On Variational Problem with Nonstandard Growth Conditions for the Restoration of Clouds Corrupted Satellite Images Pavel Khanenko1,2, Peter I. Kogut3,4 and Mykola Uvarov1,5 Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany 1 EOS Data Analytics Ukraine, Desyatynny lane, 5, 01001 Kyiv, Ukraine 2 3 Department of Differential Equations, Oles Honchar Dnipro National University, Gagarin av., 72, 49010 Dnipro, Ukraine 4 EOS Data Analytics Ukraine, Gagarin av., 103a, Dnipro, Ukraine 5 Department of Computational Physics, G. V. Kurdyumov Institute for Metal Physics, Kyiv, Ukraine Abstract Sensitivity to weather conditions, and specially to clouds, is a severe limiting factor to the use of opti- cal 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 algo- rithm for the practical implementation of the reconstruction procedure. Experimental results are very promising and they show a significant g ain over b aseline m ethods u sing t he reconstruction through linear interpolation between data available at temporally-close time instants. Keywords Risk of cloud distortion of satellite images, Risk of information loss, Image restoration, Variational ap- proach 1. Introduction A very serious obstacle to utilization of optical remote sensing satellite images is a risk of cloud and cloud shadow distortion issue (referred to as cloud contamination, hereafter). It has been reported in [1] that over 50% of all the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument aboard the Terra and Aqua satellites are covered by clouds or cloud-contaminated globally. Moreover, it is a typical situation when the measure of degradation of optical images 2𝑛𝑑 International Workshop on Computational & Information Technologies for Risk-Informed Systems, CITRisk–2021, September 16-17, 2021, Kherson, Ukraine pavel.khanenko@eosda.com (P. Khanenko); peter.kogut@eosda.com (P. I. Kogut); nikolay.uvarov@eosda.com (M. Uvarov) { https://eos.com (P. Khanenko); https://kogut.uaua.us (P. I. Kogut); https://eos.com (M. Uvarov)  0000-0003-1593-0510 (P. I. Kogut) Β© 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) 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 different 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 effort 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 different climate zones, and preserves the crop fields borders within damaged regions. In some sense, this model combines the spacial- based method with the multitemporal one. Therefore, in contrast to the standard variational- based 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 efficacy 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. 2. Preliminaries Let Ω βŠ‚ R2 be a bounded image domain with a Lipschitz boundary πœ•β„¦. With each particular image ⃗𝑒 = [𝑒1 (π‘₯), 𝑒2 (π‘₯), . . . , 𝑒𝑀 (π‘₯)]𝑑 : Ω β†’ R𝑀 , where each coordinate represents the intensity of the corresponding spectral channel, we associate the panchromatic image 𝑒 (the so-called total spectral energy of ⃗𝑒) 𝑒(π‘₯) = 𝛼1 𝑒1 (π‘₯) + . . . 𝛼𝑀 𝑒𝑀 (π‘₯). (1) Here, 𝛼1 , . . . , 𝛼𝑀 are some weight coefficients. Let 𝐷 βŠ‚ Ω be a Borel set with non empty interior and sufficiently regular boundary and such that |Ω βˆ– 𝐷| > 0. We call 𝐷 the damage region. Let ⃗𝑒0 ∈ 𝐿2 (Ω βˆ– 𝐷; R𝑀 ) be an image of interest which is assumed to be corrupted by clouds, and 𝐷 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 ⃗𝑒0 inside 𝐷. Instead of this, we assume that the texture of background surface in the damage region 𝐷 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 well- known ’chronochrome method’ [6] which essentially assumes that the background in 𝐷 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 {𝑒⃗ π‘‘βˆ’1 , . . . , βƒ—π‘’π‘‘βˆ’π‘› } be a given collection of past cloud-free images of the same region, where we set ⃗𝑒𝑑 = ⃗𝑒0 . We suppose that each cloud-free image of this time series should be well co-registered with ⃗𝑒0 ∈ 𝐿2 (Ω βˆ– 𝐷; R𝑀 ) in Ω βˆ– 𝐷 [7]. With each particular image βƒ—π‘’π‘‘βˆ’π‘˜ in this series, we associate its total spectral energy π‘’π‘‘βˆ’π‘˜ using the standard rule (1). So, each element of the time series {𝑒𝑑 , π‘’π‘‘βˆ’1 , . . . , π‘’π‘‘βˆ’π‘› } is well-defined in Ω. Let 𝑒* be a predicted total spectral energy of ⃗𝑒0 in the damage region 𝐷. This prediction can be done following the regularized regression model and the available information in the time series {𝑒𝑑 , π‘’π‘‘βˆ’1 , . . . , π‘’π‘‘βˆ’π‘› } (for the details we refer to [8]). ∫︁ [︁ 𝑛 𝛽1 βˆ‘οΈ ]︁2 β„’(𝑀) = π‘’π‘‘βˆ’1 βˆ’ π‘€π‘˜βˆ’1 π‘’π‘‘βˆ’π‘˜ 𝑑π‘₯ |𝐷| 𝐷 π‘˜=2 ∫︁ π‘›βˆ’1 𝛽1 [︁ βˆ‘οΈ ]︁2 + 𝑒𝑑 βˆ’ π‘€π‘˜ π‘’π‘‘βˆ’π‘˜ 𝑑π‘₯ + πœ†β€–π‘€β€–2Rπ‘›βˆ’1 β†’ inf, (2) |Ω βˆ– 𝐷| Ξ©βˆ–π· π‘˜=1 where πœ† > 0 is the regularization parameter, 𝛽1 > 0 and 𝛽2 > 0 are the parameters that control the importance of the prediction and estimation terms, respectively. Seeing that the prediction and estimation errors in (2) are normalized by the volume of samples contributing to each term, we can constrain the values of 𝛽1 = 𝛽 ∈ [0, 1] and 𝛽2 = 1 βˆ’ 𝛽 to control their relevance with a single parameter 𝛽. As a result, setting 𝑀0 = Argmin β„’(𝑀), the total spectral energy 𝑒* in the damage region 𝐷 can be estimated as follows 𝑒* = 𝑒 Μ‚οΈ€ in 𝐷, where π‘›βˆ’1 βˆ‘οΈ 𝑒(π‘₯) Μ‚οΈ€ = π‘€π‘˜0 π‘’π‘‘βˆ’π‘˜ (π‘₯), βˆ€ π‘₯ ∈ Ω. π‘˜=1 In order to reconstruct the texture (or geometry) of ⃗𝑒0 in the damage region 𝐷, we assume that predicted total energy 𝑒* is a function of bounded variation, i.e. 𝑒* ∈ 𝐡𝑉 (𝐷), and all spectral channels of the damaged image should share the geometry of the panchromatic image 𝑒* ∈ 𝐿2 (𝐷) in 𝐷. Hence, at most all points of almost all level sets of 𝑒* βˆˆπ΅π‘‰ (𝐷) we can define a normal vector πœƒ(π‘₯), i.e., it formally satisfies (πœƒ, 𝑒* ) = |βˆ‡π‘’* | and |πœƒ| ≀ 1 a.e. in 𝐷. 3. Problem Statement In view of the risk of cloud distortion, the problem is to reconstruct the original multi-band image ⃗𝑒0 in the damage region 𝐷 using the knowledge of its texture (geometry) on the subset 𝐷 together with the exact information about this image in Ω βˆ– 𝐷 (the undamaged region). We say that a function ⃗𝑒 = [𝑒1 , 𝑒2 , . . . , 𝑒𝑀 ]𝑑 : Ω β†’ R𝑀 is the result of restoration of a cloud corrupted image ⃗𝑒0 : 𝐷 β†’ R𝑀 if for given regularization parameters πœ‡>0, 𝛼>1, and πœ†π‘— >0, 𝑗 = 1, 2, each spectral component 𝑒𝑖 is the solution of the following constrained minimization problem with the nonstandard growth energy functional ∫︁ ∫︁ 1 πœ‡ (𝒫𝑖 ) 𝐽𝑖 (𝑣, 𝑝) := |βˆ‡π‘£(π‘₯)| 𝑝(π‘₯) 𝑑π‘₯ + |𝑣(π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)|𝛼 𝑑π‘₯ Ξ© 𝑝(π‘₯) 𝛼 Ξ©βˆ–π· ∫︁ βƒ’ βƒ’2 ∫︁ βƒ’ (︁ )︁ ⃒𝛼 βƒ’ βŠ₯ + πœ†1 βˆ‡πΊ * (𝑣 βˆ’ 𝑒 ) 𝑑π‘₯ + πœ† πœƒ , βˆ‡π‘£ βƒ’ 𝑑π‘₯ βˆ’β†’ inf , (3) βƒ’ βƒ’ βƒ’ βƒ’ 𝜎 0,𝑖 βƒ’ 2 βƒ’ Ξ©βˆ–π· 𝐷 (𝑣,𝑝)∈Ξ where β€’ Ξ stands for the set of feasible solutions to the problem (3) which we define as follows βƒ’ 𝑣 ∈ π‘Š 1,𝑝(Β·) (Ω), 𝑝 ∈ 𝐢(Ω), ⎧ βƒ’ ⎫ βŽͺ ⎨ βƒ’ βŽͺ ⎬ Ξ = (𝑣, 𝑝) βƒ’ 1 ≀ 𝛾0 ≀ 𝑣(π‘₯) ≀ 𝛾1 a.a. in Ω, βƒ’ βŽͺ βƒ’ βŽͺ ⎩ βƒ’ 𝑝(π‘₯) = F(𝑣(π‘₯)) in Ω. ⎭ Here, π‘Š 1,𝑝(Β·) (Ω) is the Sobolev-Orlicz space, F(𝑣(π‘₯)) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑣) (π‘₯)|) , and 𝑔:[0, ∞) β†’ (0, ∞) is the edge-stopping function which we take it in the form of the Cauchy law 𝑔(𝑑) = 1+(𝑑/π‘Ž) 1 2 with an appropriate π‘Ž > 0; β€’ πœƒ ∈ 𝐿∞ (𝐷, R2 ) is a given vector field such that |πœƒ(π‘₯)| ≀ 1 and (πœƒ(π‘₯), βˆ‡π‘’* (π‘₯))R2 = |βˆ‡π‘’* (π‘₯)| a.e. in 𝐷; β€’ (𝐺𝜎 * 𝑣) (π‘₯) determines the convolution of function 𝑣 with the two-dimensional Gaussian filter kernel 𝐺𝜎 , where the parameter 𝜎>0 determines the spatial size of the image details which are removed by this 2D filter; By default we assume that the functions 𝑒0,𝑖 and 𝑒* are extended by zero outside of domains Ω βˆ– 𝐷 and 𝐷, respectively. The proposed model (3) provides a completely new approach to restoration of non-smooth multi-spectral images ⃗𝑒0 with the gap in damage region. The main characteristic feature of this model is that we involve into consideration the energy functional with the nonstandard growth condition. The variable character of the exponent 𝑝(π‘₯) provides more flexibility in terms of regularity for the recovered images. Since the first term in (3) is the regularization and the second one is the so-called data fidelity, it is worth to emphasize the role of the rest terms in (3). Taking into account the fact that the magnitude F(𝑣(π‘₯)) is close to one at those points, where the spectral energy 𝑣 is slowly varying, and this value is close to zero at the edges of 𝑣, it follows that the edge information in the non-damage zone for the reconstruction is derived from the given image ⃗𝑒0 . So, in view of the estimate βƒ’ βƒ’ βƒ’ 2 2 βƒ’ |(βˆ‡πΊ * βˆ’ |(βˆ‡πΊ * ∫︁ ∫︁ 2 βƒ’ 𝜎 𝑣)| 𝜎 𝑒 0,𝑖 )| βƒ’ |𝑝(π‘₯) βˆ’ F(𝑒0,𝑖 )| 𝑑π‘₯ = π‘Ž βƒ’ βƒ’ (︁ 2 )︁ (︁ )︁ βƒ’βƒ’ 𝑑π‘₯ Ξ©βˆ–π· Ξ©βˆ–π· βƒ’ π‘Ž + |(βˆ‡πΊπœŽ * 𝑣)|2 π‘Ž2 + |(βˆ‡πΊπœŽ * 𝑒0,𝑖 )|2 βƒ’ ∫︁ 1 (︁ )︁⃒ βƒ’ ≀ 2 |(βˆ‡πΊπœŽ * 𝑣)| + |(βˆ‡πΊπœŽ * 𝑒0,𝑖 )| βƒ’ |(βˆ‡πΊπœŽ * 𝑣)| βˆ’ |(βˆ‡πΊπœŽ * 𝑒0,𝑖 )| βƒ’ 𝑑π‘₯ βƒ’ βƒ’ π‘Ž Ξ©βˆ–π· 3 (οΈƒ )οΈƒ 1 2β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) 𝛾1 |Ω| 2 ∫︁ βƒ’ βƒ’2 2 ≀ βˆ‡πΊ * (𝑣 βˆ’ 𝑒 ) 𝑑π‘₯ , βƒ’ βƒ’ 𝜎 0,𝑖 π‘Ž2 βƒ’ βƒ’ Ξ©βˆ–π· the third term is also fidelity term which forces the texture (or topological map) of minimizer 𝑒 in domain Ω βˆ– 𝐷 to stay close to the texture of a given spectral energy ⃗𝑒0,𝑖 . As for the last term in (3), we notice that since πœƒ ∈ 𝐿∞ (𝐷, R2 ) is a vector field with indicated properties, it follows that πœƒ(π‘₯) has the direction of the normal to the level lines of 𝑒* . Therefore, the counterclockwise rotation of angle πœ‹/2, denoted by πœƒβŠ₯ , represents the tangent vector to the level lines of 𝑒* . In this case, if the spectral channels 𝑒𝑖 share the geometry of the panchromatic image 𝑒* , we have (︁ )︁ βŠ₯ πœƒ , βˆ‡π‘’π‘– 2 = 0, 𝑖 = 1, . . . , 𝑀 in 𝐷. R Therefore, we impose them in the energy functional 𝐽𝑖 in the form of the last term. βˆ‡Μ‚οΈ€ 𝑒(π‘₯1 ,π‘₯2 ) In practice, at the discrete level, πœƒ can be defined by the relation πœƒ(π‘₯1 , π‘₯2 ) = |βˆ‡Μ‚οΈ€ 𝑒(π‘₯1 ,π‘₯2 )| when 𝑒(π‘₯1 , π‘₯2 ) ΜΈ= 0, and πœƒ(π‘₯1 , π‘₯2 ) = 0 when βˆ‡Μ‚οΈ€ βˆ‡Μ‚οΈ€ 𝑒(π‘₯1 , π‘₯2 ) = 0. However, a better choice would be to compute πœƒ(π‘₯1 , π‘₯2 ) by first regularizing 𝑒 Μ‚οΈ€ by the equation (οΈ‚ )οΈ‚ πœ•π‘£ βˆ‡π‘£ = div in (0, ∞) Γ— Ω, πœ•π‘‘ |βˆ‡π‘£| coupled with the initial and Neumann boundary conditions πœ•π‘£ 𝑣(0, π‘₯1 , π‘₯2 ) = 𝑒(π‘₯ Μ‚οΈ€ 1 , π‘₯2 ), for a.a. (π‘₯1 , π‘₯2 ) ∈ Ω, = 0 on πœ•β„¦. πœ•πœˆ Then, for any 𝑑 β‰₯ 0, there is a vector field πœ‰(𝑑) ∈ 𝐿∞ (Ω) with β€–πœ‰(𝑑)β€–πΏβˆž (Ξ©) ≀ 1 such that (see [9, 10] for the details) (πœ‰(𝑑), βˆ‡π‘£(𝑑)) = |βˆ‡π‘£(𝑑)| in Ω, (πœ‰(𝑑), 𝜈) = 0 on πœ•β„¦, πœ•π‘£ and = div (πœ‰(𝑑)) in the sense of distributions in (0, ∞) Γ— Ω. πœ•π‘‘ As a result, in order to characterize the texture of the cloud contaminated image ⃗𝑒0 in the damage region 𝐷, we may take πœƒ = πœ‰(𝑑) for some small value of 𝑑 > 0. As was mentioned in [9], following this way, we do not not distort the geometry of 𝑒 Μ‚οΈ€ in an essential way. So, the novelty of the model that we propose, is that the edge information for the multi- spectral restoration in Ω is accumulated in the variable exponent 𝑝(π‘₯) which we derive from the time series and initial data. 4. Existence Result In this section we show that constrained minimization problem (3) is consistent and admits at least one solution (π‘’π‘Ÿπ‘’π‘ 𝑖 , 𝑝𝑖 ) ∈ Ξ, where 𝑝𝑖 (π‘₯) = F(𝑒𝑖 (π‘₯)) in Ω. We note that because of π‘Ÿπ‘’π‘ π‘Ÿπ‘’π‘ π‘Ÿπ‘’π‘ the specific form of the energy functional 𝐽𝑖 (𝑣, 𝑝) in (3), the standard approaches are no longer applicable in its study, especially with respect to the existence of minimizers and their basic properties. It makes the minimization problem (3) rather challenging. We begin with some auxiliary results which will play a crucial role in the sequel. Lemma 4.1. Let {π‘£π‘˜ }π‘˜βˆˆN be a sequence of measurable non-negative functions π‘£π‘˜ : Ω β†’ [𝛾0 , ∞) such that {π‘£π‘˜ }π‘˜βˆˆN are uniformly bounded in 𝐿1 (Ω) and π‘£π‘˜ (π‘₯) β†’ 𝑣(π‘₯) almost every- where in Ω for some 𝑣 ∈ 𝐿1 (Ω). Let {π‘π‘˜ = 1 + 𝑔 (|(βˆ‡πΊπœŽ * π‘£π‘˜ )|)}π‘˜βˆˆN be the corresponding sequence of variable exponents. Then π‘π‘˜ (Β·) β†’ 𝑝(Β·) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑣) (Β·)|) uniformly in Ω as π‘˜ β†’ ∞, 𝛼 := 1 + 𝛿 ≀ π‘π‘˜ (π‘₯) ≀ 𝛽 := 2, βˆ€ π‘₯ ∈ Ω, βˆ€ π‘˜ ∈ N, (4) where π‘Ž2 𝛿= . π‘Ž2 + β€–πΊπœŽ β€–2𝐢 1 (Ξ©βˆ’Ξ©) supπ‘˜βˆˆN β€–π‘£π‘˜ β€–2𝐿1 (Ξ©) Proof. Since {π‘£π‘˜ }π‘˜βˆˆN is the bounded sequence in 𝐿1 (Ω), by smoothness of the Gaussian filter kernel 𝐺𝜎 , it follows that ∫︁ |(βˆ‡πΊπœŽ * π‘£π‘˜ ) (π‘₯)| ≀ |βˆ‡πΊπœŽ (π‘₯ βˆ’ 𝑦)| π‘£π‘˜ (𝑦) 𝑑𝑦 ≀ β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) β€–π‘£π‘˜ ‖𝐿1 (Ξ©) , Ξ© π‘Ž2 π‘π‘˜ (π‘₯) = 1 + π‘Ž2 + (|(βˆ‡πΊπœŽ * π‘£π‘˜ ) (π‘₯)|)2 π‘Ž2 β‰₯1+ 2 , βˆ€ π‘₯ ∈ Ω. π‘Ž + β€–πΊπœŽ β€–2𝐢 1 (Ξ©βˆ’Ξ©) β€–π‘£π‘˜ β€–2𝐿1 (Ξ©) Then 𝐿1 -boundedness of {π‘£π‘˜ }π‘˜βˆˆN guarantees the existence of a positive value 𝛿 ∈ (0, 1) such that estimate (4) holds true for all π‘˜ ∈ N. Moreover, as follows from the estimate βƒ’ βƒ’ βƒ’ 2 2 βƒ’ |(βˆ‡πΊπœŽ * π‘£π‘˜ ) (π‘₯)| βˆ’ |(βˆ‡πΊπœŽ * π‘£π‘˜ ) (𝑦)| |π‘π‘˜ (π‘₯) βˆ’ π‘π‘˜ (𝑦)| ≀ π‘Ž2 βƒ’βƒ’ (︁ βƒ’ βƒ’ )︁ (︁ )︁ βƒ’βƒ’ βƒ’ π‘Ž2 + |(βˆ‡πΊπœŽ * π‘£π‘˜ ) (π‘₯)|2 π‘Ž2 + |(βˆ‡πΊπœŽ * π‘£π‘˜ ) (𝑦)|2 βƒ’ 2β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) β€–π‘£π‘˜ ‖𝐿1 (Ξ©) ≀ ||(βˆ‡πΊπœŽ * π‘£π‘˜ ) (π‘₯)| βˆ’ |(βˆ‡πΊπœŽ * π‘£π‘˜ ) (𝑦)|| π‘Ž2 2β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) 𝛾12 |Ω| ∫︁ ≀ |βˆ‡πΊπœŽ (π‘₯ βˆ’ 𝑧) βˆ’ βˆ‡πΊπœŽ (𝑦 βˆ’ 𝑧)| 𝑑𝑧, βˆ€ π‘₯, 𝑦 ∈ Ω π‘Ž2 Ξ© and smoothness of the function βˆ‡πΊπœŽ (Β·), there exists a positive constant 𝐢𝐺 > 0 independent of π‘˜ such that 2β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) 𝛾12 |Ω|𝐢𝐺 |π‘π‘˜ (π‘₯) βˆ’ π‘π‘˜ (𝑦)| ≀ |π‘₯ βˆ’ 𝑦|, βˆ€ π‘₯, 𝑦 ∈ Ω. π‘Ž2 Setting 2β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) 𝛾12 |Ω|𝐢𝐺 𝐢 := , (5) π‘Ž2 we see that {οΈƒ βƒ’ }οΈƒ βƒ’ |β„Ž(π‘₯) βˆ’ β„Ž(𝑦)| ≀ 𝐢|π‘₯ βˆ’ 𝑦|, βˆ€ π‘₯, 𝑦, ∈ Ω, {π‘π‘˜ (Β·)} βŠ‚ S = β„Ž ∈ 𝐢 0,1 (Ω) βƒ’ βƒ’ βƒ’ 1 < 𝛼 ≀ β„Ž(Β·) ≀ 𝛽 in Ω. Since maxπ‘₯∈Ω |π‘π‘˜ (π‘₯)| ≀ 𝛽 and each element of the sequence {π‘π‘˜ }π‘˜βˆˆN has the same modulus of continuity, it follows that this sequence is uniformly bounded and equi-continuous. Hence, by Arzelà–Ascoli Theorem the sequence {π‘π‘˜ }π‘˜βˆˆN is relatively compact with respect to the strong topology of 𝐢(Ω). Taking into account that the set S is closed with respect to the uniform convergence and π‘£π‘˜ (π‘₯) β†’ 𝑣(π‘₯) almost everywhere in Ω, we deduce: π‘π‘˜ (Β·) β†’ 𝑝(Β·) uniformly in Ω as π‘˜ β†’ ∞, where 𝑝(π‘₯) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑣) (π‘₯)|) in Ω. The proof is complete. Following in many aspects the resent studies [11, 12], we give the following existence result. Theorem 4.2. For each 𝑖 = 1, . . . , 𝑀 and given πœ‡>0, πœ†1 >0, πœ†2 >0, πœƒ ∈ 𝐿∞ (𝐷, R2 ), and 𝑒0,𝑖 ∈ 𝐿2 (Ω βˆ– 𝐷), the minimization problem (3) admits at least one solution (π‘’π‘Ÿπ‘’π‘ π‘Ÿπ‘’π‘ 𝑖 , 𝑝𝑖 ) ∈ Ξ. Proof. Since Ξ ΜΈ= βˆ… and 0 ≀ 𝐽𝑖 (𝑣, 𝑝) < +∞ for all (𝑣, 𝑝) ∈ Ξ, it follows that there exists a non-negative value 𝜁 β‰₯ 0 such that 𝜁 = inf 𝐽𝑖 (𝑣, 𝑝). Let {(π‘£π‘˜ , π‘π‘˜ )}π‘˜βˆˆN βŠ‚ Ξ be a minimizing (𝑣,𝑝)∈Ξ sequence to the problem (3), i.e. (π‘£π‘˜ , π‘π‘˜ ) ∈ Ξ, π‘π‘˜ (π‘₯) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * π‘£π‘˜ ) (π‘₯)|) in Ω βˆ€ π‘˜ ∈ N, and lim 𝐽𝑖 (π‘£π‘˜ , π‘π‘˜ ) = 𝜁. π‘˜β†’βˆž So, without lost of generality, we can suppose that 𝐽𝑖 (π‘£π‘˜ , π‘π‘˜ ) ≀ 𝜁 + 1 for all π‘˜ ∈ N. From this and the initial assumptions, we deduce ∫︁ ∫︁ |π‘£π‘˜ (π‘₯)|𝛼 𝑑π‘₯ ≀ 𝛾1𝛼 𝑑π‘₯ ≀ 𝛾1𝛼 |Ω|, βˆ€ π‘˜ ∈ N, Ξ© Ξ© ∫︁ ∫︁ 1 |βˆ‡π‘£π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ ≀ 2 |βˆ‡π‘£π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ < 2(𝜁 + 1), βˆ€ π‘˜ ∈ N, (6) Ξ© 𝑝 Ξ© π‘˜ (π‘₯) where [οΈ‚ ]οΈ‚ sup sup π‘π‘˜ (π‘₯) ≀ 2. π‘˜βˆˆN π‘₯∈Ω Utilizing the fact that π‘£π‘˜ (π‘₯) ≀ 𝛾1 for almost all π‘₯ ∈ Ω, we infer the following estimate β€–π‘£π‘˜ ‖𝐿1 (Ξ©) ≀ 𝛾1 |Ω|, βˆ€ π‘˜ ∈ N. Then arguing as in Lemma 4.1 it can be shown that π‘π‘˜ ∈ 𝐢 0,1 (Ω) and 𝛼 := 1 + 𝛿 ≀ π‘π‘˜ (π‘₯) ≀ 𝛽 := 2, βˆ€ π‘₯ ∈ Ω, βˆ€ π‘˜ ∈ N, (7) π‘Ž2 with 𝛿= . (8) π‘Ž2 + β€–πΊπœŽ β€–2𝐢 1 (Ξ©βˆ’Ξ©) 𝛾12 |Ω|2 Taking this fact into account, we deduce from (6), (7), and (??) that (οΈ‚βˆ«οΈ )οΈ‚1/𝛼 𝛼 𝛼 β€–π‘£π‘˜ β€–π‘Š 1,𝛼 (Ξ©) = [|π‘£π‘˜ (π‘₯)| + |βˆ‡π‘£π‘˜ (π‘₯)| ] 𝑑π‘₯ Ξ© (οΈ‚βˆ«οΈ [︁ ]︁ )οΈ‚1/𝛼 1/𝛼 π‘π‘˜ (π‘₯) π‘π‘˜ (π‘₯) ≀ (1 + |Ω|) |π‘£π‘˜ (π‘₯)| + |βˆ‡π‘£π‘˜ (π‘₯)| 𝑑π‘₯ + 2 Ξ© )οΈ€]οΈ€1/𝛼 ≀ (1 + |Ω|) 𝛾12 |Ω| + 2𝜁 + 4 [οΈ€ (οΈ€ uniformly with respect to π‘˜ ∈ N. Therefore, there exists a subsequence of {π‘£π‘˜ }π‘˜βˆˆN , still denoted by the same index, and a function π‘’π‘Ÿπ‘’π‘ 𝑖 ∈ π‘Š 1,𝛼 (Ω) such that π‘£π‘˜ β†’ π‘’π‘Ÿπ‘’π‘ 𝑖 strongly in πΏπ‘ž (Ω) for all π‘ž ∈ [1, 𝛼* ), π‘£π‘˜ ⇀ π‘’π‘Ÿπ‘’π‘ 𝑖 weakly in π‘Š 1,𝛼 (Ω) as π‘˜ β†’ ∞, where, by Sobolev embedding theorem, 𝛼* = 2βˆ’π›Ό 2𝛼 = 2+2𝛿 1βˆ’π›Ώ > 2. Moreover, passing to a subsequence if necessary, we have (see Proposition A.3 and Lemma 4.1): 𝑖 (π‘₯) a.e. in Ω. π‘£π‘˜ (π‘₯) β†’ π‘’π‘Ÿπ‘’π‘ (9) π‘£π‘˜ ⇀ π‘’π‘Ÿπ‘’π‘ 𝑖 weakly in 𝐿 π‘π‘˜ (Β·) (Ω), βˆ‡π‘£π‘˜ ⇀ βˆ‡π‘’π‘Ÿπ‘’π‘ 𝑖 weakly in πΏπ‘π‘˜ (Β·) (Ω; R2 ), π‘π‘˜ (Β·) β†’ π‘π‘Ÿπ‘’π‘ π‘Ÿπ‘’π‘ 𝑖 (Β·) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑒𝑖 ) (Β·)|) uniformly in Ω as π‘˜ β†’ ∞, π‘Ÿπ‘’π‘ where π‘’π‘Ÿπ‘’π‘ 𝑖 ∈ π‘Š 1,𝑝 (Β·) (Ω) with π‘π‘Ÿπ‘’π‘ (π‘₯) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * π‘’π‘Ÿπ‘’π‘ 𝑖 ) (π‘₯)|) in Ω. Since 𝛾0 ≀ π‘£π‘˜ (π‘₯) ≀ 𝛾1 a.a. in Ω for all π‘˜ ∈ N, it follows from (9) that the limit function π‘’π‘Ÿπ‘’π‘ 𝑖 is also subjected the same restriction. Thus, π‘’π‘Ÿπ‘’π‘ 𝑖 is a feasible solution to minimization problem (3). Let us show that (π‘’π‘Ÿπ‘’π‘ 𝑖 , 𝑝𝑖 ) is a minimizer of this problem. With that in mind we note that π‘Ÿπ‘’π‘ in view of the obvious inequality |π‘£π‘˜ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)|𝛼 ≀ 2π›Όβˆ’1 (|π‘£π‘˜ (π‘₯)|𝛼 + |𝑒0,𝑖 (π‘₯)|𝛼 ) and the fact that 𝑒0,𝑖 ∈ 𝐿2 (Ω βˆ– 𝐷), we have: the sequence {π‘£π‘˜ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)}π‘˜βˆˆN is bounded in 𝐿𝛼 (Ω βˆ– 𝐷) and converges weakly in 𝐿𝛼 (Ω βˆ– 𝐷) to π‘’π‘Ÿπ‘’π‘ 𝑖 βˆ’ 𝑒0,𝑖 . Hence, by Proposition A.3 (see π‘Ÿπ‘’π‘ (Β·) (32)), 𝑒𝑖 βˆ’ 𝑒0,𝑖 ∈ 𝐿 π‘Ÿπ‘’π‘ 𝑝 (Ω βˆ– 𝐷) and ∫︁ ∫︁ lim inf 𝛼 |π‘£π‘˜ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)| 𝑑π‘₯ β‰₯ |π‘’π‘Ÿπ‘’π‘ 𝛼 𝑖 (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)| 𝑑π‘₯. (10) π‘˜β†’βˆž Ξ©βˆ–π· Ξ©βˆ–π· As for the rest terms in (3), in view of the strong convergence π‘£π‘˜ β†’ π‘’π‘Ÿπ‘’π‘π‘– in πΏπ‘ž (Ω) with π‘ž > 2, we have ∫︁ βƒ’ βƒ’2 ∫︁ (οΈ‚βˆ«οΈ )οΈ‚2 π‘Ÿπ‘’π‘ βƒ’ π‘Ÿπ‘’π‘ βƒ’βˆ‡πΊπœŽ * (π‘£π‘˜ βˆ’ 𝑒𝑖 )βƒ’ 𝑑π‘₯ ≀ |βˆ‡πΊπœŽ (π‘₯ βˆ’ 𝑦)| |π‘£π‘˜ (𝑦) βˆ’ 𝑒𝑖 (𝑦)| 𝑑𝑦 𝑑π‘₯ βƒ’ Ξ© Ξ© Ξ© ∫︁ (οΈ‚βˆ«οΈ )οΈ‚2βˆ’ 2 π‘ž π‘ž ≀ |βˆ‡πΊπœŽ (π‘₯ βˆ’ 𝑦)| π‘žβˆ’1 𝑑𝑦 𝑑π‘₯ β€–π‘£π‘˜ βˆ’ π‘’π‘Ÿπ‘’π‘ 2 𝑖 β€–πΏπ‘ž (Ξ©) Ξ© Ξ© 3βˆ’ 2 𝑖 β€–πΏπ‘ž (Ξ©) β†’ 0 as π‘˜ β†’ ∞. ≀ β€–πΊπœŽ ‖𝐢 1 (Ξ©βˆ’Ξ©) |Ω| π‘ž β€–π‘£π‘˜ βˆ’ π‘’π‘Ÿπ‘’π‘ 2 Hence, ∫︁ βƒ’ βƒ’2 ∫︁ βƒ’ βƒ’2 lim inf βƒ’βˆ‡πΊπœŽ * (π‘£π‘˜ βˆ’ 𝑒0,𝑖 )βƒ’ 𝑑π‘₯ = π‘Ÿπ‘’π‘ βƒ’βˆ‡πΊπœŽ * (𝑒𝑖 βˆ’ 𝑒0,𝑖 )βƒ’ 𝑑π‘₯, (11) βƒ’ βƒ’ βƒ’ βƒ’ π‘˜β†’βˆž Ξ©βˆ–π· Ξ©βˆ–π· ∫︁ βƒ’ (︁ )︁ ⃒𝛼 ∫︁ βƒ’ (︁ )︁ ⃒𝛼 βƒ’ βŠ₯ βƒ’ βŠ₯ lim inf βƒ’ πœƒ , βˆ‡π‘£π‘˜ βƒ’ 𝑑π‘₯ β‰₯ βƒ’ πœƒ , βˆ‡π‘’π‘Ÿπ‘’π‘ βƒ’ 𝑑π‘₯ 𝑑π‘₯, (12) βƒ’ βƒ’ 𝑖 π‘˜β†’βˆž 𝐷 𝐷 As a result, utilizing relations (10), (11), (12), and the lower semicontinuity property (32), we finally obtain 𝜁 = inf 𝐽𝑖 (𝑣, 𝑝) = lim 𝐽𝑖 (π‘£π‘˜ , π‘π‘˜ ) = lim inf 𝐽𝑖 (π‘£π‘˜ , π‘π‘˜ ) β‰₯ 𝐽𝑖 (π‘’π‘Ÿπ‘’π‘ π‘Ÿπ‘’π‘ 𝑖 , 𝑝𝑖 ). (𝑣,𝑝)∈Ξ π‘˜β†’βˆž π‘˜β†’βˆž Thus, (π‘’π‘Ÿπ‘’π‘ 𝑖 , 𝑝𝑖 ) is a minimizer to the problem (3), whereas its uniqueness remains as an π‘Ÿπ‘’π‘ open question. 5. On Relaxation of the Restoration Problem It is clear that because of the nonstandard energy functional and its non-convexity, constrained minimization problem (3) is not trivial in its practical implementation. The main difficulty in its study comes from the state constraints 1 ≀ 𝛾0 ≀ 𝑣(π‘₯) ≀ 𝛾1 a.a. in Ω, 𝑝(π‘₯) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑒) (π‘₯)|) that we impose on the set of feasible solutions Ξ. This motivates us to pass to some relaxation. In view of this, we propose the following iteration procedure which is based on the concept of relaxation of extremal problems and their variational convergence [13, 14, 15, 16]. At the first step we set up 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑒0,𝑖 ) (π‘₯)|) , if π‘₯βˆˆβ„¦ βˆ– 𝐷, {οΈ‚ }οΈ‚ 𝑝0 (π‘₯) = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑒* ) (π‘₯)|) , if π‘₯∈𝐷, 𝑒0 = Argmin 𝐽𝑖 (𝑣, 𝑝0 (Β·)). π‘£βˆˆβ„¬π‘0 (Β·) Then, for each π‘˜ β‰₯ 1, we set (︁⃒(︁ )︁ βƒ’)︁ π‘π‘˜ (π‘₯) = 1 + 𝑔 βƒ’ βˆ‡πΊπœŽ * 𝑒 π‘˜βˆ’1 (π‘₯)βƒ’ , βˆ€ π‘₯ ∈ Ω, π‘’π‘˜ = Argmin 𝐽𝑖 (𝑣, π‘π‘˜ (Β·)). (13) βƒ’ βƒ’ π‘£βˆˆβ„¬π‘π‘˜ (Β·) Here, ℬ𝑝(Β·) = 𝑣 ∈ π‘Š 1,𝑝(Β·) (Ω) : 1≀𝛾0 ≀ 𝑣(π‘₯) ≀ 𝛾1 a.a. in Ω . {οΈ€ }οΈ€ Before proceeding further, we set {οΈ‚ βƒ’ }οΈ‚ βƒ’ |β„Ž(π‘₯) βˆ’ β„Ž(𝑦)| ≀ 𝐢|π‘₯ βˆ’ 𝑦|, βˆ€ π‘₯, 𝑦 ∈ Ω, S = β„Ž ∈ 𝐢(Ω) βƒ’ βƒ’ 𝛼 := 1 + 𝛿 ≀ β„Ž(π‘₯) ≀ 𝛽 := 2, βˆ€ π‘₯ ∈ Ω, where 𝐢 > 0 and 𝛿 > 0 are defined by (5) and (8), respectively. Arguing as in the proof of Theorem 1 and using the convexity arguments, it can be shown 0,𝑝(Β·) 0,𝑝(Β·) that, for each 𝑝(Β·) ∈ S, there exists a unique element 𝑒𝑖 ∈ ℬ𝑝(Β·) such that 𝑒𝑖 = Argminπ‘£βˆˆβ„¬π‘(Β·) 𝐽𝑖 (𝑣, 𝑝(Β·)). Moreover, it can be shown that, for given 𝑖 = 1, . . . , 𝑀 , πœ‡ > 0, πœ†1 >0, πœ†2 > 0, 𝑒* ∈ π‘Š 1,𝛼 (𝐷), and ⃗𝑒0 ∈ 𝐿2 (β„¦βˆ–π·, R𝑀 ), the sequence π‘’π‘˜ ∈ π‘Š 1,π‘π‘˜ (Β·) (Ω) π‘˜βˆˆN {οΈ€ }οΈ€ is compact with respect to the weak topology of π‘Š 1,𝛼 (Ω), whereas the exponents {π‘π‘˜ }π‘˜βˆˆN are compact with respect to the strong topology of 𝐢(Ω). We say that a pair (Μ‚οΈ€ Μ‚οΈ€ is a weak solution to the original problem (3) if 𝑒𝑖 , 𝑝) 𝑒 ̂︀𝑖 = Argmin 𝐽𝑖 (𝑣, 𝑝(Β·)), Μ‚οΈ€ ̂︀𝑖 ∈ ℬ𝑝(Β·) 𝑒 Μ‚οΈ€ , 𝑝(π‘₯) Μ‚οΈ€ = 1 + 𝑔 (|(βˆ‡πΊπœŽ * 𝑒 ̂︀𝑖 ) (π‘₯)|) , βˆ€ π‘₯ ∈ Ω. π‘£βˆˆβ„¬π‘(Β·) Μ‚οΈ€ Our main result can be stated as follows: Theorem 5.2. Let πœ‡>0, πœ†1 >0, πœ†2 >0, 𝑒* βˆˆπ΅π‘‰ (𝐷), and ⃗𝑒0 ∈𝐿2 (β„¦βˆ–π·, 𝑀 {οΈ€ π‘˜R ) }οΈ€be given data. Then, for each 𝑖 ∈ {1, . . . , 𝑀 }, the sequence of approximated solutions (𝑒 , π‘π‘˜ ) π‘˜βˆˆN possesses the asymptotic properties: π‘’π‘˜ (π‘₯) β†’ 𝑒(π‘₯) ΜƒοΈ€ a.e. in Ω, π‘’π‘˜ ⇀ 𝑒 ΜƒοΈ€ in 𝐿𝛼 (Ω), and βˆ‡π‘’π‘˜ ⇀ βˆ‡ΜƒοΈ€ 𝑒 in 𝐿𝛼 (Ω; R2 ), π‘π‘˜ β†’ 𝑝̃︀ = F(ΜƒοΈ€ 𝑒(π‘₯)) strongly in 𝐢(Ω) as π‘˜ β†’ ∞, where (ΜƒοΈ€ ΜƒοΈ€ is a weak solution to the original problem (3), that is, 𝑒, 𝑝) ΜƒοΈ€ ∈ ℬ𝑝(Β·) 𝑒 ΜƒοΈ€ , 𝑒 ΜƒοΈ€ = Argmin 𝐽𝑖 (𝑣, 𝑝(Β·)), ΜƒοΈ€ π‘£βˆˆβ„¬π‘(Β·) ΜƒοΈ€ and, in addition, the following variational property holds true [οΈƒ ]οΈƒ lim 𝐽𝑖 (π‘’π‘˜ , π‘π‘˜ (Β·)) = lim inf 𝐽𝑖 (𝑣, π‘π‘˜ (Β·)) = inf 𝐽𝑖 (𝑣, 𝑝(Β·)) ΜƒοΈ€ = 𝐽𝑖 (ΜƒοΈ€ 𝑒, 𝑝(Β·)). ΜƒοΈ€ (14) π‘˜β†’βˆž π‘˜β†’βˆž π‘£βˆˆβ„¬π‘π‘˜ (Β·) π‘£βˆˆβ„¬π‘(Β·) ΜƒοΈ€ Proof. Let’s assume the converse β€” namely, there is a function π‘’βˆ™ ∈ ℬ𝑝(Β·) ΜƒοΈ€ such that 𝐽𝑖 (π‘’βˆ™ , 𝑝(Β·)) ΜƒοΈ€ = inf 𝐽𝑖 (𝑣, 𝑝(Β·)) ΜƒοΈ€ < 𝐽𝑖 (ΜƒοΈ€ 𝑒, 𝑝(Β·)). ΜƒοΈ€ (15) π‘£βˆˆβ„¬π‘(Β·) ΜƒοΈ€ Using the procedure of the direct smoothing, we set (οΈ‚ )οΈ‚ π‘₯ βˆ’ 𝑧 ΜƒοΈβˆ™ ∫︁ 1 π‘’πœ€ (π‘₯) = 2 𝐾 𝑒 (𝑧) 𝑑𝑧, πœ€ R2 𝜁(πœ€) where πœ€ > 0 is a small parameter, 𝐾 is a positive compactly supported smooth function with properties ∫︁ 𝐾 ∈ 𝐢0∞ (R2 ), 𝐾(π‘₯) 𝑑π‘₯ = 1, and 𝐾(π‘₯) = 𝐾(βˆ’π‘₯), R2 and 𝑒 ΜƒοΈβˆ™ is zero extension of π‘’βˆ™ outside of Ω. Since π‘’βˆ™ ∈ π‘Š 1,̃︀𝑝(Β·) (Ω) and 𝑝(π‘₯) ΜƒοΈ€ β‰₯ 𝛼 = 1 + 𝛿 in Ω), it follows that π‘’βˆ™ ∈ π‘Š 1,𝛼 (Ω). Then π‘’πœ€ ∈ 𝐢0∞ (R2 ) for each πœ€ > 0, π‘’πœ€ β†’ π‘’βˆ™ in 𝐿𝛼 (Ω), βˆ‡π‘’πœ€ β†’ βˆ‡π‘’βˆ™ in 𝐿𝛼 (Ω; R2 ) (16) by the classical properties of smoothing operators (see [17]). From this we deduce that π‘’πœ€ (π‘₯) β†’ π‘’βˆ™ (π‘₯) a.e. in Ω. (17) Moreover, taking into account the estimates ∫︁ ∫︁ π‘’πœ€ (π‘₯) = ΜƒοΈβˆ™ (π‘₯ βˆ’ 𝜁(πœ€)𝑦) 𝑑𝑦 ≀ 𝛾1 𝐾 (𝑦) 𝑒 𝐾 (𝑦) 𝑑𝑦 = 𝛾1 , R 2 R 2 ∫︁ ∫︁ π‘’πœ€ (π‘₯) β‰₯ βˆ™ 𝐾 (𝑦) 𝑒 (π‘₯ βˆ’ 𝜁(πœ€)𝑦) 𝑑𝑦 β‰₯ 𝛾0 ̃︁ 𝐾 (𝑦) 𝑑𝑦 β‰₯ 𝛾0 , π‘¦βˆˆπœ(πœ€)βˆ’1 (π‘₯βˆ’Ξ©) π‘¦βˆˆπœ(πœ€)βˆ’1 (π‘₯βˆ’Ξ©) we see that each element π‘’πœ€ is subjected to the pointwise constraints 𝛾0 ≀ π‘’πœ€ (π‘₯) ≀ 𝛾1 a.a. in Ω, βˆ€ πœ€ > 0. Since, for each πœ€ > 0, π‘’πœ€ ∈ π‘Š 1,π‘π‘˜ (Β·) (Ω) for all π‘˜ ∈ N, it follows that π‘’πœ€ ∈ β„¬π‘π‘˜ (Β·) , i.e., each ⟨ element of the sequence ⟩ {π‘’πœ€ }πœ€>0 is a feasible solution to all approximating problems inf π‘£βˆˆβ„¬π‘ (Β·) 𝐽𝑖 (𝑣, π‘π‘˜ (Β·)) . Hence, π‘˜ 𝐽𝑖 (π‘’π‘˜ , π‘π‘˜ (Β·)) ≀ 𝐽𝑖 (π‘’πœ€ , π‘π‘˜ (Β·)), βˆ€ πœ€ > 0, βˆ€ π‘˜ = 0, 1, . . . (18) Further we notice that lim inf 𝐽𝑖 (π‘’π‘˜ , π‘π‘˜ (Β·)) β‰₯ 𝐽𝑖 (ΜƒοΈ€ 𝑒, 𝑝(Β·)) ΜƒοΈ€ (19) π‘˜β†’βˆž by Proposition A.3 and Fatou’s lemma, and ∫︁ ∫︁ 1 πœ‡ lim 𝐽𝑖 (π‘’πœ€ , π‘π‘˜ (Β·)) = lim |βˆ‡π‘’πœ€ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ + |π‘’πœ€ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)|𝛼 𝑑π‘₯ π‘˜β†’βˆž π‘˜β†’βˆž Ξ© π‘π‘˜ (π‘₯) 𝛼 Ξ©βˆ–π· ∫︁ βƒ’ βƒ’2 ∫︁ βƒ’ (︁ )︁ ⃒𝛼 βƒ’ βŠ₯ + πœ†1 βƒ’βˆ‡πΊπœŽ * (π‘’πœ€ βˆ’ 𝑒0,𝑖 )βƒ’ 𝑑π‘₯ + πœ†2 βƒ’ πœƒ , βˆ‡π‘’πœ€ βƒ’ 𝑑π‘₯. (20) βƒ’ βƒ’ βƒ’ Ξ©βˆ–π· 𝐷 Since 1 1 |βˆ‡π‘’πœ€ (π‘₯)|π‘π‘˜ (π‘₯) β†’ |βˆ‡π‘’πœ€ (π‘₯)|𝑝(π‘₯) ΜƒοΈ€ uniformly in Ω as π‘˜ β†’ ∞, π‘π‘˜ (π‘₯) 𝑝(π‘₯) ΜƒοΈ€ it follows from the Lebesgue dominated convergence theorem and (20) that lim 𝐽𝑖 (π‘’πœ€ , π‘π‘˜ (Β·)) = 𝐽𝑖 (π‘’πœ€ , 𝑝(Β·)), ΜƒοΈ€ βˆ€πœ€ > 0. (21) π‘˜β†’βˆž As a result, passing to the limit in (18) and utilizing properties (19)–(21), we obtain ∫︁ ∫︁ 1 πœ‡ 𝐽𝑖 (ΜƒοΈ€ 𝑒, 𝑝(Β·)) ΜƒοΈ€ ≀ 𝐽𝑖 (π‘’πœ€ , 𝑝(Β·)) ΜƒοΈ€ = |βˆ‡π‘’πœ€ (π‘₯)|𝑝(π‘₯) ΜƒοΈ€ 𝑑π‘₯ + |π‘’πœ€ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)|𝛼 𝑑π‘₯ Ξ© 𝑝(π‘₯) ΜƒοΈ€ 𝛼 Ξ©βˆ–π· ∫︁ βƒ’ βƒ’2 ∫︁ βƒ’ (︁ )︁ ⃒𝛼 βƒ’ βŠ₯ + πœ†1 βƒ’βˆ‡πΊπœŽ * (π‘’πœ€ βˆ’ 𝑒0,𝑖 )βƒ’ 𝑑π‘₯ + πœ†2 βƒ’ πœƒ , βˆ‡π‘’πœ€ βƒ’ 𝑑π‘₯, (22) βƒ’ βƒ’ βƒ’ Ξ©βˆ–π· 𝐷 for all πœ€ > 0. Taking into account the pointwise convergence (see (17) and property (16)) |βˆ‡π‘’πœ€ (π‘₯)|𝑝(π‘₯) ΜƒοΈ€ β†’ |βˆ‡π‘’βˆ™ (π‘₯)|𝑝(π‘₯) ΜƒοΈ€ , |π‘’πœ€ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)|𝛼 β†’ |π‘’βˆ™ (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)|𝛼 , βƒ’ βƒ’2 βƒ’ βƒ’2 βˆ™ βˆ‡πΊ * (𝑒 βˆ’ 𝑒 ) β†’ βˆ‡πΊ * (𝑒 βˆ’ 𝑒0,𝑖 βƒ’ , ) βƒ’ βƒ’ βƒ’ βƒ’ βƒ’ 𝜎 πœ€ 0,𝑖 βƒ’ βƒ’ 𝜎 βƒ’ (︁ )︁ ⃒𝛼 βƒ’ (︁ )︁ ⃒𝛼 βƒ’ βŠ₯ βƒ’ βŠ₯ βˆ™ βƒ’ πœƒ , βˆ‡π‘’ β†’ πœƒ , βˆ‡π‘’ βƒ’ βƒ’ πœ€ βƒ’ βƒ’ βƒ’ as πœ€ β†’ 0, and the fact that, for πœ€ small enough, |βˆ‡π‘’πœ€ (π‘₯)|𝑝(π‘₯) ΜƒοΈ€ ≀ (1 + |βˆ‡π‘’βˆ™ (Β·)|)𝑝(Β·) ΜƒοΈ€ ∈ 𝐿1 (Ω) a.e. in Ω, [︁ ]︁ |π‘’πœ€ (Β·) βˆ’ 𝑒0,𝑖 (Β·)|𝛼 ≀ 2 (1 + |π‘’βˆ™ (Β·)|)𝛼 + 2 (1 + |𝑒0,𝑖 (Β·)|)2 ∈ 𝐿1 (Ω) a.e. in Ω βˆ– 𝐷, βƒ’ βƒ’2 βˆ‡πΊ * (𝑒 βˆ’ 𝑒 )(π‘₯) βƒ’ ≀ β€–πΊπœŽ β€–2𝐢 1 (Ξ©βˆ’Ξ©) |Ω|2 𝛾02 = const, βˆ€ π‘₯ ∈ Ω, βƒ’ βƒ’ βƒ’ 𝜎 πœ€ 0,𝑖 βƒ’ (︁ )︁ ⃒𝛼 βƒ’ βŠ₯ βˆ™ 𝑝(Β·) πœƒ , βˆ‡π‘’ πœ€ βƒ’ ≀ β€–πœƒβ€–πΏβˆž (𝐷,R2 ) (1 + |βˆ‡π‘’ (Β·)|) ∈ 𝐿1 (Ω) a.e. in Ω, βƒ’ ΜƒοΈ€ βƒ’ we can pass to the limit in (22) as πœ€ β†’ 0 by the Lebesgue dominated convergence theorem. This yields 𝐽𝑖 (ΜƒοΈ€ 𝑒, 𝑝(Β·)) ΜƒοΈ€ ≀ lim 𝐽𝑖 (π‘’πœ€ , 𝑝(Β·)) ΜƒοΈ€ = 𝐽𝑖 (π‘’βˆ™ , 𝑝(Β·)). ΜƒοΈ€ πœ€β†’0 Combining this inequality with (22) and (15), we finally get 𝐽𝑖 (π‘’βˆ™ , 𝑝(Β·)) ΜƒοΈ€ = inf 𝐽𝑖 (𝑣, 𝑝(Β·)) ΜƒοΈ€ < 𝐽𝑖 (𝑒* , 𝑝(Β·)) ΜƒοΈ€ ≀ 𝐽𝑖 (π‘’βˆ™ , 𝑝(Β·)), ΜƒοΈ€ π‘£βˆˆβ„¬π‘(Β·),𝑖 ΜƒοΈ€ that leads us into conflict with the initial assumption. Thus, 𝐽𝑖 (ΜƒοΈ€ 𝑒, 𝑝(Β·)) ΜƒοΈ€ = inf 𝐽𝑖 (𝑣, 𝑝(Β·)) ΜƒοΈ€ (23) π‘£βˆˆβ„¬π‘(Β·) ΜƒοΈ€ and, therefore, (ΜƒοΈ€ ΜƒοΈ€ is a weak solution to the original problem (3). As for the variational 𝑒, 𝑝) property (14), it is a direct consequence of (23) and (21). 6. Optimality Conditions To characterize the solution 𝑒0,𝑝(Β·) ∈ ℬ𝑝(Β·) of the approximating optimization problem ⟨ ⟩ inf π‘£βˆˆβ„¬π‘(Β·) 𝐽𝑖 (𝑣, 𝑝(Β·)) , we check that the functional 𝐹𝑝(Β·) is GΓ’teaux differentiable, that is, 𝐽𝑖 (𝑒0,𝑝(Β·) + 𝑑𝑣, 𝑝(Β·)) βˆ’ 𝐽𝑖 (𝑒0,𝑝(Β·) , 𝑝(Β·)) ∫︁ (︁ )︁ lim = |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’2 βˆ‡π‘’0,𝑝(Β·) (π‘₯), βˆ‡π‘£(π‘₯) 𝑑π‘₯ 𝑑→0 𝑑 Ξ© ∫︁ βƒ’ βƒ’π›Όβˆ’2 βƒ’ 0,𝑝(Β·) +πœ‡ ⃒𝑒 (π‘₯) βˆ’ 𝑒0,𝑖 (π‘₯)βƒ’ 𝑒0,𝑝(Β·) (π‘₯)𝑣(π‘₯) 𝑑π‘₯ βƒ’ Ξ©βˆ–π· ∫︁ ∫︁ βƒ’ (︁ )︁ βƒ’π›Όβˆ’1 (︁ )︁ βƒ’ βŠ₯ +2πœ†1 Ξ›(π‘₯)𝑣(π‘₯) 𝑑π‘₯ + π›Όπœ†2 βƒ’ πœƒ , βˆ‡π‘’0,𝑝(Β·) βƒ’ πœƒβŠ₯ , βˆ‡π‘£ 𝑑π‘₯, (24) βƒ’ Ξ© Ξ© for all 𝑣 ∈ π‘Š 1,𝑝(Β·) (Ω), where ∫︁ ∫︁ (︁ )︁ Ξ›(π‘₯) = (βˆ‡πΊπœŽ (𝑦 βˆ’ 𝑧), βˆ‡πΊπœŽ (𝑦 βˆ’ π‘₯)) 𝑒0,𝑝(Β·) (𝑧) βˆ’ 𝑒0,𝑖 (𝑧) πœ’Ξ©βˆ–π· (𝑦) 𝑑𝑧 𝑑𝑦. Ξ© Ξ© To this end, we note that |βˆ‡π‘’0,𝑝(Β·) (π‘₯) + π‘‘βˆ‡π‘£(π‘₯)|𝑝(π‘₯) βˆ’ |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯) 𝑝(π‘₯)𝑑 (︁ )︁ β†’ |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’2 βˆ‡π‘’0,𝑝(Β·) (π‘₯), βˆ‡π‘£(π‘₯) as 𝑑 β†’ 0 almost everywhere in Ω. Since, by convexity, |πœ‰|𝑝 βˆ’ |πœ‚|𝑝 ≀ 2𝑝 |πœ‰|π‘βˆ’1 + |πœ‚|π‘βˆ’1 |πœ‰ βˆ’ πœ‚|, (οΈ€ )οΈ€ it follows that βƒ’ βƒ’ βƒ’ |βˆ‡π‘’0,𝑝(Β·) (π‘₯) + π‘‘βˆ‡π‘£(π‘₯)|𝑝(π‘₯) βˆ’ |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯) βƒ’ βƒ’ βƒ’ 𝑝(π‘₯)𝑑 βƒ’ βƒ’ βƒ’ βƒ’ (︁ )︁ ≀ 2 |βˆ‡π‘’0,𝑝(Β·) (π‘₯) + π‘‘βˆ‡π‘£(π‘₯)|𝑝(π‘₯)βˆ’1 + |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’1 |βˆ‡π‘£(π‘₯)| (︁ )︁ ≀ const |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’1 + |βˆ‡π‘£(π‘₯)|𝑝(π‘₯)βˆ’1 |βˆ‡π‘£(π‘₯)|. (25) Taking into account that by (33) (οΈ‚βˆ«οΈ )οΈ‚ 1β€² 𝛽 0,𝑝(Β·) 𝑝(π‘₯)βˆ’1 0,𝑝(Β·) 𝑝(π‘₯) ‖𝑒 (π‘₯)| ‖𝐿𝑝′ (Β·) (Ξ©) ≀ ‖𝑒 (π‘₯)| 𝑑π‘₯ + 1 Ξ© by (??) (︁ )︁ 1β€² ‖𝑒0,𝑝(Β·) |2𝐿𝑝(Β·) (Ξ©) + 2 𝛽 ≀ , by (33) ∫︁ |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’1 |βˆ‡π‘£(π‘₯)| 𝑑π‘₯ ≀ 2‖𝑒0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’1 ‖𝐿𝑝′ (Β·) (Ξ©) ‖𝑣(π‘₯)|‖𝐿𝑝(Β·) (Ξ©) , Ξ© by (??) and 𝑝(π‘₯) 𝑑π‘₯ ≀ ‖𝑣‖2 + 1, we see that the right hand side of inequality (25) is βˆ«οΈ€ Ξ© |𝑣(π‘₯)| 𝐿𝑝(Β·) (Ξ©) an 𝐿1 (Ω) function. Therefore, |βˆ‡π‘’0,𝑝(Β·) (π‘₯) + π‘‘βˆ‡π‘£(π‘₯)|𝑝(π‘₯) βˆ’ |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯) ∫︁ 𝑑π‘₯ Ξ© 𝑝(π‘₯)𝑑 ∫︁ (︁ )︁ β†’ |βˆ‡π‘’0,𝑝(Β·) (π‘₯)|𝑝(π‘₯)βˆ’2 βˆ‡π‘’0,𝑝(Β·) (π‘₯), βˆ‡π‘£(π‘₯) 𝑑π‘₯ as 𝑑 β†’ 0 Ξ© by the Lebesgue dominated convergence theorem. Utilizing the similar arguments to the rest terms in (3), we deduce that the representation (24) for the GΓ’teaux differential of 𝐽𝑖 (Β·, 𝑝(Β·)) at the point 𝑒0,𝑝(Β·) ∈ ℬ𝑝(Β·) is valid. Thus, in order to derive some optimality conditions for the minimizing element 𝑒0,𝑝(Β·) ∈ ℬ𝑝(Β·) to the problem inf 𝐽𝑖 (𝑣, 𝑝(Β·)), we note that ℬ𝑝(Β·) is a nonempty convex subset of π‘£βˆˆβ„¬π‘(Β·) π‘Š 1,𝑝(Β·) (Ω) and the objective functional 𝐽𝑖 (Β·, 𝑝(Β·)) : ℬ𝑝(Β·) β†’ 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 π‘π‘˜ (Β·) ∈ S be an exponent given by the iterative rule (13). Then the unique minimizer π‘’π‘˜ ∈ β„¬π‘π‘˜ (Β·) to the approximating problem inf π‘£βˆˆβ„¬π‘ (Β·) 𝐽𝑖 (𝑣, π‘π‘˜ (Β·)) is characterized by π‘˜ ∫︁ (οΈ‚βƒ’ )οΈ‚ ∫︁ βƒ’ π‘˜ βƒ’π‘π‘˜ (π‘₯)βˆ’2 π‘˜ βƒ’ (︁ )︁ π‘˜ βƒ’βˆ‡π‘’ (π‘₯)βƒ’ βˆ‡π‘’ (π‘₯), βˆ‡π‘£(π‘₯) βˆ’ βˆ‡π‘’ (π‘₯) 𝑑π‘₯ + 2πœ†1 Ξ›(π‘₯) 𝑣(π‘₯) βˆ’ π‘’π‘˜ (π‘₯) 𝑑π‘₯ Ξ© Ξ© ∫︁ βƒ’ βƒ’π›Όβˆ’2 (︁ )︁ βƒ’ π‘˜ π‘˜ π‘˜ +πœ‡ 𝑒 (π‘₯) βˆ’ 𝑒 (π‘₯) 𝑒 (π‘₯) 𝑣(π‘₯) βˆ’ 𝑒 (π‘₯) 𝑑π‘₯ βƒ’ βƒ’ 0,𝑖 βƒ’ Ξ©βˆ–π· ∫︁ βƒ’ (︁ )︁ βƒ’π›Όβˆ’1 (︁ )︁ βƒ’ βŠ₯ + π›Όπœ†2 βƒ’ πœƒ , βˆ‡π‘’0,𝑝(Β·) βƒ’ πœƒβŠ₯ , βˆ‡π‘£ βˆ’ βˆ‡π‘’π‘˜ 𝑑π‘₯ β‰₯ 0, βˆ€ 𝑣 ∈ β„¬π‘π‘˜ (Β·) . βƒ’ Ξ© 7. Numerical Experiments 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. Figure 1: Given collection of past cloud-free images. Date of generation: (left) - 2019/06/15, (right) - 2019/07/01 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. Figure 2: The could contaminated image with date of generation 2019/07/17 8. Conclusion 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 𝑝(π‘₯), where the variable exponent 𝑝(π‘₯) 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 ⃗𝑒, we are going to restore, can have rather rich structure in the damage region 𝐷. 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. Figure 3: Result of the restoration of image in Fig 2 by the proposed method. Appendix A. On Orlicz Spaces Let 𝑝(Β·) be a measurable exponent function on Ω such that 1 ≀ 𝛼 ≀ 𝑝(π‘₯) ≀ 𝛽 < ∞ a.e. in Ω, 𝑝(Β·) where 𝛼 and 𝛽 are given constants. Let 𝑝′ (Β·) = 𝑝(Β·)βˆ’1 be the corresponding conjugate exponent. It is clear that 𝛽 𝛼 1≀ ≀ 𝑝′ (π‘₯) ≀ a.e. in Ω, π›½βˆ’1 ⏟ βˆ’βž 1 𝛼 ⏟ ⏞ 𝛽′ 𝛼′ where 𝛽 β€² and 𝛼′ stand for the conjugates of constant exponents. Denote by 𝐿𝑝(Β·) (Ω) the set of all measurable functions 𝑓 (π‘₯) on Ω such that Ξ© |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯ < ∞. Then 𝐿𝑝(Β·) (Ω) is a reflexive βˆ«οΈ€ separable Banach space with respect to the Luxemburg norm (see [20] for the details) ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) = inf πœ† > 0 : πœŒπ‘ (πœ†βˆ’1 𝑓 ) ≀ 1 , (26) {οΈ€ }οΈ€ where πœŒπ‘ (𝑓 ) := Ξ© |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯. βˆ«οΈ€ β€² It is well-known that 𝐿𝑝(Β·) (Ω) is reflexive provided 𝛼 > 1, and its dual is 𝐿𝑝 (Β·) (Ω), that is, any continuous functional 𝐹 = 𝐹 (𝑓 ) on 𝐿𝑝(Β·) (Ω) has the form (see [21, Lemma 13.2]) ∫︁ β€² 𝐹 (𝑓 ) = 𝑓 𝑔 𝑑π‘₯, with 𝑔 ∈ 𝐿𝑝 (Β·) (Ω). Ξ© As for the infimum in (26), we have the following result. Proposition A.1. The infimum in (26) is attained if πœŒπ‘ (𝑓 ) > 0. Moreover if πœ†* := ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) > 0, then πœŒπ‘ (πœ†βˆ’1 * 𝑓 ) = 1. Taking this result and condition 1 ≀ 𝛼 ≀ 𝑝(π‘₯) ≀ 𝛽 into account, we see that βƒ’ 𝑓 (π‘₯) ⃒𝑝(π‘₯) ∫︁ ∫︁ βƒ’ βƒ’ ∫︁ 1 𝑝(π‘₯) 1 |𝑓 (π‘₯)| 𝑑π‘₯ ≀ βƒ’ βƒ’ 𝑑π‘₯ ≀ 𝛼 |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯, πœ†π›½* βƒ’ πœ†* βƒ’ πœ†* Ξ© Ξ© Ξ© ∫︁ ∫︁ 1 𝑝(π‘₯) 1 𝛽 |𝑓 (π‘₯)| 𝑑π‘₯ ≀ 1 ≀ 𝛼 |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯. πœ†* Ξ© πœ† * Ξ© Hence, (see [20] for the details) ∫︁ 𝛼 ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) ≀ |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯ ≀ ‖𝑓 ‖𝛽𝐿𝑝(Β·) (Ξ©) , if ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) > 1, Ξ© ∫︁ 𝛽 ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) ≀ |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯ ≀ ‖𝑓 ‖𝛼𝐿𝑝(Β·) (Ξ©) , if ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) < 1, Ξ© and, therefore, ∫︁ ‖𝑓 ‖𝛼𝐿𝑝(Β·) (Ξ©) βˆ’ 1 ≀ |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯ ≀ ‖𝑓 ‖𝛽𝐿𝑝(Β·) (Ξ©) + 1, βˆ€ 𝑓 ∈ 𝐿𝑝(Β·) (Ω), Ξ© ∫︁ ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) = |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯, if ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) = 1. Ξ© The following estimates are well-known: if 𝑓 ∈ 𝐿𝑝(Β·) (Ω) then ‖𝑓 ‖𝐿𝛼 (Ξ©) ≀ (1 + |Ω|)1/𝛼 ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) , β€² 𝛽 ‖𝑓 ‖𝐿𝑝(Β·) (Ξ©) ≀ (1 + |Ω|)1/𝛽 ‖𝑓 ‖𝐿𝛽 (Ξ©) , 𝛽′ = , βˆ€ 𝑓 ∈ 𝐿𝛽 (Ω). π›½βˆ’1 Let {π‘π‘˜ }π‘˜βˆˆN βŠ‚ 𝐢 0,𝛿 (Ω), with some 𝛿 ∈ (0, 1], be a given sequence of exponents. Hereinafter in this subsection we assume that 1 ≀ 𝛼 ≀ π‘π‘˜ (π‘₯) ≀ 𝛽 < ∞ a.e. in Ω for π‘˜ = 1, 2, . . . , and π‘π‘˜ (Β·) β†’ 𝑝(Β·) in 𝐢(Ω) as π‘˜ β†’ ∞. (27) We associate with this sequence the following collection π‘“π‘˜ ∈ 𝐿 𝑝 (Β·) (Ω) π‘˜βˆˆN . The characteris- {οΈ€ }οΈ€ π‘˜ tic feature of this set of functions is that each element 𝑓 }οΈ€π‘˜ lives in the corresponding Orlicz space 𝑝 (Β·) (Ω). We say that the sequence π‘“π‘˜ ∈ 𝐿 𝑝 (Β·) (Ω) π‘˜βˆˆN is bounded if (see [22, Section 6.2]) {οΈ€ 𝐿 π‘˜ π‘˜ ∫︁ lim sup |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ < +∞. (28) π‘˜β†’βˆž Ξ© Definition A.2. A bounded sequence π‘“π‘˜ ∈ πΏπ‘π‘˜ (Β·) (Ω) π‘˜βˆˆN is weakly convergent in the {οΈ€ }οΈ€ variable Orlicz space πΏπ‘π‘˜ (Β·) (Ω) to a function 𝑓 ∈ 𝐿𝑝(Β·) (Ω), where 𝑝 ∈ 𝐢 0,𝛿 (Ω) is the limit of {π‘π‘˜ }π‘˜βˆˆN βŠ‚ 𝐢 0,𝛿 (Ω) in the uniform topology of 𝐢(Ω), if ∫︁ ∫︁ lim π‘“π‘˜ πœ™ 𝑑π‘₯ = 𝑓 πœ™ 𝑑π‘₯, βˆ€ πœ™ ∈ 𝐢0∞ (R2 ). π‘˜β†’βˆž Ξ© Ξ© 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 πΏπ‘π‘˜ (Β·) -norm with respect to the weak convergence in πΏπ‘π‘˜ (Β·) {οΈ€ (Ω). Proposition A.3. If a bounded sequence π‘“π‘˜ ∈ πΏπ‘π‘˜ (Β·) (Ω) π‘˜βˆˆN converges weakly in 𝐿𝛼 (Ω) }οΈ€ to 𝑓 for some 𝛼 > 1, then 𝑓 ∈ 𝐿𝑝(Β·) (Ω), π‘“π‘˜ ⇀ 𝑓 in variable πΏπ‘π‘˜ (Β·) (Ω), and ∫︁ ∫︁ lim inf |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ β‰₯ |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯. (29) π‘˜β†’βˆž Ξ© Ξ© Proof. In view of the fact that βƒ’βˆ«οΈ ∫︁ βƒ’ βƒ’ βƒ’ |π‘“π‘˜ (π‘₯)| 𝑝 (π‘₯) 𝑝(π‘₯) 𝑝 (π‘₯) βƒ’ π‘˜ 𝑑π‘₯ βˆ’ |π‘“π‘˜ (π‘₯)| π‘˜ 𝑑π‘₯βƒ’βƒ’ Ξ© π‘π‘˜ (π‘₯) βƒ’ Ξ© ∫︁ 1 ≀ β€–π‘π‘˜ βˆ’ 𝑝‖𝐢(Ξ©) |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ Ξ© π‘˜π‘ (π‘₯) β€–π‘π‘˜ βˆ’ 𝑝‖𝐢(Ξ©) ∫︁ by (28) ≀ |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ β†’ 0 as π‘˜ β†’ ∞, 𝛼 Ξ© we see that ∫︁ ∫︁ π‘π‘˜ (π‘₯) 𝑝(π‘₯) lim inf |π‘“π‘˜ (π‘₯)| 𝑑π‘₯ = lim inf |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯. π‘˜β†’βˆž Ξ© π‘˜β†’βˆž Ξ© π‘π‘˜ (π‘₯) β€² Using the Young inequality π‘Žπ‘ ≀ |π‘Ž|𝑝 /𝑝 + |𝑏|𝑝 /𝑝′ , we have ∫︁ ∫︁ ∫︁ 𝑝(π‘₯) 𝑝(π‘₯) π‘β€²π‘˜ (π‘₯) |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ β‰₯ 𝑝(π‘₯)π‘“π‘˜ (π‘₯)πœ™(π‘₯) 𝑑π‘₯ βˆ’ β€² (π‘₯) |πœ™(π‘₯)| 𝑑π‘₯, (30) 𝑝 Ξ© π‘˜ (π‘₯) Ξ© 𝑝 Ξ© π‘˜ for π‘β€²π‘˜ (π‘₯) = π‘π‘˜ (π‘₯)/(π‘π‘˜ (π‘₯) βˆ’ 1) and any πœ™ ∈ 𝐢0∞ (R2 ). Then passing to the limit in (30) as π‘˜ β†’ ∞ and utilizing property (27) and the fact that ∫︁ ∫︁ β€² lim π‘“π‘˜ (π‘₯)πœ™(π‘₯) 𝑑π‘₯ = 𝑓 (π‘₯)πœ™(π‘₯) 𝑑π‘₯ for all πœ™ ∈ 𝐿𝛼 (Ω), (31) π‘˜β†’βˆž Ξ© Ξ© we obtain ∫︁ ∫︁ ∫︁ π‘π‘˜ (π‘₯) 𝑝(π‘₯) β€² lim inf |π‘“π‘˜ (π‘₯)| 𝑑π‘₯ β‰₯ 𝑝(π‘₯)𝑓 (π‘₯)πœ™(π‘₯) 𝑑π‘₯ βˆ’ β€² |πœ™(π‘₯)|𝑝 (π‘₯) 𝑑π‘₯. π‘˜β†’βˆž Ξ© Ξ© Ξ© 𝑝 (π‘₯) β€² Since the last inequality is valid for all πœ™ ∈ 𝐢0∞ (R2 ) and the set 𝐢0∞ (R2 ) is dense in 𝐿𝑝 (Β·) (Ω), β€² it follows that this relation holds true for πœ™ ∈ 𝐿𝑝 (Β·) (Ω). So, taking πœ™ = |𝑓 (π‘₯)|𝑝(π‘₯)βˆ’2 𝑓 (π‘₯), we arrive at the announced inequality (29). As an consequence of (29) and estimate (??), we get: 𝑓 ∈ 𝐿𝑝(Β·) (Ω). To end of the proof, it remains to observe that relation (31) holds true for πœ™ ∈ 𝐢0∞ (R2 ) as well. From this the weak convergence π‘“π‘˜ ⇀ 𝑓 in the variable Orlicz space πΏπ‘π‘˜ (Β·) (Ω) follows. Remark A.4. Arguing in a similar manner and using, instead of (30), the estimate ∫︁ ∫︁ ∫︁ 1 π‘π‘˜ (π‘₯) 1 𝑝′ (π‘₯) lim inf |π‘“π‘˜ (π‘₯)| 𝑑π‘₯ β‰₯ 𝑝(π‘₯)𝑓 (π‘₯)πœ™(π‘₯) 𝑑π‘₯ βˆ’ β€² (π‘₯) |πœ™(π‘₯)| 𝑑π‘₯, π‘˜β†’βˆž 𝑝 Ξ© π‘˜ (π‘₯) Ξ© 𝑝 Ξ© π‘˜ it is easy to show that the lower semicontinuity property (29) can be generalized as follows ∫︁ ∫︁ 1 1 lim inf |π‘“π‘˜ (π‘₯)|π‘π‘˜ (π‘₯) 𝑑π‘₯ β‰₯ |𝑓 (π‘₯)|𝑝(π‘₯) 𝑑π‘₯. (32) π‘˜β†’βˆž Ξ© π‘π‘˜ (π‘₯) Ξ© 𝑝(π‘₯) We need the following result that leads to the analog of the HΓΆlder inequality in Lebesgue spaces with variable exponents (for the details we refer to [20]). β€² Proposition A.6. If 𝑓 ∈ 𝐿𝑝(Β·) (Ω)𝑁 and 𝑔 ∈ 𝐿𝑝 (Β·) (Ω)𝑁 , then (𝑓, 𝑔) ∈ 𝐿1 (Ω) and ∫︁ (𝑓, 𝑔) 𝑑π‘₯ ≀ 2‖𝑓 ‖𝐿𝑝(Β·) (Ξ©)𝑁 ‖𝑔‖𝐿𝑝′ (Β·) (Ξ©)𝑁 . (33) Ξ© Appendix B. Sobolev Spaces with Variable Exponent We recall here the well-known facts concerning the Sobolev spaces with variable exponent. Let 𝑝(Β·) be a measurable exponent function on Ω such that 1 < 𝛼 ≀ 𝑝(π‘₯) ≀ 𝛽 < ∞ a.e. in Ω, where 𝛼 and 𝛽 are given constants. We associate with it the so-called Sobolev-Orlicz space {οΈ‚ ∫︁ [︁ ]︁ }οΈ‚ 1,𝑝(Β·) 1,1 𝑝(π‘₯) 𝑝(π‘₯) π‘Š (Ω) := 𝑦 ∈ π‘Š (Ω) : |𝑦(π‘₯)| + |βˆ‡π‘¦(π‘₯)| 𝑑π‘₯ < +∞ Ξ© and equip it with the norm β€–π‘¦β€–π‘Š 1,𝑝(Β·) (Ξ©) = ‖𝑦‖𝐿𝑝(Β·) (Ξ©) + β€–βˆ‡π‘¦β€–πΏπ‘(Β·) (Ξ©;R2 ) . 0 It is well-known that, in general, unlike classical Sobolev spaces, smooth functions are not 1,𝑝(Β·) necessarily dense in π‘Š = π‘Š0 (Ω). Hence, with the given variable exponent 𝑝 = 𝑝(π‘₯) (1 < 𝛼 ≀ 𝑝 ≀ 𝛽) it can be associated another Sobolev space, 𝐻 = 𝐻 1,𝑝(Β·) (Ω) as the closure of the set 𝐢 ∞ (Ω) in π‘Š 1,𝑝(Β·) (Ω)-norm. Since the identity π‘Š = 𝐻 is not always valid, it makes sense to say that an exponent 𝑝(π‘₯) is regular if 𝐢 ∞ (Ω) is dense in π‘Š 1,𝑝(Β·) (Ω). The following result reveals an important property ensuring the regularity of exponent 𝑝(π‘₯). Proposition B.1. Assume that there exists 𝛿 ∈ (0, 1] such that 𝑝 ∈ 𝐢 0,𝛿 (Ω). Then the set ∞ 𝐢 (Ω) is dense in π‘Š 1,𝑝(Β·) (Ω), and, therefore, π‘Š = 𝐻. Proof. Let 𝑝 ∈ 𝐢 0,𝛿 (Ω) be a given exponent. Since lim |𝑑|𝛿 log(|𝑑|) = 0 with an arbitrary 𝛿 ∈ (0, 1], (34) 𝑑→0 it follows from the HΓΆlder continuity of 𝑝(Β·) that [οΈƒ ]οΈƒ 𝛿 |π‘₯ βˆ’ 𝑦|𝛿 |𝑝(π‘₯) βˆ’ 𝑝(𝑦)| ≀ 𝐢|π‘₯ βˆ’ 𝑦| ≀ sup βˆ’1 πœ”(|π‘₯ βˆ’ 𝑦|), βˆ€ π‘₯, 𝑦 ∈ Ω, π‘₯,π‘¦βˆˆΞ© log(|π‘₯ βˆ’ 𝑦| ) where πœ”(𝑑) = 𝐢/ log(|𝑑|βˆ’1 ), and 𝐢 > 0 is some positive constant. Then property (34) implies that 𝑝(Β·) is a log-HΓΆlder continuous function. So, to deduce the density of 𝐢 ∞ (Ω) in π‘Š 1,𝑝(Β·) (Ω) it is enough to refer to Theorem 13.10 in [21]. References [1] M. D. King, S. Platnick, P. Menzel, S. A. Ackerman, P. A. Hubanks, Spatial and temporal distribution of clouds observed by modis onboard the terra and aqua satellites, IEEE Trans, Geosci. Remote Sens, 51, 2013), pp. 3826–3852. doi:10.1109/TGRS.2012.2227333 [2] H. 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