Parabolic problems with data measure

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.

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 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 L^p(0, T; W^1,p(Ω)) by an inhomogeneous Sobolev space W^1,xL_M built from an Orlicz space L_M instead of L^p, 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

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 R^N. 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 R^N, T > 0 and set Q = Ω × ]0, T[. Let m ≥ 1 be an integer and let M be an N-function. 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 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 ∈ W^{m, x}L_M(Q), then the function : t ↦ u(t) = u(t,.) is defined on [0, T] with values in W^mL_M(Ω). If, further, u ∈ W^m,xE_M(Q) then the concerned function is a W^mE_M(Ω)-valued and is strongly measurable. Furthermore, the following imbedding holds: W^m,xE_M(Q) ⊂ L¹(0,T; W^mE_M(Ω)). The space W^m,xL_M(Q) is not in general separable, if u ∈ W^m,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 W^m,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 W^m,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 W^m,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: W^1,xL_M(Q) and .

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

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(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 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 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,

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(R^N+1) which are zero in a neighborhood of (0,T) × ∂Ω and {T} × Ω.

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.

The authors declare that they have no competing interests.

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