Abstract
The article deals with the existence of solutions of some unilateral problems in the OrliczSobolev spaces framework when the righthand side is a Radon measure.
Mathematics Subject Classification: 35K86.
Keywords:
unilateral problem; radon measure; OrliczSobolev spaces1 Introduction
We deal with boundary value problems
where
T > 0 and Ω is a bounded domain of R^{N}, with the segment property. a : Ω × R × R^{N }→ R^{N }is a Carathéodory function (that is, measurable with respect to x in Ω for every (t, s, ξ) in R × R × R^{N }, and continuous with respect to (s, ξ) in R × R^{N }for almost every x in Ω) such that for all ξ, ξ* ∈ R^{N }, ξ ≠ ξ*,
where c (x,t) belongs to , P is an Nfunction 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 ∈ L^{1}(0,T;L^{1}(Ω)) and u_{0 }∈ L^{1}(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 L^{p}(0, T; W^{1,p}(Ω)) by an inhomogeneous Sobolev space W^{1,x}L_{M }built from an Orlicz space L_{M }instead of L^{p}, where the Nfunction 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 Nfunction, 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 nondecreasing, right continuous, with a(0) = 0, a(t) > 0 for t > 0 and a(t) → ∞ as t → ∞. The Nfunction conjugate to M is defined by , where is given by (see [3,4]).
The Nfunction M is said to satisfy the Δ_{2 }condition if, for some k > 0:
when this inequality holds only for t ≥ t_{0 }> 0, M is said to satisfy the Δ_{2 }condition near infinity.
Let P and Q be two Nfunctions. P ≪ Q means that P grows essentially less rapidly than Q; i.e., for each ε > 0,
Let Ω be an open subset of R^{N}. The Orlicz class (resp. the Orlicz space L_{m}(Ω)) is defined as the set of (equivalence classes of) realvalued 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 Δ_{2 }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 Δ_{2 }condition, for all t or for t large, according to whether Ω has infinite measure or not.
We now turn to the OrliczSobolev space. W^{1}L_{M}(Ω) (resp. W^{1}E_{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, W^{1}L_{M}(Ω) and W^{1}E_{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 W^{1}E_{M}(Ω) and the space as the closure of in W^{1}L_{M}(Ω). We say that u_{n }converges to u for the modular convergence in W^{1}L_{M}(Ω) if for some for all α ≤ 1. This implies convergence for . If M satisfies the Δ_{2 }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 Nfunction 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 R^{N}, T > 0 and set Q = Ω × ]0, T[. Let m ≥ 1 be an integer and let M be an Nfunction. For each α ∈ IN^{N }, denote by the distributional derivative on Q of order α with respect to the variable x ∈ R^{N}. The inhomogeneous OrliczSobolev 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 αorder derivatives, α ≤ m. We shall also consider the weak topologies and . If u ∈ W^{m, x}L_{M}(Q), then the function : t ↦ u(t) = u(t,.) is defined on [0, T] with values in W^{m}L_{M}(Ω). If, further, u ∈ W^{m,x}E_{M}(Q) then the concerned function is a W^{m}E_{M}(Ω)valued and is strongly measurable. Furthermore, the following imbedding holds: W^{m,x}E_{M}(Q) ⊂ L^{1}(0,T; W^{m}E_{M}(Ω)). The space W^{m,x}L_{M}(Q) is not in general separable, if u ∈ W^{m,x}L_{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 L^{1}(0,T). The space is defined as the (norm) closure in W^{m,x}E_{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 W^{m,x}L_{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 W^{m,x}L_{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 F_{0 }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 OrliczSobolev spaces of order 1: W^{1,x}L_{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 R^{N}, 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
Lemma 3.2 Under the hypotheses (1.1)(1.3), if f, u_{0 }are regular functions and f, u_{0 }≥ 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(u_{0}_{2})t  s, where u_{0}(x) = u(x, 0).
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]).
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 Nfunction, we define K as the set of Nfunction D satisfying the following conditions:
i) M(D^{1}(s)) is a convex function,
iii) There exists an Nfunction 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(R^{N+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 Nfunction satisfying the conditions (i)(ii)(iii).
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 L^{1}(Q) and converge to μ in M_{b}(Q).
Let be a sequence in D(Ω) which is bounded in L^{1}(Ω) and converge to u_{0 }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 C^{2 }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 L^{1 }(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,
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 Nfunction 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 ω_{μ,j }have the following properties:
Set v = (T_{k}(u_{n})  ω_{μ,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,
Concerning I_{2},
and
About I_{3},
Set Φ(s) = s^{2}/2, Φ ≥ 0,then
So,
We are interested now with the terms of (1)(4).
About (1):
recall that ρ_{m}(u_{n}) = 1 on {u_{n} ≤ k}.
By using the inequality (1.3), we can deduce the existence of some measurable function h_{k }such that
since
Then,
Following the same way as in J_{2}, one has
Concerning the terms J_{4 }:
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(R^{N+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 OrliczSobolev spaces when the righthand side is a Radon measure.
Competing interests
The authors declare that they have no competing interests.
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