Sensitivity to weather conditions, and specially to clouds, is a severe limiting factor to the use of optical remote sensing for Earth monitoring applications. Typically, the optical satellite images are often corrupted because of poor weather conditions. As a rule, the measure of degradation of optical images is such that one can not rely even on the brightness inside of the damaged regions. As a result, some subdomains of such images become absolutely invisible. So, there is a risk of information loss in optical remote sensing data. In view of this, we propose a new variational approach for exact restoration of multispectral satellite optical images. We discuss the consistency of the proposed variational model, give the scheme for its regularization, derive the corresponding optimality system, and discuss the algorithm for the practical implementation of the reconstruction procedure. Experimental results are very promising and they show a significant g ain o ver b aseline m ethods u sing t he r econstruction through linear interpolation between data available at temporally-close time instants.
is such that we can not rely even on the brightness inside of the damaged regions. As a result, there is a risk of information loss, some subdomains of such images become absolutely invisible. So, there is a great deal of missing information in optical remote sensing data, and a huge gap still exists between the satellite data we acquire and the data we require. Therefore, the reconstruction of missing information in remote sensing data becomes an active research field.
Many solutions have been developed to remove the clouds from multispectral images. (for the technical review, we refer to [2]). Formally, the present traditional algorithms can be primarily classified into four categories: 1) spatial-based methods, without any other auxiliary information source; 2) spectral-based methods, which extract the complementary information from other spectra; 3) multitemporal-based methods, which extract the complementary information from other data acquired at the same position and at diferent time periods; and 4) hybrid methods, which extract the complementary information by a combination of the three previous approaches. In parallel to the traditional approach, data-driven machine learning algorithms are actively developing since 2014 [3]. However, the necessity of large datasets and volatility to errors in input data limits its performance.
Our main efort in this research is to develop a new variational model for the exact restoration of the damaged multi-band optical satellite images that will meet demands from the agro application, that is, it must be applicable for large areas in diferent climate zones, and preserves the crop fields borders within damaged regions. In some sense, this model combines the spacialbased method with the multitemporal one. Therefore, in contrast to the standard variationalbased methods that are often optimased for a specific region or significantly blur textures, (see, for instance, [4] and the references therein), we focus on the global texture reconstruction inside damage regions. With that in mind we assume that the texture of a corrupted image can be predicted through a number of past cloud-free images of the same region from the time series. In order to describe the texture of background surface in the damage region, we follow the paper [5], where the authors experimentally checked the hypothesis that the essential geometric contents of a color image is contained in the level lines of the corresponding total spectral energy of such image.
We also pay much attention to the faithfulness of the reconstruction problem in the framework of the proposed model and supply this approach by the rigorous mathematical substantiation. The experiments undertaken in this study confirmed the eficacy of the proposed method and revealed that it can acquire plausible visual performance and satisfactory quantitative accuracy for agro scenes with rather complicated texture of background surface.
Let Ω ⊂ R2 be a bounded image domain with a Lipschitz boundary Ω . With each particular
image ⃗ = [1(), 2(), . . . , ()] : Ω → R , where each coordinate represents the
intensity of the corresponding spectral channel, we associate the panchromatic image (the
so-called total spectral energy of ⃗)
() = 11() + . . . ().
(
Let ⊂ Ω be a Borel set with non empty interior and suficiently regular boundary and such that |Ω ∖ | > 0. We call the damage region. Let ⃗0 ∈ 2(Ω ∖ ; R ) be an image of interest which is assumed to be corrupted by clouds, and is the zone of missing information.
As it was mentioned before, we deal with the case where we have no information about the original image ⃗0 inside . Instead of this, we assume that the texture of background surface in the damage region can be predicted with some accuracy by a number of past cloud-free images of the same region from the time series of satellite images. Unlike the wellknown ’chronochrome method’ [6] which essentially assumes that the background in is stationary in wide sense, we admit that the image time series follows smooth variations over land (background), the time-series data are strictly chronological, and display regular fluctuations.
Let {⃗−1 , . . . , ⃗− } be a given collection of past cloud-free images of the same region,
where we set ⃗ = ⃗0. We suppose that each cloud-free image of this time series should be well
co-registered with ⃗0 ∈ 2(Ω ∖ ; R ) in Ω ∖ [7]. With each particular image ⃗− in this
series, we associate its total spectral energy − using the standard rule (
Let * be a predicted total spectral energy of ⃗0 in the damage region . This prediction can be done following the regularized regression model and the available information in the time series {, −1 , . . . , − } (for the details we refer to [8]).
ℒ() = 1 ∫︁ [︁ || −1 −
∑︁ −1 − ]︁2 =2 + 1
∫︁
|Ω ∖ | Ω∖
[︁
−
∑−1︁ − ]︁2 + ‖ ‖2R−1
=1
→ inf, (
As a result, setting 0 = Argmin ℒ(), the total spectral energy * in the damage region can be estimated as follows * = in , where ̂︀
−1 ̂︀() = ∑︁ 0− (), =1
In order to reconstruct the texture (or geometry) of ⃗0 in the damage region , we assume that predicted total energy * is a function of bounded variation, i.e. * ∈ (), and all spectral channels of the damaged image should share the geometry of the panchromatic image * ∈ 2() in . Hence, at most all points of almost all level sets of * ∈ () we can define a normal vector (), i.e., it formally satisfies (, * ) = |∇* | and | | ≤ 1 a.e. in .
say that a function ⃗ = [1, 2, . . . , ] : Ω In view of the risk of cloud distortion, the problem is to reconstruct the original multi-band image ⃗0 in the damage region using the knowledge of its texture (geometry) on the subset together with the exact information about this image in Ω ∖ (the undamaged region). We → R is the result of restoration of a cloud corrupted image ⃗0 : → R if for given regularization parameters >0 , > 1, and >0, = 1, 2, each spectral component is the solution of the following constrained minimization problem with the nonstandard growth energy functional () (, ) :=
Ω () |∇()|() + ∫︁ ∫︁ 1 ⃒ Ω∖ ⃒⃒ ∇ * ( − 0,)⃒⃒ + 2 Ω∖ |() − 0,()| Cauchy law () = 1+(1/)2 with an appropriate > 0; and :[0, ∞) → (0, ∞) is the edge-stopping function which we take it in the form of the • ∈ ∞(, R2) is a given vector field such that
| ()| ≤ 1 and ( (), ∇* ())R2 = |∇* ()| a.e. in ; • ( * ) () determines the convolution of function with the two-dimensional Gaussian iflter kernel , where the parameter >0 determines the spatial size of the image details which are removed by this 2D filter;
By default we assume that the functions 0, and * are extended by zero outside of domains Ω ∖ and , respectively.
The proposed model (
2 2‖ ‖1(Ω−Ω) 1|Ω| 23 (︃ ∫︁
⃒ Ω∖ ⃒⃒ ∇ * ( − 0,)⃒⃒ the third term is also fidelity term which forces the texture (or topological map) of minimizer in domain Ω ∖ to stay close to the texture of a given spectral energy ⃗0,.
As for the last term in (
R2
= 0, = 1, . . . , in .
Therefore, we impose them in the energy functional in the form of the last term. to compute (1, 2) by first regularizing by the equation
In practice, at the discrete level, can be defined by the relation (1, 2) = |∇∇̂̂︀︀((11,,22))| when ∇̂︀(1, 2) ̸= 0, and (1, 2) = 0 when ∇̂︀(1, 2) = 0. However, a better choice would be = div ︂( ̂︀ ∇ )︂ |∇| in (0, ∞) × Ω, coupled with the initial and Neumann boundary conditions Then, for any ≥ [9, 10] for the details) (0, 1, 2) = ̂︀(1, 2), for a.a. (1, 2) ∈ Ω, = 0 on Ω. 0, there is a vector field () ∈ ∞(Ω)
with ‖ ()‖∞(Ω) ≤ 1 such that (see and ( (), ∇()) = |∇()| in Ω,
( (), ) = 0 on Ω, = div ( ()) in the sense of distributions in (0, ∞) × Ω.
As a result, in order to characterize the texture of the cloud contaminated image ⃗0 in the damage region , we may take = () for some small value of > 0. As was mentioned in [9], following this way, we do not not distort the geometry of in an essential way.
So, the novelty of the model that we propose, is that the edge information for the multispectral restoration in Ω is accumulated in the variable exponent () which we derive from ̂︀ the time series and initial data.
In this section we show that constrained minimization problem (
We begin with some auxiliary results which will play a crucial role in the sequel.
Lemma 4.1. Let {}∈N be a sequence of measurable non-negative functions : Ω → [ 0, ∞) such that {}∈N are uniformly bounded in 1(Ω) and () → () almost everywhere in Ω for some ∈ 1(Ω). Let { = 1 + (|(∇ * )|)}∈N be the corresponding sequence of variable exponents. Then (·) → (·) = 1 + (|(∇ * ) (·)|) := 1 + ≤ () ≤ := 2,
|∇ ( − )| () ≤ ‖ ‖1(Ω−Ω) ‖‖1(Ω), () = 1 + ∫︁ ,
Then 1-boundedness of {}∈N guarantees the existence of a positive value ∈ (
Moreover, as follows from the estimate
⃒⃒ |(∇ * ) ()|2 − |(∇ * ) ()|2 |() − ()| ≤ 2 ⃒⃒⃒⃒ (︁ 2 + |(∇ * ) ()|2︁) (︁ ⃒⃒⃒ 2 + |(∇ * ) ()|2︁) ⃒⃒⃒ Ω and smoothness of the function ∇ (·) , there exists a positive constant > 0 independent of such that
|∇ ( − ) − ∇ ( − )| , ∀ , ∈ Ω |() − ()| ≤
||(∇ * ) ()| − |(∇ * ) ()||
non-negative value ≥ 0 such that =
sequence to the problem (
inf Since max∈Ω |()| ≤ and each element of the sequence {}∈N has the same modulus of continuity, it follows that this sequence is uniformly bounded and equi-continuous. Hence, by Arzelà–Ascoli Theorem the sequence {}∈N is relatively compact with respect to the strong topology of (Ω) . Taking into account that the set S is closed with respect to the uniform convergence and () → () almost everywhere in Ω , we deduce: (·) Ω as →
∞, where () = 1 + (|(∇ * ) ()|) in Ω. The proof is complete.
Following in many aspects the resent studies [11, 12], we give the following existence result.
0, ∈ 2(Ω ∖ ), the minimization problem (
}︃ ⃒ ⃒ 1 < ≤ ℎ(·) ≤ in Ω.
(
sup () ≤ 2. ‖‖1(Ω) ≤ 1|Ω|,
∀ ∈ N.
:= 1 + ≤ () ≤ := 2,
|()| ≤
1 ≤ 1 |Ω|,
∀ ∈ N,
|∇()|() ≤ 2
Ω () |∇()|() < 2( + 1),
∀ ∈ N,
(
Moreover, passing to a subsequence if necessa r−y, we ha− ve (see Proposition A.3 and Lemma 4.1): () → () a.e. in Ω. ⇀ weakly in (·) (Ω),
weakly in (·) (Ω; ∇ ⇀ ∇
R2), (·) → (·) = 1 + (|(∇ * ) (·)|) uniformly in Ω as → ∞, where ∈ 1,(·) (Ω) with () = 1 + (|(∇ * ) ()|) in Ω.
Since 0 ≤ () ≤ 1 a.a. in Ω for all ∈ N, it follows from (9) that the limit function
is a feasible solution to minimization problem
is also subjected the same restriction. Thus,
(
Let us show that (, ) is a minimizer of this problem. With that in mind we note that in view of the obvious inequality |() − 0,()| ≤ 2−1 (|()| + |0,()| ) and the fact that 0, ∈ 2(Ω ∖ ), we have: the sequence {() − 0,()}∈N is bounded in (Ω ∖ ) and converges weakly in (Ω ∖ ) to − 0,. Hence, by Proposition A.3 (see (32)), − 0, ∈ (·) (Ω ∖ ) and lim inf →∞ ∫︁ Ω∖ ∫︁
Ω∖ |() − 0,()| ≥
() − 0,()| .
|
(10)
As for the rest terms in (
Ω ⃒ lim inf →∞
Ω∖ ⃒⃒ ∇ * ( − 0,)⃒⃒ = lim inf →∞ Ω∖ ⃒⃒ ∇ * ( − 0,)⃒⃒ , As a result, utilizing relations (10), (11), (12), and the lower semicontinuity property (32), we =
inf (, ) = lim (, ) = lim inf (, ) ≥ ( , ).
) is a minimizer to the problem (
Thus, (,
open question. ℎ()| ≤ | − |, ∀ , ∈ Ω, ℎ() ≤ := 2,
It is clear that because of the nonstandard energy functional and its non-convexity, constrained
minimization problem (
∈ℬ0(·) 0 = Argmin (, 0(·)). = Argmin (, (·)).
(13) ∈ℬ(·) ︁( ⃒ (︁
⃒ ︂{ Then, for each ≥
1, we set () = 1 + ⃒ ∇ * −1 )︁ ()⃒ , ⃒ ⃒ )︁ Here, ℬ(·) = {︀ ∈ 1,(·) (Ω) : 1≤
Before proceeding further, we set
0 ≤ () ≤ 1 a.a. in Ω }︀ .
where > 0 and > 0 are defined by (
Arguing as in the proof of Theorem 1 and using the convexity arguments, it can be shown
that, for each (·)
Argmin∈ℬ(·)
∈ S, there exists a unique element
0,(·)
∈ ℬ(·) such that
0,(·)
(, (·)) . Moreover, it can be shown that, for given = 1, . . . , , >
compact with respect to the strong topology of (Ω).
1>0, 2 > 0, * ∈ 1, (), and ⃗0 ∈ 2(Ω ∖ , R ), the sequence {︀
∈ 1,(·) (Ω) }︀
is compact with respect to the weak topology of 1, (Ω) , whereas the exponents {}∈N are
∈N
=
0,
We say that a pair (̂︀, ) is a weak solution to the original problem (
̂︀ ∈ℬ̂︀(·) ̂︀ = Argmin (, ̂︀(·)), ̂︀ ∈ ℬ̂︀(·) , () = 1 + (|(∇ * ̂︀) ()|) , ∀ ∈ Ω.
̂︀ the asymptotic properties: Our main result can be stated as follows:
Then, for each ∈ {1, . . . , }, the sequence of approximated solutions {︀ (, ) ∈N
︀}
possesses
Let >0, 1>0, 2>0, * ∈ (), and ⃗0∈2(Ω∖,
R ) be given data.
where (̃︀, ̃︀) is a weak solution to the original problem (
() → ̃︀() a.e. in Ω, ⇀ ̃︀ in (Ω),
and ∇ ⇀ ∇̃︀ in (Ω; R2), → ̃︀ = F(()) strongly in (Ω) as → ∞,
̃︀ ̃︀ ∈ ℬ̃︀(·) , ̃︀ = Argmin (, ̃︀(·)), and, in addition, the following variational property holds true
[︃ →∞ lim (, (·)) = lim →∞ ∈ℬ(·) inf (, (·)) = inf
Proof. Let’s assume the converse — namely, there is a function ∙ ∈ ℬ̃︀(·) such that inf
(, ̃︀(·)) < (̃︀, ̃︀(·)).
Using the procedure of the direct smoothing, we set where > 0 is a small parameter, is a positive compactly supported smooth function with properties ∈ 0∞(R2),
() = 1, and () = (− ), and ̃︁∙ is zero extension of ∙ outside of Ω. by the classical properties of smoothing operators (see [17]). From this we deduce that () → ∙ () a.e. in Ω.
Moreover, taking into account the estimates →∞ ∫︁ ∫︁ ⃒ 1 ̃︀ 1 ∫︁ ∫︁
R2 ∈ ()−1 (−Ω) (16) (17) (18) (19) (21) we see that each element is subjected to the pointwise constraints 0 ≤ () ≤ 1 a.a. in Ω, ∀ > 0. ⟨ inf∈ℬ(·)
(, (·)) . Hence, Since, for each > 0, ∈ 1,(·) (Ω) for all ∈
N, it follows that ∈ ℬ(·) , i.e., each element of the sequence {}>0 is a feasible solution to all approximating problems (, (·)) ≤ (, (·)), ∀ > 0, ∀ = 0, 1, . . .
Further we notice that by Proposition A.3 and Fatou’s lemma, and →∞ lim inf (, (·)) ≥ (̃︀, ̃︀(·)) →∞ lim (, (·)) = lim Ω () |∇()|() +
Ω∖ ⃒⃒ ∇ * ( − 0,)⃒⃒ + 2 Ω∖ ∫︁ ⃒ (︁ ⃒ ⃒ ⊥, ∇ ⃒⃒ () |∇()|() → () |∇()|̃︀() it follows from the Lebesgue dominated convergence theorem and (20) that →∞ lim (, (·)) = (, ̃︀(·)), ∀ > 0.
As a result, passing to the limit in (18) and utilizing properties (19)–(21), we obtain (̃︀, ̃︀(·)) ≤ (, ̃︀(·)) = ∫︁
1 Ω ̃︀() |∇()|̃︀ () + ∫︁ Ω∖ |() − 0,()| Ω∖ ⃒
⃒ 0,)⃒ + 2 ⃒ ⊥, ∇ ⃒ , for all > 0. Taking into account the pointwise convergence (see (17) and property (16)) 0,(·) ()() as → 0, and the fact that, for small enough, |∇()|̃︀ [︁ 2 (1 + |∙ (·)|)
+ 2 (1 + |0,(·)|) 2 ‖1(Ω−Ω) || Ω 2 02 = const, ∈ 1(Ω) a.e. in Ω
∖ , we can pass to the limit in (22) as → 0 by the Lebesgue dominated convergence theorem. This (̃︀, ̃︀(·)) ≤ →0 lim (, ̃︀(·)) = (∙ , ̃︀(·)).
Combining this inequality with (22) and (15), we finally get
that leads us into conflict with the initial assumption. Thus,
(̃︀, ̃︀(·)) =
inf
and, therefore, (, ) is a weak solution to the original problem (
|∇()|̃︀
To characterize the solution 0,(·) ⟨ lim →0 ⟩ + ∫︁ Ω∖ ⃒ ⃒⃒ 0,(·) () − = ∫︁ (︁
Ω 0,()⃒ ⃒
of the approximating optimization problem inf∈ℬ(·)
(, (·)) , we check that the functional (·) is Gâteaux diferentiable, that is, (0,(·) + , (·)) − (0,(·) , (·)) |∇0,(·) ()|()−2 ︁) for all ∈ 1,(·) (Ω), where ∫︁ Ω ⃒ ⊥, ∇0,(·) )︁ ⃒⃒ −1 ︁( ⊥, ∇ ︁) , Λ() = (∇ ( − ), ∇ ( − )) 0,(·) () 0,() ︁) Ω∖ () .
To this end, we note that |∇0,(·) () + ∇()|() − |∇
0,(·) ()|() () almost everywhere in Ω. Since, by convexity, it follows that ⃒ ⃒ ⃒ ⃒ ⃒ |∇0,(·) () + ∇()|() − |∇
0,(·) ()|() ⃒⃒ () → |∇0,(·) ()| ()−2 ∇0,(·) (), ∇() as →
0 ︁) | | − | | ≤ 2 (︀ | | 1 − + | | −1 )︀ | −
|, ︁( ︁( ≤ ≤ ≤ by the Lebesgue dominated convergence theorem.
Taking into account that ‖0,(·) ()| ()−1
‖′(·) (Ω) ∫︁ |∇0,(·) () + ∇()| ()−1 + |∇0,(·) ()|
()−1 )︁ ≤ const |∇0,(·) ()| ()−1 + |∇()| |∇()|.
(25) by (33) (︂ ∫︁ by (??) (︁ by (33) Ω ‖0,(·) ()|
() ‖0,(·) 2 |(·) (Ω) + 2 + 1
1 ︁) ′ , Ω | ()|() by (??) ≤ ‖
2 ‖(·) (Ω) function. Therefore, |∇0,(·) () + ∇()|() − |∇ 0,(·) ()|()
() → ∫︁ ︁( + 1, we see that the right hand side of inequality (25) is |∇0,(·) ()| ()−2 ︁) inf ∈ℬ(·)
Utilizing the similar arguments to the rest terms in (
Thus, in order to derive some optimality conditions for the minimizing element 0,(·) ∈ ℬ(·) to the problem
(, (·)) , we note that ℬ(·) is a nonempty convex subset of 1,(·) (Ω) and the objective functional (·, (·)) : ℬ(·) → R is strictly convex. Hence, the well known classical result (see [18, Theorem 1.1.3]) and representation (24) lead us to the following conclusion.
Theorem 6.1. Let (·) ∈ S be an exponent given by the iterative rule (13). Then the unique minimizer ∈ ℬ(·) to the approximating problem inf∈ℬ(·) (, (·)) is characterized by ∫︁ (︂ ⃒ Ω ⃒⃒ ∇()⃒⃒ ⃒ ()−2 ∇(), ∇() − ∇ () + 2 1 ︂) ∫︁ Ω Λ() (︁ () − () ︁) +
∫︁ + 2 Ω∖ ⃒
In order to illustrate the proposed algorithm for the restoration of satellite multi-spectral images we have provided some numerical experiments. As input data we have used a series of Sentinel-2 L2A images over the Dnipro Airport area, Ukraine (see Fig. 1, 2). This region represents a typical agricultural area with medium sides fields of various shapes. As a final result, we obtain in Fig. 3. Comparing the restored image and the contaminated one we could see that the texture of original image is well preserved. However, overall colors of different fields are shifted due to colorization part of algorithm. This problem has to be addressed in the following research.
We propose a novel model for the restoration of satellite multi-spectral images. This model is based on the solutions of special variational problems with nonstandard growth objective functional. Because of the risk of information loss in optical images (see [19] for the details), we do not impose any information about such images inside the damage region, but instead we assume that the texture of these images can be predicted through a number of past cloud-free images of the same region from the time series. So, the characteristic feature of variational problems, which we formulate for each spectral channel separately, is the structure of their objective functionals. On the one hand, we involve into consideration the energy functionals with the nonstandard growth (), where the variable exponent () is unknown a priori and it directly depends on the texture of an image that we are going to restore. On the other hand, the texture of an image ⃗, we are going to restore, can have rather rich structure in the damage region . In order to identify it, we push forward the following hypothesis: the geometry of each spectral channels of a cloud corrupted image in the damage region is topologically close to the geometry of the total spectral energy that can be predicted with some accuracy by a number of past cloud-free images of the same region. As a result, we impose this requirement in each objective functional in the form of a special fidelity term. In order to study the consistency of the proposed collection of non-convex minimization problems, we develop a special technique and supply this approach by the rigorous mathematical substantiation.
Let (·) be a measurable exponent function on Ω such that 1 ≤ ≤ () ≤ < where and are given constants. Let ′(·) = (·)−1 be the corresponding conjugate exponent. measurable functions () on Ω such that ∫︀ Ω | ()|() < ∞. Then (·) (Ω) is a reflexive where ′ and ′ stand for the conjugates of constant exponents. Denote by (·) (Ω) the set of all separable Banach space with respect to the Luxemburg norm (see [20] for the details) 1 ≤ − 1 ≤ ′() ≤ − 1 ‖ ‖(·) (Ω) = inf {︀ > 0 : ( −1 ) ≤ 1︀} , (26) ⏟ ′⏞ ∫︁
Ω ∫︁ ∫︁ Ω Ω ∫︁ Ω ∫︁ ‖ ‖(·) (Ω) − 1 ≤ | ()|() ≤ ‖ ‖(·) (Ω) + 1,
∀ ∈ (·) (Ω), ‖ ‖(·) (Ω) =
| ()|() , if ‖ ‖(·) (Ω) = 1. 1 ∫︁ * Ω | ()|() ≤ 1 ∫︁ * Ω
| ()|() , ∫︁ ⃒⃒ () ⃒⃒ () Ω ⃒⃒ * ⃒ ⃒ ≤
* 1 ∫︁ Ω 1 ∫︁
Ω | ()|() ≤ 1 ≤
| ()|() .
| ()|() ≤ ‖ ‖(·) (Ω), if ‖ ‖(·) (Ω) > 1, | ()|() ≤ ‖ ‖(·) (Ω), if ‖ ‖(·) (Ω) < 1, where ( ) := ∫︀Ω | ()|() .
It is well-known that (·) (Ω) is reflexive provided > 1, and its dual is ′(·) (Ω) , that is, any continuous functional = ( ) on (·) (Ω) has the form (see [21, Lemma 13.2]) ( ) = ,
with ∈ ′(·) (Ω).
if * := ‖ ‖(·) (Ω) > 0, then ( −*1 ) = 1.
As for the infimum in (26), we have the following result.
Proposition A.1. The infimum in (26) is attained if ( ) > 0. Moreover
Taking this result and condition 1 ≤ ≤ () ≤ into account, we see that Hence, (see [20] for the details) and, therefore, ‖ ‖(·) (Ω) ≤ ‖ ‖(·) (Ω) ≤ The following estimates are well-known: if ∈ (·) (Ω) then
‖ ‖ (Ω) ≤ (1 + |Ω|) 1/ ‖ ‖(·) (Ω), ‖ ‖(·) (Ω) ≤ (1 + |Ω|) 1/ ′ ‖ ‖ (Ω), ′ = ∀ ∈ (Ω).
− 1 , in this subsection we assume that
Let {}∈N ⊂
0, (Ω) , with some ∈ (0, 1], be a given sequence of exponents. Hereinafter 1 ≤ ≤ () ≤ <
∞ a.e. in Ω for = 1, 2, . . . , and (·) → (·) in (Ω) as → ∞. (·) (Ω). We say that the sequence {︀ ∈ (·) (Ω) }︀ We associate with this sequence the following collection {︀ ∈ (·) (Ω) }︀ tic feature of this set of functions is that each element lives in the corresponding Orlicz space ∈N. The characteris∈N is bounded if (see [22, Section 6.2]) (27) (28) (29)
A bounded sequence {︀ ∈ (·) (Ω) }︀ ∈N is weakly convergent in the {}∈N ⊂ 0, (Ω) in the uniform topology of (Ω), if variable Orlicz space (·) (Ω) to a function ∈ (·) (Ω), where ∈ 0, (Ω) is the limit of ∫︁ lim
For our further analysis, we need some auxiliary results (we refer to [21, Lemma 13.3] for comparison). We begin with the lower semicontinuity property of the variable (·) -norm with respect to the weak convergence in (·) (Ω).
Proposition A.3. If a bounded sequence {︀ ∈ (·) (Ω) }︀ to for some > 1, then ∈ (·) (Ω), ⇀ in variable (·) (Ω), and
∈N converges weakly in (Ω) lim inf →∞ ∫︁ Ω ∫︁ Proof. In view of the fact that ⃒ ⃒ ⃒ ∫︁ ⃒ Ω |()|() − lim inf →∞ ∫︁ |()|() = lim inf →∞
Ω () |()|() .
∫︁ lim Using the Young inequality ≤ | |/ + ||′/′, we have for ′() = ()/(() − 1) and any ∈ 0∞(R2).
Then passing to the limit in (30) as → ∞ and utilizing property (27) and the fact that ()() = ()() for all ∈ ′(Ω), (31) Since the last inequality is valid for all it follows that this relation holds true for ∈ 0∞(R2) and the set 0∞(R2) is dense in ′(·) (Ω) , ∈ ′(·) (Ω) . So, taking = | ()|()−2 (), we arrive at the announced inequality (29). As an consequence of (29) and estimate (??), we get:
To end of the proof, it remains to observe that relation (31) holds true for well. From this the weak convergence ⇀ in the variable Orlicz space (·) (Ω) ∈ 0∞(R2) as follows.
Remark A.4. Arguing in a similar manner and using, instead of (30), the estimate ∫︁
1 lim inf →∞ Ω () |()|() ≥ ∫︁ Ω () ()() − ∫︁
1 Ω ′() |( )|′() , it is easy to show that the lower semicontinuity property (29) can be generalized as follows ∫︁
1 lim inf →∞ Ω () |()|() ≥ ∫︁
1 Ω () | ()|() .
We need the following result that leads to the analog of the Hölder inequality in Lebesgue spaces with variable exponents (for the details we refer to [20]).
Proposition A.6. If ∈ (·) (Ω) and ∈ ′(·) (Ω) , then (, ) ∈ 1(Ω) and ∫︁ Ω
(, ) ≤ 2‖ ‖(·) (Ω) ‖‖′(·) (Ω) . |()|() ≥ () ()() −
Appendix B. Sobolev Spaces with Variable Exponent We recall here the well-known facts concerning the Sobolev spaces with variable exponent. Let (·) be a measurable exponent function on Ω such that 1 < ≤ () ≤ < where and are given constants. We associate with it the so-called Sobolev-Orlicz space ∈ 1,1(Ω) : ︂} ∫︁ [︁|()|() + |∇()|()]︁ < +∞ Ω (32) (33) and equip it with the norm ‖‖01,(·) (Ω) = ‖‖(·) (Ω) + ‖∇‖(·) (Ω;R2).
It is well-known that, in general, unlike classical Sobolev spaces, smooth functions are not necessarily dense in = 01,(·) (Ω) . Hence, with the given variable exponent = () (1 < ≤ ≤ ) it can be associated another Sobolev space,
= 1,(·) (Ω) as the closure of the set ∞(Ω) in 1,(·) (Ω)-norm.
Since the identity = is not always valid, it makes sense to say that an exponent () is regular if ∞(Ω) is dense in 1,(·) (Ω).
The following result reveals an important property ensuring the regularity of exponent ().
Proposition B.1. Assume that there exists ∈ (0, 1] such that ∈ 0, (Ω). Then the set ∞(Ω) is dense in 1,(·) (Ω), and, therefore, = .
Proof. Let ∈ 0, (Ω) be a given exponent. Since lim || log(||) = 0 →0 with an arbitrary ∈ (0, 1], (34) it follows from the Hölder continuity of (·) that |() − ()| ≤ | − | ≤ [︃
| − | sup ,∈Ω log(| − |−1 ) ]︃ (| − |), ∀ , ∈ Ω, where () = / log(||−1 ), and > 0 is some positive constant.
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