily show as in [6] that when Ω has the segment property, then each element u of the closure of with respect of the weak * topology is limit, in Wm,xL M (Q), of some subsequence for the modular convergence; i.e., there exists λ > 0 such that for all |α| ≤ m,
this implies that (u i ) converges to u in Wm,xL M (Q) for the weak topology . Consequently, , and this space will be denoted by .
Furthermore, . Poincaré's inequality also holds in , i.e., there is a constant C > 0 such that for all one has . Thus both sides of the last inequality are equivalent norms on . We have then the following complementary system:
F being the dual space of . It is also, except for an isomorphism, the quotient of by the polar set , and will be denoted by , and it is shown that . This space will be equipped with the usual quotient norm where the infimum is taken on all possible decompositions . The space F0 is then given by and is denoted by .
We can easily check, using Lemma 4.4 of [6], that each uniformly lipschitzian mapping F, with F(0) = 0, acts in inhomogeneous Orlicz-Sobolev spaces of order 1: W1,xL M (Q) and .
3 Main results
First, we give the following results which will be used in our main result.
3.1 Useful results
Hereafter, we denote by the real number defined by is the measure of the unit ball of RN, and for a fixed t ∈ [0, T], we denote μ(θ) = meas{(x,t) : |u(x, t)| > θ}.
Lemma 3.1 [8] Let , and let fixed t ∈ [0, T], then we have
and where is defined by
Lemma 3.2 Under the hypotheses (1.1)-(1.3), if f, u0 are regular functions and f, u0 ≥ 0, then there exists at least one positive weak solution of the problem
such that
Proof
Let u be a continuous function, we say that u satisfies (*) if: there exists a continuous and increasing function β such that ||u(t) - u(s)||2 ≤ β(||u0||2)|t - s|, where u0(x) = u(x, 0).
Let .
Let us consider the set , where C is a closed convex of . It is easy to see that is a closed convex (since all its elements satisfy (*) ).
We claim that the problem
has a weak solution which is unique in the sense defined in [9].
Indeed, let us consider the approximate problem
where the functional Φ is defined by Φ : X → R ∪ {+ ∞} such that
The existence of such u n ∈ X was ensured by Kacur et al. [10].
Following the same proof as in [9], we can prove the existence of a solution u of the problem (E') as limit of u n (for more details see [9]).
Lemma 3.3 Let such that and .
Then, there exists a smooth function (v j ) such that
v j → v for the modular convergence in ,
for the modular convergence in .
For the proof, we use the same technique as in [11] in the parabolic case.
3.2 Existence result
Let M be a fixed N-function, we define K as the set of N-function D satisfying the following conditions:
i) M(D-1(s)) is a convex function,
ii) ,
iii) There exists an N-function H such that and near infinity.
Theorem 3.1 Under the hypotheses (1.1)-(1.5), The problem (P) has at least one solution u in the following sense:
for all φ ∈ D(RN+1) which are zero in a neighborhood of (0, T) × ∂ Ω and {T} × Ω.
Remark 3.1 (1) If ψ = - ∞ in the problem (P), then the above theorem gives the same regularity as in the elliptic case.
(2) An improved regularity is reached for all N-function satisfying the conditions (i)-(ii)-(iii).
For example, , for all .
In the sequel and throughout the article, we will omit for simplicity the dependence on x and t in the function a(x, t, s, ξ) and denote ϵ(n, j, μ, s, m) all quantities (possibly different) such that
and this will be in the order in which the parameters we use will tend to infinity, that is, first n, then j, μ, s, and finally m. Similarly, we will write only ϵ(n), or ϵ(n, j),... to mean that the limits are made only on the specified parameters.
3.2.1 A sequence of approximating problems
Let (f n ) be a sequence in D(Q) which is bounded in L1(Q) and converge to μ in M b (Q).
Let be a sequence in D(Ω) which is bounded in L1(Ω) and converge to u0 in M b (Ω).
We define the following problems approximating the original (P):
Lemma 3.4 Under the hypotheses (1.1)-(1.3), there exists at least one solution u n of the problem (P n ) such that a.e. in Q.
For the proof see Lemma 3.2.
3.2.2 A priori estimates
Lemma 3.5 There exists a subsequence of (u n ) (also denoted (u n )), there exists a measurable function u such that:
Proof:
Recall that u n ≥ 0 since f n ≥ 0.
Let h > 0 and consider the following test function v = T h (u n - T k (u n )) in (P n ), we obtain
We have
So,
Now, let us fix k > ||ψ||∞, we deduce the fact that: .
Let h to tend to zero, one has
Let us use as test function in (P n ),v = T k (u n ), then as above, we obtain
Then (T k (u n ) n ) is bounded in , and then there exist some such that
T k (u n ) ⇀ ω k , weakly in for , strongly in E M (Q) and a.e in Q.
Let consider the C2 function defined by
Multiplying the approximating equation by , we get in the distributions sense. We deduce then that η k (u n ) being bounded in and in . By Corollary 1 of [12], η k (u n ) is compact in L1 (Q).
Following the same way as in [2], we obtain for every k > 0,
Using now the estimation (3.1) and Fatou's lemma to obtain
Let fixed a t ∈ [0, T]. We argue now as for the elliptic case, the problem becomes:
We denote g n := nT n ((u n - ψ)-).
Let φ be a truncation defined by
for all θ, h > 0.
Using v = φ (u n ) as a test function in the approximate elliptic problem , we obtain by using the same technique as in [8]
here and below C denote positive constants not depending on n.
By using Lemma 3.1, we obtain (supposing -μ'(θ) > 0 which does not affect the proof) and following the same way as in [8], we have for D ∈ K,
Let denote , then
Using Lemma 3.1, denoting one has
Remark also that and using Lemma 3.2, we have .
Combining the inequalities (3.5) and (3.6) we obtain,
and since the function is absolutely continuous, we get
Then, the sequence (u n ) is bounded in and we deduce that for all N-function D ∈ K.
3.3 Almost everywhere convergence of the gradients
Lemma 3.6 The subsequence (u n ) obtained in Lemma 3.5 satisfies:
Proof:
Let m > 0, k > 0 such that m > k. Let ρ m be a truncation defined by
where v j ∈ D(Q) such that v j ≥ ψ and v j → T k (u) with the modular convergence in (for the existence of such function see [11] since ).
ω μ is the mollifier function defined in Landes [13], the function ωμ,jhave the following properties:
Set v = (T k (u n ) - ωμ,j) ρ m (u n ) as test function, we have
Let us recall that for , there exists a smooth function u nσ (see [14]) such that
Remark also that,
and it is easy to see that .
Concerning I2,
As in I1, we obtain ,
and
thus by using the fact that .
So, .
About I3,
Set Φ(s) = s2/2, Φ ≥ 0,then
So,
and easily we deduce, .
Finally we conclude that: .
We are interested now with the terms of (1)-(4).
About (1):
recall that ρ m (u n ) = 1 on {|u n | ≤ k}.
Let .
By using the inequality (1.3), we can deduce the existence of some measurable function h k such that
since
and strongly in .
Then,
Following the same way as in J2, one has
Concerning the terms J4 :
Letting n → ∞, then
Taking now the limits j → ∞ and after μ → ∞ in the last equality, we obtain
Then,
About (2):
Since (u n ) is bounded in and using (iii), we have (a(., u n ,∇u n )) is bounded in L H (Q), then
so,
About (4):
Since u ≥ ψ, then T k (u) ≥ T k (ψ) and there exist a smooth function v j ≥ T k (ψ) such that v j → T k (u) for the modular convergence in .
Taking into account now the estimation of (1), (2), (4)and (5), we obtain
On the other hand,
each term of the last right hand side is of the form ϵ(n, j, s), which gives
Following the same technique used by Porretta [2], we have for all r < s :
Thus, as in the elliptic case (see [7]), there exists a subsequence also denoted by u n such that
We deduce then that,
Lemma 3.7 For all k > 0,
Proof:
We have proved that
≤ ϵ (n, j, μ, s, m) (see (3.8)).
We can also deduce that
Then
then,
Letting n → ∞, we deduce
Using the same argument as above, we obtain
and Vitali's theorem and (1.1) gives
3.3.1 The convergence of the problems (P n ) and the completion of the proof of Theorem 3.1
The passage to the limit is an easy task by using the last steps, then
then,
for all φ ∈ D(RN+1) which are zero in a neighborhood of (0,T) × ∂Ω and {T} × Ω.
4 Conclusion
In this article, we have proved the existence of solutions of some class of unilateral problems in the Orlicz-Sobolev spaces when the right-hand side is a Radon measure.
References
Boccardo L, Dall'Aglio A, Gallouet T, Orsina L: Nonlinear parabolic equations with measures data. J Funct Anal 1997, 147(1):237-258. 10.1006/jfan.1996.3040
Porretta A: Existence results for nonlinear parabolic equations via strong convergence of truncations. Ann Mat Pura Appl 1999, IV(CLXXVII):143-172.
Krasnoselśkii M, Rutickii Ya: Convex Functions and Orlicz Spaces. Volume 3. Nodhoff Groningen; 1969.
Adams R: Sobolev Spaces. Academic Press, New York; 1975.
Gossez JP: Some approximation properties in Orlicz-Sobolev spaces. Studia Math 1982, 74: 17-24.
Gossez JP: Nonlinear elliptic boundary value problems for equations with rapidely or (slowly) increasing coefficients. Trans Am Math Soc 1974, 190: 163-205.
Benkirane A, Elmahi A: Almost everywhere convergence of the gradients of solutions to elliptic equations in orlicz spaces and application. Nonlinear Anal TMA 1997, 11(28):1769-1784.
Talenti G: Nonlinear elliptic equations, rearrangements of functions and orlicz spaces. Ann Mat Pura Appl 1979, 120(4):159-184.
Achchab B, Agouzal A, Debit N, Kbiri Alaoui M, Souissi A: Nonlinear parabolic inequalities on a general convex. J Math Inequal 2010, 4(2):271-284.
Kacur J: Nonlinear Parabolic Boundary Value Problems in the Orlicz-Sobolev Spaces. Partial Differential Equations, Banach Center Publications. Volume 10. PWN Polish Scientific Publishers, Warsaw; 1983.
Gossez JP, Mustonen V: Variationnel inequality in orlicz-sobolev spaces. Nonlinear Anal Theory Appl 1987, 11: 379-392. 10.1016/0362-546X(87)90053-8
Elmahi A, Meskine D: Strongly nonlinear parabolic equations with natural growth terms and L1data in Orlicz spaces. Portugaliae Math 2005., 62: Fas.2
Landes R: On the existence of weak solutions for quasilinear parabolic initial-boundary value problems. Proc R Soc Edinburgh A 1981, 89: 217-137. 10.1017/S0308210500020242
Elmahi A, Meskine D: Parabolic equations in orlicz space. J Lond Math Soc 2005, 72(2):410-428. 10.1112/S0024610705006630
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Alaoui, M.K. Parabolic problems with data measure. Bound Value Probl 2011, 46 (2011). https://doi.org/10.1186/1687-2770-2011-46
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DOI: https://doi.org/10.1186/1687-2770-2011-46