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Article

Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces

1
Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan
2
Research Team on Intelligent Decision Support Systems, Department of Artificial Intelligence Methods and Applied Mathematics, Faculty of Computer Science and Information Technology, West Pomeranian University of Technology, Szczecin ul. Zołnierska 49, 71-210 Szczecin, Poland
*
Author to whom correspondence should be addressed.
Fractal Fract. 2022, 6(3), 128; https://doi.org/10.3390/fractalfract6030128
Submission received: 9 January 2022 / Revised: 11 February 2022 / Accepted: 15 February 2022 / Published: 23 February 2022

Abstract

:
In many physical models, internal energy will run out without external energy sources. Therefore, finding optimal energy sources and studying their behavior are essential issues. In this article, we study the following variational problem: G ϵ = sup Ω G ( u ) ϵ q ( x ) d x : u L p ( . ) ϵ , u = 0   on   Ω , with the help of Γ -convergence, where G : R R is upper semicontinuous, non zero in the L 1 sense, 0 G ( u ) c | u | q ( . ) , Ω is a bounded open subset of R n , n 3 , 1 < p ( . ) < n , and p ( . ) q ( . ) p ( . ) . For special choices of G, we can study Bernoulli’s free-boundary and plasma problems in variable exponent Lebesgue spaces.

1. Introduction

The aim of this paper is to study the asymptotic behavior of low energy extremals { u ϵ } W 0 1 , p ( . ) ( Ω ) for which the following supremum is attained:
G ϵ = sup Ω G ( u ) ϵ q ( x ) d x : u W 0 1 , p ( . ) ( Ω ) , u L p ( . ) ϵ , u = 0   on   Ω ,
with Γ -convergence, when ϵ 0 . Here, Ω is a bounded open subset of R n , n 3 , and G : R R is a upper semicontinuous map, non zero in the L 1 sense, and satisfies the following growth condition:
0 G ( u ) c | u | q ( . ) ,   for   all   u W 0 1 , p ( . ) ( Ω ) .
Additionally, the exponents p , q : Ω ¯ R satisfy the following conditions:
(C1)
1 < p = inf p sup p = p + < n over Ω ¯ ;
(C2)
p ( . ) q ( . ) p ( . ) , where p ( x ) = n p ( x ) n p ( x ) , for x Ω ¯ ;
(C3)
p and q are Log-Hölder continuous;
(C4)
The set C = { x Ω ¯ : p ( x ) = q ( x ) } .
We refer to the next section for the details of variable exponent Sobolev space. In our study, due to Condition (C4), the embedding W 1 , p ( . ) ( Ω ) L q ( . ) ( Ω ) is not compact.
For a constant exponent, Problem (1) and its variants were extensively studied; see [1,2,3,4,5,6] and the references therein. For p = 2 , G ( s ) = | s | 2 : s R , Lions [7] used the concentration–compactness principle to prove that maximizing sequence of (1) either converges to a single point of compact; the concentration-compactness principle was also used to prove the existence of a solution for many kinds of elliptic PDEs with critical growth (see [8,9,10,11]). Later on, Flucher and Müller [2] generalized Lions’ concentration–compactness principle to study Problem (1) for a constant exponent and discussed its various applications.
Bonder and Silva [12] and Yongqiang [13] proved the concentration/compactness principle for variable exponent Lebesgue spaces, independently, to deal with the following nonlinear elliptic PDE with critical growth at infinity:
p ( . ) v = g ( . , v ) ,
where p ( x ) v : = div ( | v | p ( x ) 2 v ) is known as a p ( x ) -Laplacian operator. Critical growth is known as
| g ( . , s ) | c ( 1 + | s | q ( . ) ) .
Many authors have studied models such as electrorheological fluids with variants of the above elliptic PDE with different boundary conditions (Neumann, Dirichlet, and nonlinear); see [14,15,16,17] and references therein.
Recently, Bashir et al. [18] extended the work of Bonder and Silva [12] and proved generalized concentration/compactness principles for variable exponent Lebesgue spaces. In addition, they initiated the work on Problem (1) and proved the existence of low energy extremals. Γ -convergence is a significant variational convergence to study the asymptotic behavior of extremals of a sequence of functionals; typically, the Γ -convergence limit retains the desired features of a sequence.
This approach has already been used in similar scenarios. Bonder et al. [19] used Γ -convergence to determine the subcritical approximation of the best constant associated with critical Sobolev embedding W 1 , p ( . ) ( Ω ) L p ( . ) ( Ω ) . Briane et al. [20] studied the asymptotic behavior of the sequence of equi-coercive nonlinear energies defined on vector-valued functions. Amaziane et al. [21] worked with homogenization and minimization problems for a class of variational functionals, using Γ -convergence in the framework of Sobolev spaces with continuous variable exponents. Carbotti et al. [22] proved the local minimality of half spaces in Carnot groups and provided a lower bound of Γ -lim inf of the recalled energy in the form of horizontal parameters. Kolditz and Mang [23] studied the numerical solutions of quasi-static phase-field fracture problems based upon the Γ -convergence of parameters. For a comprehensive study on Γ -convergence, we refer to [24].
In Section 2, we present some preliminary concepts. Section 4 deals with the proofs of the main results of Theorems 1 and 2. In Section 5, we give some concluding remarks and possible future lines of research.

2. Preliminaries

To work in the framework of variable exponent Lebesgue spaces, we present some preliminary concepts here; for a detailed study, see [25]. Variable exponent Lebesgue space with exponent p ( . ) is defined as
L p ( . ) ( Ω ) = u L l o c 1 ( Ω ) : Ω | u ( x ) | p ( . ) d x < ,
with the following norm
u L p ( . ) ( Ω ) = inf σ > 0 : Ω | u ( x ) / σ | p ( . ) d x 1 .
The variable exponent Sobolev space is defined as
W 1 , p ( . ) ( Ω ) = u W l o c 1 , 1 ( Ω ) : u L p ( . ) ( Ω ) , | u | L p ( . ) ,
with the norm
u W 1 , p ( . ) ( Ω ) = u L p ( . ) ( Ω ) + u L p ( . ) ( Ω ) .
The closure of C c ( Ω ) in W 1 , p ( . ) ( Ω ) is denoted as W 0 1 , p ( . ) ( Ω ) . For 1 < inf p ( . ) sup p ( . ) < , the spaces L p ( . ) ( Ω ) , W 1 , p ( . ) ( Ω ) and W 0 1 , p ( . ) ( Ω ) are reflexive and separable Banach spaces.
Proposition 1.
Let p and q be exponents satisfying conditions (C1)–(C4). Then, we have the continuous embedding
W 1 , p ( . ) ( Ω ) L q ( . ) ( Ω ) ,
Moreover, the above embedding is compact if inf x Ω ¯ ( p ( x ) q ( x ) ) > 0 .
Proposition 2
(Poincaré inequality). For all u in W 0 1 , p ( . ) ( Ω ) , we have
| | u | | L p ( . ) ( Ω ) K | | u | | L p ( . ) ( Ω ) .
Recall the notations:
q + = sup Ω q ( x ) , q = inf Ω q ( x ) .
Let ρ ( u ) : = Ω | u ( x ) | p ( x ) d x , then the following proposition is useful to deal with the variable exponent norm.
Proposition 3
([26]). For u L p ( x ) ( Ω ) and the sequence { u n } n N L p ( x ) ( Ω ) ,
u 0 u L p ( x ) ( Ω ) = λ ρ u λ = 1 .
u L p ( x ) ( Ω ) < 1 ( = 1 ; > 1 ) ρ ( u ) < 1 ( = 1 ; > 1 ) .
u L p ( x ) ( Ω ) > 1 u L p ( x ) ( Ω ) p ρ ( u ) u L p ( x ) ( Ω ) p + .
u L p ( x ) ( Ω ) < 1 u L p ( x ) ( Ω ) p + ρ ( u ) u L p ( x ) ( Ω ) p .
lim n u n L p ( x ) ( Ω ) = 0 lim n ρ ( u n ) = 0 .
lim n u n L p ( x ) ( Ω ) = lim n ρ ( u n ) = .
Remark 1.
The norms | | u | | L p ( . ) ( Ω ) and | | u | | W 1 , p ( . ) ( Ω ) are equivalent for the Sobolev space W 0 1 , p ( . ) ( Ω ) .
In the end, we present two lemmas from [18] for use in a later section.
Lemma 1.
Consider G ( p ( . ) , q ( . ) , R N ) and ϵ < 1 . Then,
(a) 
G ϵ ( p ( . ) , q ( . ) , Ω ) G ;
(b) 
G = lim ϵ 0 G ϵ ( p ( . ) , q ( . ) , Ω ) .
Lemma 2.
Let v in W 0 1 , p ( . ) ( Ω ) with v L p ( . ) ( Ω ) ϵ . Take x 0 Ω , u = v / ϵ , δ > 0 , r < R satisfying r R k ( δ ) . Let
p R + = sup x B R ( x 0 ) p ( x ) ,   p R = inf x B R ( x 0 ) p ( x ) ,
q R + = sup x B R ( x 0 ) q ( x ) ,   q R = inf x B R ( x 0 ) q ( x ) .
For G satisfying the growth condition (2), the following inequality holds:
B r ( x 0 ) G ( v ) ϵ q ( x ) d x G ϵ max B R ( x 0 ) | u | p ( x ) d x + δ max u L p ( . ) p R + , u L p ( . ) p R q R + p R , B R ( x 0 ) | u | p ( x ) d x + δ max u L p ( . ) p R + , u L p ( . ) p R q R p R + ,

3. Proposed Theorems

First, we fix some notions to present the main result:
  • Best Sobolev constant,
    S ( p ( . ) , q ( . ) , Ω ) = sup Ω | u | q ( . ) d x : u W 0 1 , p ( . ) ( Ω ) , u L p ( . ) ( Ω ) 1 ,
  • Generalized Sobolev constant,
    G ( p ( . ) , q ( . ) , Ω ) : = sup Ω G ( u ) d x : u W 0 1 , p ( . ) ( Ω )   and   u L p ( . ) ( Ω ) 1 ,
  •  
    G 0 +   : = lim sup s 0 G ( s ) | s | q + , G 0   : = lim inf s 0 G ( s ) | s | q + , G +   : = lim sup | s | G ( s ) | s | q , G   : = lim inf | s | G ( s ) | s | q ,
  • Moreover, denote G 0 : = G 0 + = G 0 and G : = G + = G . In the case of equality, we assume that the following function exists:
    T ( x ) : = lim s 0 G ( s ) s q ( x ) : x Ω .
To apply Γ -convergence, we consider the following functional:
H ϵ ( u ) : = Ω G ( ϵ u ) ϵ q ( x ) d x .
Problem (1) is rewritten as
G ϵ = sup H ϵ ( u ) : u W 0 1 , p ( . ) ( Ω ) , u L p ( . ) 1 .
However, for any sequence { u ϵ } in W 0 1 , p ( . ) ( Ω ) , any functional defined on W 0 1 , p ( . ) ( Ω ) alone cannot retain the desired properties of H ϵ ( u ϵ ) . In this kind of scenario, the idea is to work on the ordered pair ( u ϵ , | u ϵ | p ( . ) d x ) and | u ϵ | p ( . ) d x is taken as a measure. Therefore, we consider the following space
X ( Ω ) = ( u , μ ) W 0 1 , p ( . ) ( Ω ) × M ( Ω ¯ ) : μ ( Ω ¯ ) 1 , μ = | u | p ( . ) d x + μ ¯ + i I μ i δ x i ,
where M ( Ω ¯ ) is the set of all finite nonnegative finite Borel measures on Ω ¯ and μ ¯ is a non-atomic measure in the decomposition of μ . Furthermore, we define the convergence in this space as
( u ϵ , μ ϵ ) T ( u , μ ) u ϵ u , weakly   in   L q ( . ) ( Ω ) ; μ ϵ μ , in   the   sense   of   measure   in   M ( Ω ¯ ) .
We extend H ϵ to X ( Ω ) as follows:
H ϵ ( u , μ ) = Ω G ( ϵ u ) ϵ q ( x ) d x , if   μ = | u | p ( . ) d x ; 0 , otherwise .
In our investigation, we prove the following functional as the Γ -convergence limit of H ϵ
H ¯ ( u , μ ) = Ω T ( x ) | u | q ( x ) d x + G i I μ i q ( x i ) p ( x i ) ,
where { x i } i I C are the atoms of a limiting measure. Our main result is the following.
Theorem 1.
Let p and q be the exponents satisfying (C1)–(C4). The sequence of functionals { H ϵ } Γ-converges to the functional H ¯ , i.e., the following two statements hold true for every ( u , μ ) X ( Ω ) :
1. 
For every sequence of pairs { ( u ϵ , μ ϵ ) } in X ( Ω ) with ( u ϵ , μ ϵ ) T ( u , μ ) , we have the inequality
lim sup ϵ 0 H ϵ ( u ϵ , μ ϵ ) H ¯ ( u , μ ) ,
2. 
There exists a sequence of pairs { ( u ϵ , μ ϵ ) } in X ( Ω ) with ( u ϵ , μ ϵ ) T ( u , μ ) , which satisfies the following inequality:
lim inf ϵ 0 H ϵ ( u ϵ , μ ϵ ) H ¯ ( u , μ ) .
Remark 2.
In Theorem 1, what we call Γ-convergence is known generally as Γ + -convergence; see [24]. Γ-convergence is a proper variational convergence to study the convergence of maximizers. More explicitly, if { u ϵ } is the sequence of low energy extremals (maximizers) of H ϵ (i.e., H ϵ ( u ϵ ) = G ϵ or more generally, H ϵ ( u ϵ ) = G ϵ ( 1 + o ( 1 ) ) for ϵ 0 + ), u ϵ u in w L q ( . ) ( Ω ) and | u ϵ | p ( . ) d x μ in M ( Ω ¯ ) , then ( u , μ ) is a maximizer of H ¯ . Furthermore,
G ϵ max X ( Ω ) H ¯ .
In fact, we prove that max H ¯ = G .
Theorem 2.
Let { ( u ϵ , μ ϵ ) } X ( Ω ) , be a maximizing sequence for the functional H ϵ with μ ϵ = | u ϵ | p ( . ) d x . Then as ϵ 0 , this sequence concentrates at a single point in the sense that ( u ϵ , μ ϵ ) T ( 0 , δ x ) , where x Ω ¯ . Further, for each x Ω ¯ , there exists a maximizing sequence which concentrates at x.
In general, for Problem (1), we can study many intersecting scenarios as applications. For example, with G ( s ) = 0 , s < 1 ; 1 , s > 1 . we can work on Bernoulli’s free boundary problem as the following volume functional:
G ϵ V = sup { ϵ u 1 } ϵ p ( x ) d x : u W 0 1 , p ( . ) ( Ω ) , u L p ( . ) 1 .
Equivalently, we can write as
G ϵ V = max A ϵ p ( x ) d x : A Ω , cap ( A ) ϵ ,
where cap ( A ) is the harmonic capacity of the subset A with respect to Ω , known as follows:
cap ( A ) = inf u L p ( . ) ( Ω ) : u W 0 1 , p ( . ) ( Ω ) , u 1   a . e .   on   A .
Bernoulli’s free boundary problem naturally arises in electrostatics, fluid dynamics, optimal insulation, and electrochemistry. We intend to work on such problems in the future.

4. Proofs of Theorems

The first task is to determine the possible Γ -convergence limit, in this regard: for the generalized concentration/compactness principle ([18], Theorem 3) plays a vital role, but during our work on establishing the Γ -convergence, we realize that there is a need to refine the generalized concentration/compactness principle and find more sharp bounds. Therefore, we prove a refined version (of [18], Theorem 3) before proving the main result.
Theorem 3.
Let p and q be exponents satisfying (C1)–(C4). Let { v ϵ } be a sequence in W 0 1 , p ( . ) ( Ω ) with v ϵ L p ( . ) ( Ω ) ϵ . Take u ϵ : = v ϵ / ϵ , if the following hold:
  • u ϵ u weakly in W 0 1 , p ( . ) ( Ω ) ,
  • | u ϵ | p ( . ) d x μ in M ( Ω ¯ ) ,
  • ϵ q ( . ) G ( v ϵ ) d x ν in M ( Ω ¯ ) .
Then, for a countable index set I,
μ = | u | p ( . ) d x + μ ¯ + i I μ i δ x i ,   μ ( Ω ¯ ) 1 ,
ν = h   d x + i I ν i δ x i ,   ν ( Ω ¯ ) G ,
where { x i } i I C , μ ¯ is a positive non-atomic measure in M ( Ω ¯ ) and h L 1 ( Ω ) . Moreover, the atomic and regular parts satisfy the following generalized Sobolev-type inequalities:
ν i G μ i q ( x i ) p ( x i ) ,
ν ( Ω ¯ ) G max μ ( Ω ¯ ) q + p , μ ( Ω ¯ ) q p + ,
Ω h d x G max Ω | u | p ( x ) d x + η ¯ ( Ω ¯ ) q + p , Ω | u | p ( x ) d x + η ¯ ( Ω ¯ ) q p + ,
h T | u | q ( . )   a . e .   in   Ω ,
Ω h d x G 0 + S max Ω | u | p ( x ) d x q + p , Ω | u | p ( x ) d x q p + .
Remark 3.
Note that the refinement is done in Inequalities (19) and (22) in comparison to Ref. [18], Theorem 3. Thus we only need to prove these inequalities.
Proof. 
For y Ω ¯ by Lemma 2
ϵ q ( . ) G ( v ϵ ) d x ( B ¯ r ( y ) ) G ϵ max | u ϵ | p ( . ) d x ( B ¯ R ( y ) ) + δ q R + p R , | u ϵ | p ( . ) d x ( B ¯ R ( y ) ) + δ q R p R + ,
By Lemma 1, G ϵ G when ϵ 0 . Hence,
ν ( y ) ν ( B ¯ r ( y ) ) G max μ ( B ¯ R ( y ) ) + δ q R + p R , μ ( B ¯ R ( y ) ) + δ q R p R + ,
Now, p R and p R + converge to p ( y ) , whereas q R and q R + converge to q ( y ) , when R 0 . Hence, taking r 0 , R 0 and δ 0
ν ( y ) G μ ( y ) q ( y ) p ( y ) .
In particular, for atoms { x i } ,
ν i G μ i q ( x i ) p ( x i ) ,
where ν i = ν ( { x i } ) and μ i = μ ( { x i } ) .
Now, for the regular part, there is a subsequence which satisfies
| u ϵ | q ( . ) d x ν = | u | q ( x ) d x + i I ν i δ x i   or   | u ϵ u | q ( x ) d x i i ν i δ x i .
For A Ω and s > 0
A h d x ν ( A ) lim inf ϵ 0 A G ( v ϵ ) ϵ q ( x ) d x , lim sup ϵ 0 A { | u ϵ | < s } G ( ϵ u ϵ ) ϵ q ( x ) | u ϵ | q ( x ) | u ϵ | q ( x ) + c lim sup ϵ 0 A { | u ϵ | s } | u ϵ | q ( x ) d x .
By Egoroff’s theorem, without loss of generality, we assume that lim s 0 s q ( x ) G ( s ) converges uniformly to T on Ω . Thus,
A h d x lim sup ϵ 0 A { | u ϵ | < s } ( T + o ( 1 ) ) | u ϵ | q ( x ) d x + c lim sup ϵ 0 A { | u ϵ | s } | u | q ( x ) d x + A | u ϵ u | q ( x ) d x .
On the other hand,
lim s | { x A : | u ϵ | s } | = 0 ,
therefore,
A h d x A T d ν + c x i A ¯ ν i A T | u | q ( x ) d x + ( G 0 + + c ) x i A ¯ ν i .
By the Radon–Nikodym theorem, we deduce that h T | u | q ( . ) a.e. in Ω .
The above theorem greatly helps us to pinpoint Γ -convergence limit H ¯ and prove the main result.
Proof of lim sup inequality Theorem 1.
Let ( u ϵ , | u ϵ | p ( . ) d x ) τ ( u , μ ) in X, i.e., u ϵ u and | u ϵ | p ( . ) d x μ in measure. Let
H = lim sup ϵ 0 H ϵ ( u ϵ , μ ϵ ) .
Then, there exists a subsequence still called u ϵ such that ϵ q ( . ) G ( ϵ u ϵ ) ν in measure and
lim ϵ 0 H ϵ ( u ϵ , μ ϵ ) = H .
By Theorem 3,
H = Ω d ν = Ω h d x + i ν i .
In addition, (19) and (22) give us
H Ω T | u | q ( . ) d x + G i μ i q ( x i ) p ( x i ) = H ¯ ( u , μ ) .

Proof of lim inf Inequality

In the proof of the lim inf inequality of Theorem 1(2), we discuss two cases: one in which μ has no atoms (Proposition 4) and second in which μ has a finite number of atoms (Propositions 5 and 6).
Proposition 4.
Let ( u , μ = | u | p ( . ) d x + μ ¯ ) X ( Ω ) , then
lim ϵ 0 H ϵ ( u ϵ , μ ϵ ) = H ¯ ,
whenever, ( u ϵ , | u ϵ | p ( . ) d x ) τ ( u , μ ) in X.
Proof. 
Fix x Ω , let us consider a sequence { ( u ϵ , μ ϵ ) } ϵ > 0 from X ( Ω ) , with μ ϵ = | u ϵ | p ( . ) d x and ( u ϵ , | u ϵ | p ( . ) d x ) T ( u , μ ) . Then there is a subsequence | u ϵ | q ( . ) ν in M ( Ω ¯ ) ; furthermore, μ and ν have the same atoms and thus, ν is also atomless and ν = | u | q ( . ) . In particular, Ω | u ϵ | q ( x ) d x Ω | u | q ( x ) d x . In addition, u ϵ is weakly convergent to u in L q ( . ) ( Ω ) . Combining the two facts, we deduce that u ϵ u strongly in L q ( . ) ( Ω ) . Hence, there exists a subsequence { ( u ϵ , μ ϵ ) } ϵ > 0 such that u ϵ u a.e. on Ω and lim ϵ 0 H ϵ ( u ϵ , μ ϵ ) = lim inf ϵ 0 H ϵ ( u ϵ , μ ϵ ) . For a particular element x Ω , if u ( x ) 0 then u ϵ ( x ) 0 , as ϵ 0 , so
ϵ q ( x ) G ( ϵ u ϵ ( x ) ) = ( ϵ | u ϵ ( x ) | ) q ( x ) G ( ϵ u ϵ ( x ) ) | u ϵ ( x ) | q ( x ) ,
and also we have
lim ϵ 0 ( ϵ | u ϵ ( x ) | ) q ( x ) G ( ϵ u ϵ ( x ) ) | u ϵ ( x ) | q ( x ) = T ( x ) | u ( x ) | q ( x ) .
Now we discuss the case when u ( x ) = 0 . By using the growth condition on G ( s ) , we have the inequality
ϵ q ( x ) G ( ϵ u ϵ ) c | u ϵ | q ( x ) .
which implies
ϵ q ( x ) G ( ϵ u ϵ ) 0 = T ( x ) | u ( x ) | q ( x ) .
Finally, using the Lebesgue dominated convergence theorem, we obtain
lim ϵ 0 Ω ϵ q ( x ) G ( ϵ u ϵ ( x ) ) = Ω T ( x ) | u ( x ) | q ( x ) d x ,
lim ϵ 0 H ϵ ( u ϵ , μ ϵ ) = Ω T ( x ) | u ( x ) | q ( x ) d x .
In order to discuss the atomic part, we first prove the lemma below.
Lemma 3.
For every pair ( u , μ ) X ( Ω ) , assume q / p + > 1 , then we have the inequality H ¯ ( u , μ ) G . Additionally, H ¯ ( u , μ ) = G ( u , μ ) = ( 0 , δ x 0 ) , for some x 0 Ω ¯ .
Proof. 
H ¯ ( u , μ ) = Ω T ( x ) | u | q ( x ) d x + G i μ i q ( x i ) p ( x i ) , G 0 + Ω | u | q ( x ) d x + G i μ i q ( x i ) p ( x i ) , G 0 + S max { u L p ( x ) ( Ω ) q + , u L p ( x ) ( Ω ) q } + G i μ i q ( x i ) p ( x i ) ,
since
1 μ ( Ω ¯ ) Ω | u | p ( . ) d x + i μ i ,
thus
max { u L p ( x ) ( Ω ) q + , u L p ( x ) ( Ω ) q } = u L p ( x ) ( Ω ) q Ω | u | p ( . ) d x q / p + .
Hence,
H ¯ ( u , μ ) G Ω | u | p ( . ) d x + G i μ i q ( x i ) p ( x i ) ,
H ¯ ( u , μ ) G Ω | u | p ( . ) d x + G i μ i G ,
and for the case when u = 0 , and μ = δ x 0 for x 0 Ω ¯ , we have the equality H ¯ ( u , μ ) = G . □
Proposition 5.
For every open set Ω 0 Ω , for every x Ω 0 ¯ and for every ( u , μ ) X with ( u , μ ) = ( 0 , δ x ) , there exists the Γ-limit of the sequence { H ϵ } such that its restriction to Ω 0 satisfies
( Γ lim ϵ 0 + H ϵ ) ( 0 , δ x ) = G   in   X ( Ω 0 ) .
Proof. 
Let us fix the open set Ω 0 Ω and also take ϵ s 0 as s . Using the compactness property of Γ -convergence, there exists a subsequence which we denote by { ϵ s } with a functional H : X ( Ω 0 ) R + , and
H ϵ s ( Ω 0 ) Γ H ( Ω 0 ) .
Additionally, as s , Lemma 1 gives
sup X ( Ω 0 ) H ϵ s = G ϵ s ( Ω 0 ) G = max X ( Ω 0 ) H ,
By Lemma 3 for ( u , μ ) ( 0 , δ x 0 ) in Ω 0 we obtain
H ( u , μ ) lim sup ϵ s 0 H ( u ϵ s , μ ϵ s ) H ¯ ( u , μ ) < G ,
Hence, there exists some x 0 Ω 0 ¯ such that H ( 0 , δ x 0 ) = G .
Take a family of spheres B r c Ω 0 , with center c Q n Ω 0 and radius r Q . We can find by diagonalization a subsequence denoted by { ϵ s } , a functional H such that H ϵ s ( B r c ) Γ H ( B r c ¯ ) for each r Q and for each c Q n Ω 0 , and H ϵ s ( Ω 0 ) Γ H ( Ω 0 ¯ ) . Following the similar method as before, for each sphere B r c , we can find x r c B r c such that
H ( 0 , δ x r c ) = G .
Now as r 0 , then x r c c and the fact that Γ -limit is upper semicontinuous on Ω 0 ¯ , we obtain
lim sup r 0 H ( 0 , δ x r c ) H ( 0 , δ q ) ,
and by combining, we obtain
G = lim r 0 H ( 0 , δ x r c ) lim sup r 0 H ( 0 , δ x r c ) H ( 0 , δ c ) ,
which implies
H ( 0 , δ c ) = G .
Again, using the fact that Γ -limit is upper semicontinuous and the density of Q n Ω 0 in Ω 0 ¯ , for each x Ω 0 ¯ with ( u , μ ) = ( 0 , δ x ) , we have
H ( 0 , δ x ) = G .
The above process is independent of the selection of sequence; hence, for each pair ( u , μ ) = ( 0 , δ x ) where x Ω 0 ¯ , there always exists a Γ -limit H and H ( 0 , δ x ) = G . □
Proposition 6.
Let ( u , μ ) = ( 0 , i = 0 N μ i δ x i ) be a pair in X ( Ω ) . Then
H ϵ Γ H ¯ .
Proof. 
We will prove our result for N = 1 , μ = i = 0 1 μ i δ x i , where μ i ( 0 , 1 ) with i = 0 1 μ i 1 . We introduce first Ω i = Ω B r i ( x i ) , where d i s t ( Ω 0 , Ω 1 ) > 0 for i { 0 , 1 } . By Proposition 5, there are sequences ( u ϵ i , μ ϵ i ) in X ( Ω i ) such that ( u ϵ i , μ ϵ i ) T ( 0 , δ x i ) , where μ ϵ i = | u ϵ i | p ( x ) d x and H ¯ ( 0 , δ x i , Ω i ¯ ) = G .
lim inf ϵ 0 + H ϵ ( u ϵ i , μ ϵ i , Ω i ) H ¯ ( 0 , δ x i , Ω i ¯ ) = G ,
which gives
lim ϵ 0 + Ω i ϵ q ( x ) G ( ϵ u ϵ i ) d x = G , i { 0 , 1 } .
Let us set u ϵ = ( μ 0 ) 1 / p ( x 0 ) u ϵ 0 + ( μ 1 ) 1 / p ( x 1 ) u ϵ 1 with μ ϵ = | u ϵ | p ( . ) . Due to the fact that x 0 x 1 and that the supports of u ϵ 0 and u ϵ 1 are disjoint for small ϵ , it is to be noted that the functions u ϵ for every given ϕ C ( Ω ¯ ) satisfy that
Ω ϕ | u | p ( . ) d x = Ω 0 μ 0 p ( x ) / p ( x 0 ) ϕ | u ϵ 0 | p ( . ) d x + Ω 1 μ 1 p ( x ) / p ( x 1 ) ϕ | u ϵ 1 | p ( . ) d x ,
lim ϵ 0 Ω ϕ | u | p ( . ) d x = μ 0 ϕ ( x 0 ) + μ 1 ϕ ( x 1 ) = Ω ϕ d μ ,
which implies specifically that
lim ϵ 0 Ω | u | p ( . ) d x = μ 0 + μ 1 = μ ( Ω ¯ ) < 1 .
Thus, we obtain that the pair ( u ϵ , | u ϵ | p ( . ) d x ) is the part of the set X ( Ω ) for ϵ sufficiently small and further ( u ϵ , | u ϵ | p ( . ) d x ) T ( 0 , μ ) when ϵ 0 . Finally, again using the fact that supports of u ϵ 0 and u ϵ 1 are disjoint, we obtain
H ϵ ( u ϵ , | u ϵ | p ( . ) ) = Ω G ( ϵ u ϵ ) ϵ q ( . ) d x , = Ω 0 μ 0 q ( . ) / p ( x 0 ) G ( ϵ μ 0 1 / p ( x 0 ) u ϵ 0 ) μ 0 q ( . ) / p ( x 0 ) ϵ q ( . ) d x + Ω 1 μ 1 q ( . ) / p ( x 1 ) G ( ϵ μ 1 1 / p ( x 1 ) u ϵ 1 ) μ 1 q ( . ) / p ( x 1 ) ϵ q ( . ) d x , μ 0 q ( x 0 ) p ( x 0 ) G + μ 1 q ( x 1 ) p ( x 1 ) G = H ¯ ( u , μ ) .
The next lemma shows that the proof of the lim inf inequality of Theorem 1(2) for measures of disjoint support is sufficient for the general case.
Lemma 4.
If the lim inf inequality of Theorem 1(2) holds for ( u , μ ) X ( Ω ) which satisfies
  • μ ( Ω ¯ ) < 1 ,
  • μ = | u | p ( . ) d x + μ ¯ + i = 0 n μ i δ x i ,
  • d i s t ( s u p p ( | u | + μ ¯ ) ¯ , i = 0 n { x i } ) > 0 ,
then the lim inf inequality of Theorem 1(2) holds for every ( u , μ ) X ( Ω ) .
Proof. 
Take ( u , μ ) X ( Ω ) such that μ = | u | p ( . ) d x + μ ¯ + i = 0 n μ i δ x i and μ ( Ω ¯ ) < 1 . Our target is to make a sequence which satisfies the following conditions:
  • ( u σ , μ σ ) X ( Ω ) and μ σ = | u σ | p ( . ) d x + μ ¯ σ + i = 0 n μ i δ x i ,
  • μ σ ( Ω ¯ ) < 1 as σ goes to 0,
  • d i s t ( s u p p ( | u σ | + μ ¯ σ ) ¯ , i = 0 n { x i } ) > 0 ,
  • u σ u in L q ( . ) ( Ω ) and ( u σ , μ σ ) T ( u , μ ) ,
  • lim σ 0 H ( u σ , μ σ ) = H ¯ ( u , μ ) .
Define B σ ( x i ) Ω = B σ i for σ > 0 and i { 0 , , N } and choose cut off functions ϕ σ , with 0 ϕ σ 1 , ϕ σ = 1 in Ω \ i = 0 n B 2 σ i , ϕ σ = 0 in i = 0 n B σ i , | ϕ σ | 1 / σ . And further set
( u σ , μ σ ) = u ϕ σ , | u ϕ σ | p ( . ) d x + μ ¯ ϕ σ + i = 0 n μ i δ x i , = u , | u | p ( . ) d x + μ ¯ + i = 0 n μ i δ x i ,
By similar steps as in [6], we have the following deductions: u σ W 0 1 , p ( . ) ( Ω ) and μ σ M ( Ω ¯ ) , d i s t ( s u p p ( | u σ | + μ ¯ σ ) ¯ , i = 0 n { x i } ) σ > 0 .
Ω | u ϕ σ u | q ( . ) d x i = 0 n B 2 σ | u | q ( . ) d x 0 ,
as σ 0 . Hence u σ u in L q ( . ) ( Ω ) . Further for every ψ C 0 ( Ω ¯ ) , we have Ω ¯ ψ d μ σ Ω ¯ ψ d μ which implies μ σ μ as measures in the space M ( Ω ¯ ) when σ goes to 0. Combining them, we have ( u σ , μ σ ) T ( u , μ ) , and as μ ( Ω ¯ ) < 1 , which implies that μ σ satisfies condition (2). In particular ( u σ , μ σ ) X ( Ω ) . The last condition is satisfied by the definition of the Γ -limit and u σ u in L q ( . ) ( Ω ) . Thus, we have
lim inf σ 0 H σ ( u σ , μ σ ) H ¯ ( u σ , μ σ ) ,
when σ > 0 . By the upper semicontinuity of the Γ -limit, we obtain
lim inf σ 0 H σ ( u σ , μ σ ) lim sup σ 0 lim inf σ 0 H σ ( u σ , μ σ ) lim σ 0 H ¯ ( u σ , μ σ ) = H ¯ ( u , μ ) ,
where ( u , μ ) X ( Ω ) and μ ( Ω ¯ ) < 1 , μ = | u | p ( . ) d x + μ ¯ + i = 0 n μ i δ x i .
Now consider a pair ( u , μ ) X ( Ω ) such that μ ( Ω ¯ ) < 1 . Define a sequence ( u m , μ m ) such that
u m = u , μ m = | u | p ( . ) + μ ¯ + i = 0 m μ i δ x i .
m 0 .
Obviously, ( u m , μ m ) X ( Ω ) and ( u m , μ m ) T ( u , μ ) , as m + . Using the above steps for the pair ( u m , μ m ) , we obtain
lim inf m + H m ( u m , μ m ) H ¯ ( u m , μ m ) ,
for m > 0 . Again, applying the upper semicontinuity of the Γ -limit, we have
lim inf m + H m ( u m , μ m ) lim sup m + lim inf m + H m ( u m , μ m ) lim m + H ( u m , μ m ) = H ¯ ( u , μ ) .
Lastly, we study for general pairs ( u , μ ) X ( Ω ) for ρ > 0 , defined by
u ρ = u 1 + ρ ,   μ ρ = μ ( 1 + ρ ) p ( . ) .
which then becomes
μ ρ = | u ρ | p ( . ) + μ ¯ ( 1 + ρ ) p ( . ) + i = 0 + μ i ( 1 + ρ ) p ( . ) δ x i .
It is clear that ( u ρ , μ ρ ) X ( Ω ) and μ ρ ( Ω ) < 1 , ( u ρ , μ ρ ) T ( u , μ ) , as ρ 0 . On the lines of the previous steps, we obtain
lim inf ρ 0 H ρ ( u ρ , μ ρ ) H ¯ ( u ρ , μ ρ ) = Ω T ( x ) ( 1 + ρ ) q ( . ) | u | q ( . ) + G ( 1 + ρ ) q ( . ) i μ i q ( x i ) p ( x i ) .
Again applying the upper semicontinuity of the Γ -limit, we have
lim inf ρ 0 H ρ ( u ρ , μ ρ ) lim sup ρ 0 lim inf ρ 0 H ρ ( u ρ , μ ρ ) lim ρ 0 H ( u ρ , μ ρ ) = H ¯ ( u , μ ) .
Now we are in a position to prove the main result.
Proof of Theorem 1.
To complete the proof, it remains to show the lim inf inequality lim inf ϵ 0 H ϵ ( u ϵ , μ ϵ ) H ¯ for the pair ( u , μ ) X ( Ω ) , satisfying the conditions of Lemma 4, given as follows:
  • μ ( Ω ¯ ) < 1 ,
  • μ = | u | p ( . ) d x + μ ¯ + i = 0 n μ i δ x i ,
  • d i s t ( s u p p ( | u | + u ¯ ) ¯ , i = 0 n { x i } ) > 0 .
Let us denote μ A = | u | p ( . ) d x + μ ¯ and μ B = i = 0 n μ i δ x i = μ μ A . In addition, consider two open sets A and B in Ω , with s u p p ( | u | + μ ¯ ) ¯ A ¯ , s u p p ( μ B ) ¯ B ¯ and A ¯ B ¯ = .
In view of Proposition 4, there exists a sequence of pairs ( u ϵ A , μ ϵ A ) X ( Ω ) , where u ϵ A W 0 1 , p ( . ) ( A ) , μ ϵ A = | u ϵ A | p ( . ) d x with μ ϵ A ( Ω ¯ ) < 1 for small ϵ , and
( u ϵ A , μ ϵ A ) T ( u , μ A ) when ϵ 0 ,
and also,
H ϵ ( u ϵ A , μ ϵ A ) H ¯ ( u , μ A ) when ϵ 0 .
In the same way, using Proposition 6, it follows that there exists a sequence of pairs ( u ϵ B , μ ϵ B ) X ( Ω ) , where u ϵ B W 0 1 , p ( . ) ( B ) , μ ϵ B = | u ϵ B | p ( . ) d x and μ ϵ B ( Ω ¯ ) < 1 for ϵ small enough, and
( u ϵ B , μ ϵ B ) T ( 0 , μ B ) when ϵ 0 ,
and also,
H ϵ ( u ϵ B , μ ϵ B ) H ¯ ( 0 , μ B ) when ϵ 0 .
Set now u ϵ = u ϵ A + u ϵ B with μ ϵ = μ ϵ A + μ ϵ B . Since s u p p ( u ϵ A ) s u p p ( u ϵ B ) = , ( u ϵ , μ ϵ ) X ( Ω ) with u ϵ W 0 1 , p ( . ) , μ ϵ = | u ϵ | p ( . ) = | u ϵ A | p ( . ) + | u ϵ B | p ( . ) and for small ϵ we have
μ ϵ ( Ω ¯ ) = Ω | u ϵ A | p ( . ) d x + Ω | u ϵ B | p ( . ) d x = μ A ( Ω ¯ ) + μ B ( Ω ¯ ) = μ ( Ω ¯ ) < 1 .
By (24) and (25), we have finally
H ϵ ( u ϵ , μ ϵ ) = Ω ϵ q ( x ) G ( ϵ u ϵ ) d x = A ϵ q ( x ) G ( ϵ u ϵ A ) d x + B ϵ q ( x ) G ( ϵ u ϵ B ) d x ,
and taking the limit as ϵ 0 ,
lim ϵ 0 H ϵ ( u ϵ , μ ϵ ) = Ω T ( x ) | u | q ( x ) d x + G i = 0 n μ i q ( x i ) p ( x i ) = H ¯ ( u , μ ) .
Now we give the proof of the second main result, which follows directly from the Γ -convergence result and Lemma 3.
Proof Theorem 2.
We saw in the Γ -convergence result and the Lemma 3 that, if { ( u ϵ , μ ϵ ) } is the maximizing sequence for H ϵ , then { ( u ϵ , μ ϵ ) } converges to ( 0 , δ x ) , where x Ω ¯ and every pair ( 0 , δ x ) is a maximizer for the Γ -limit H ¯ . □

5. Conclusions

We establish the Γ -convergence of low energy extremals for a class of general functions in settings of variable exponent Lebesgue spaces. This work will help to determine low energy solutions. Normally, they form a spike near concentration points in the domain.
In the future, we can work on the geometric shape of these spikes with the help of Robin functions. Another line of research is to study these problems in fractional Sobolev spaces [27,28,29] and fractional-order systems [30,31].

Author Contributions

A.S. (Adil Siddique), A.S. (Andrii Shekhovtsov), Z.B. and W.S. equally contributed to this research work. All authors have read and agreed to the published version of the manuscript.

Funding

The work was supported by the National Science Center, Decision number UMO-2018/29/B/HS4/02725.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Acknowledgments

The authors are grateful to the editor and anonymous reviewers for their valuable suggestions, which helped us to improve this manuscript significantly.

Conflicts of Interest

The authors declare no conflict of interest.

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Siddique, A.; Shekhovtsov, A.; Bashir, Z.; Sałabun, W. Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces. Fractal Fract. 2022, 6, 128. https://doi.org/10.3390/fractalfract6030128

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Siddique A, Shekhovtsov A, Bashir Z, Sałabun W. Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces. Fractal and Fractional. 2022; 6(3):128. https://doi.org/10.3390/fractalfract6030128

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Siddique, Adil, Andrii Shekhovtsov, Zia Bashir, and Wojciech Sałabun. 2022. "Asymptotic Analysis of Low Energy Extremals with Γ-Convergence in Variable Exponent Lebesgue Spaces" Fractal and Fractional 6, no. 3: 128. https://doi.org/10.3390/fractalfract6030128

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