Generalization of Poincaré inequality in a Sobolev Space with exponent constant to the case of Sobolev space with a variable exponent

In this work, we study the Poincaré inequality in Sobolev spaces with variable exponent. As a consequence of this result we show the equivalent norms over such cones. The approach we adopt in this work avoids the difficulty arising from the possible lack of density of the space C∞0 (Ω). ©2021 All rights reserved.


Introduction
In recent years the whole world has undergone various changes in several fields, notably in computing and physics and especially in mechanics, flexible electronics and robotics. As far as we are concerned, we deal with the Poincaré inequality in Sobolev spaces with variable exponent. A multitude of results has emerged in recent years. This growing interest is because many concrete problems can be solved. Sobolev spaces make a special framework of our study, but the problem that we often encounter is that some properties which are verified in a classic Sobolev space but are not verified in a Sobolev space with variable exponent.
Still, this does not provide a satisfactory generalization of the classical Poincaré inequality, because the equality which holds for a constant exponent: The previous inequality does not correct, see for example [8].
If such a generalized Poincaré inequality can be established for all functions in the space D(Ω), it cannot be extended in this case by means of a density argument to all functions in the space W 1,p (Ω) whose traces vanish on ∂Ω := Γ . The purpose of this research note is to now more information about the Poincaré inequality for functions in D(Ω) and then extending it by a density argument. We obtain the Poincaré inequality as an immediate corollary of an equivalence of norms in cones of the space W 1,p (Ω) (see Theorems 3.1 and 3.2). There are many questions in this way like does the Poincaré inequality applied when we have space like Thus, avoiding the difficulty arising from the possible lack of density of the space D(Ω) in this space. secondly, it can be extended at no extra cost to the space Where Γ 0 ⊂ Γ and we have dΓ − meas(Γ 0 ) > 0.

Preliminaries
Given two normed vector spaces V and W, the notation V → W, respectively V ⊂⊂ W, means that V ⊂ W the canonical injection from V into W is continuous, respectively compact. The notation D(Ω) denotes the space of functions that are infinitely differentiables in Ω and whose support is a compact subset of Ω. Given a real number p 1, the notations L p (Ω), W 1,p (Ω) and denote the usual Lebesgue and Sobolev spaces; "usual" means here that the exponent p 1 is a constant.
if and only if for all x ∈ Ω.
(c) Let p(·) ∈ Ω such that p − 1. For each element x ∈ Ω, we define Suppose that there exists a function q(·) ∈ C(Ω) satisfying Then the following injection: is compact. Thus, in particular, the trace operator Finally, we state several results and properties of traces of functions in the space W 1,p(·) (Ω)

Poincaré Inequality in a Cone in W 1,p(·) (Ω)
We now establish some results extended for the usual Sobolev spaces to Sobolev spaces with variable exponents established by Jebelean and Precup [5].
Recall that, as a domain in R N , the open set Ω is in particular connected. So, a function u ∈ W 1,1 (Ω) that satisfies ∂ i u = 0 a.e. in Ω, 1 i N, is a constant function.

Idea of the proof
We use absurd reasoning to prove inequality. Hence w 0,p(·) 1 µ(U) |v| 1,p(·) , ∀w ∈ U, is the best possible constant in this Poincaré inequality.

Idea of the proof
We use the previous theorem, and we prove that C is the best possible constant in this Poincaré inequality.

Applications
Our first application is applied to the case where the cone U in W 1,p(·) (Ω) is a subspace, associated with an homogeneous Dirichlet condition. Idea of the proof 1. It suffices to verify that the above set U satisfies the assumptions of Theorem 3.1; 2. To show that U is sequentially weakly closed in W 1,p(·) (Ω); 3. It remains to show that |v| 1,p(·) is a norm over the space U. Therefore its trace on Γ is a constant function that takes the same value, and this value is zero since the trace vanishes on Γ 0 and dΓ − meas(Γ 0 ) > 0.
When Γ = Γ 0 , the above theorem shows that the Poincaré inequality holds. Then, there exists a constant C > 0 such that v 0,p(·) C|v| 1,p(·) , ∀v ∈ U, But, as indicated in the introduction, the space may be only strictly contained in the space. Hence in this case, establishing the Poincaré inequality for all functions in D(Ω) would yield the same inequality for all functions in W 1.p (Ω), but not for all functions inW 1,p(·) (Ω).
Our second application is to "genuine" cones U (i.e., which are not subspaces) of a specific form.

Theorem 4.3.
Let Ω a subset of R N . Let p( . ) ∈ C(Ω) and q( . ) ∈ C(Ω) with q − > 1 and such that where the function p ∂ : Ω −→ R is defined as follows Let f : Ω × R −→ R be a Carathéodory function such that, for all λ 0 there exists two constants C − λ and C + λ with the property that for all x ∈ Ω and all s ∈ R.
that there exist a non-negative function a ∈ L q (·) (Ω) and a positive constant b such that |f(x, s)| a(x) + b|s| q(x)−1 for all x ∈ Ω and all s ∈ R.
Finally, assume that the set does not contain nonzero constant functions. Then there exists a constant C = C(U) such that Idea of the proof.
1. Verify that the above set U satisfies the assumption of Theorem 3.1.
2. Verify that U is sequentially weakly closed. By using this inequality |f(x, s)| a(x) + b|s| q(x) q 1 (x) for almost all x ∈ Ω and all s ∈ R, the associated Nemytskii operator N f , defined for any function w ∈ L q(·) (Ω) maps L q(·) (Ω) into L q 1 (·) (Ω), and is continuous. The mapping is thus continuous, as composed of the continuous mappings On another hand, if w k converges to w as k converges to +∞ in L q(·) (Ω), Hence Ω f(x, w(x))dx = 0.
Remark 4.5. Theorem 2.1(c) shows that the same conclusion holds if the set U is of the more general form where h is a given non-negative function in the space L q(·) (Ω).

Conclusion and perspectives
This work has many advantages, but the most important thing is to put this result into practice and to broaden our knowledge. Our goal is to find the Poincaré inequality in the Sobolev spaces of the weighted variable exponent and the resolution of certain systems using certain methods in these enlarged spaces.