In this paper we develop a novel approach to the analysis and study of one class optimization problems with non-coercive objective functionals. With this in mind we introduce a special class fo anisotropic functional spaces. We give a precise definition of such spaces and show that they can be considered as a natural generalization of the standard Sobolev spaces. Bases on this concept, we relax of a special class of non-coercive minimization problems in Sobolev spaces W1,2(Ω), provide a rigorous mathematical analysis of the proposed relaxed version, establish sufficient conditions of its solvability, show that the objective functional is coercive, and derive the corresponding optimality conditions. To demonstrate the validity of the obtained results, we apply the proposed approach to the relaxation of the well-know variational model for removing multiplicative noise in image processing.

In this article, we discuss the existence and uniqueness of mild, variational, and the so-called non-variational solutions to an initial-boundary value problem (IBVP) for linear wave equation with strong interior degeneracy of the coefficient in the principal part of the elliptic operator. The objective is to provide a well-posedness analysis of the IBVP and find out how the density property of smooth functions in the corresponding weighted Sobolev space affects the uniqueness of its solutions. We show that, in general, the uniqueness of solutions may be violated if the 'degree of degeneracy' corresponds to the strong degeneracy case.

Mostly motivated by the crop field classification problem and the automated computational methodology for extracting agricultural crop fields from satellite data, we proposed in a bounded variation (BV) space a new approach to the piecewise smooth approximation of the slope-based vegetation indices and the closely related crop field segmentation problem of multi-band satellite images.

In this paper, we analyze the numerical aspects of the practical implementation of the generalized active contour model, that has been recently proposed in the literature, for extracting agricultural crop fields with a high level of inhomogeneity from satellite data. We also derive the corresponding Euler–Lagrange equation and discuss its relaxation method.

Mostly motivated by the practical applications especially in the field of satellite remote sensing of agricultural territories, we develop a novel approach to the domain decomposition basing on the anisotropic version of the Chan-Vese active contour model. With that in mind we propose a new statement of this problem in variable Sobolev spaces, provide a rigorous mathematical analysis of the proposed optimization problem, establish sufficient conditions of its consistency and solvability, show that the objective functional is Gateaux differentiable, and derive the corresponding optimality conditions. To illustrate the validity of the obtained results, we give some examples of numerical simulations with the real satellite images.

We study a Dirichlet optimal control problem for a quasilinear monotone elliptic equation with the so-called weighted p(x)-Laplace operator. The coefficient of the p(x)-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the admissible controls. In order to handle the inherent degeneracy of the p(x)-Laplacian, we use a special two-parametric regularization scheme. We derive existence and uniqueness of variational V -solutions to the underlying boundary value problem and the corresponding optimal control problem. Further we discuss the asymptotic behaviour of the solutions to regularized problems on each (ε, k)-level as the parameters tend to zero and infinity, respectively. The characteristic feature of the considered OCP is the fact that the exponent p(x) is assumed to be Lebesgue-measurable, and we do not impose any additional assumptions on p(x) like to be a Lipschitz function or satisfy the so-called log-H‥older continuity condition

Владислав Федорович Бабенко (до 75-річчя від дня народження)

We study the bilinear forms on the space of measurable $p$-integrable functions which are generated by skew-symmetric matrices with unbounded coefficients. We give an example showing that if a skew-symmetric matrix contains a locally unbounded $L^q$-elements, then the corresponding quadratic forms can be alternating. These questions are closely related to the existence issues of the Nuemann boundary value problem for $p$-Laplace elliptic equations with non-symmetric and locally unbounded anisotropic diffusion matrices.

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous fusion and de- noising of color images with different spatial resolution. The characteristic feature of the proposed model is that we deal with a constrained minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

We propose a new variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to the simultaneous contrast enhancement and denoising of color images. The characteristic feature of the proposed model is that we deal with a constrained non-convex minimization problem that lives in variable Sobolev-Orlicz spaces where the variable exponent is unknown a priori and it depends on a particular function that belongs to the domain of the objective functional. In contrast to the standard approach, we do not apply any spatial regularization to the image gradient. We discuss the consistency of the variational model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

In this paper, the problem of restoration of cloud contaminated optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of optical multi-band images utilising Synthetic Aperture Radar (EOS – Spatial Data Analytics, GIS Software, Satellite Imagery – is a cloud-based platform to derive remote sensing data and analyse satellite imagery for business and science purposes) images of the same regions. We prove existence of solutions, propose an alternating minimisation method for computing them, prove convergence of this method to weak solutions of the original problem and derive optimality conditions.

We study a coupled two-level variational model in Sobolev-Orlicz spaces with non-standard growth conditions of the objective functional and discuss its consistence and solvability issues. At the first level, we deal with the so-called temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection-diffusion equation with two types of control functions — distributed $L^2$-control and $BV$-bounded control in coefficients. At the second level, we have a constrained minimization problem with the nonstandard growth energy functional that lives in a variable Sobolev-Orlicz space. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with non-standard growth in the objective functional, is unknown a priori and it depends on the solution of the first-level optimal control problem.

In this paperwe discuss a newvariational approach to the Date Fusion problem of multi-spectral satellite images from Sentinel-2 and MODIS that have been captureed at different resolution level and, arguably, on different days. The crucial assumption to our approach is that the MODIS image has to be cloud-free whereas the images from Sentinel-2 can be corrupted by clouds or noise. We formulate the data fusion problem as the two-level optimization problem. We discuss the well thoroughness and consistency of the proposed variational models. We also derive some optimality conditions and supply our approach by results of numerical simulations with the real satellite images.

In this paper we discuss a new variational approach to the Date Fusion problem of multi-spectral satellite images from Sentinel-2 and MODIS that have been captured at different resolution level and, arguably, on different days. The crucial point of our approach that the MODIS image is cloud-free whereas the images from Sentinel-2 can be corrupted by clouds or noise.

In this work, we study a sparse optimal control problem involving a quasilinear parabolic equation with variable order of nonlinearity as a state equation and with a pointwise control constraints. We show that in the case if the cost functional contains the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the compensated compactness technique and the consept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation.

We propose a new variational model in Sobolev–Orlicz spaces with non-standard growth conditions of the objective functional and discuss its applications to image processing. The characteristic feature of the proposed model is that the variable exponent, which is associated with non-standard growth, is unknown a priori and it depends on a particular function that belongs to the domain of objective functional. So, we deal with a constrained minimization problem that lives in variable Sobolev–Orlicz spaces. In view of this, we discuss the consistency of the proposed model, give the scheme for its regularization, derive the corresponding optimality system, and propose an iterative algorithm for practical implementations.

We discuss the optimal control formulation for enhancement and denoising of satellite multiband images and propose to take it in the form of an L1 control problem for a quasi-linear parabolic equation with a nonlocal p[u] Laplacian and with a cost functional of a tracking type. The main characteristic features of the considered parabolic problem is that the variable exponent p(t,x) and the diffusion anisotropic tensor D(t,x) are not predefined well a priori; instead, these characteristics nonlocally depend on the form of the solution of this problem (i.e., pu=p(t,x,u) and Du=D(t,x,u)). We prove the existence of optimal pairs with sparse L1 controls used for the indirect approach and a special family of approximation problems.