Nonstandard Growth Conditions Research Articles

Gradient higher integrability for double phase problems on metric measure spaces

We study local and global higher integrability properties for quasiminimizers of a class of double phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure space with a doubling measure and a Poincaré inequality. The main novelty is an intrinsic approach to double phase Sobolev-Poincaré inequalities.

Blow-up and Decay for a Pseudo-parabolic Equation with Nonstandard Growth Conditions

Blow-up and Decay for a Pseudo-parabolic Equation with Nonstandard Growth Conditions

Singular orthotropic functionals with nonstandard growth conditions

Singular orthotropic functionals with nonstandard growth conditions

Quasilinear elliptic systems with nonstandard growth conditions in Orlicz-Sobolev spaces

Quasilinear elliptic systems with nonstandard growth conditions in Orlicz-Sobolev spaces

The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions

The generalized anisotropic dynamical Wentzell heat equation with nonstandard growth conditions

Upper semicontinuity of pullback D$\mathcal {D}$‐attractors for nonlinear parabolic equation with nonstandard growth condition

AbstractThis paper is devoted to the well‐posed problem and the existence of pullback ‐attractors for a class of nonlinear parabolic equation with nonstandard growth condition. First, by making use of Galerkin's method and monotone operator method, the existence of solutions is proved in Orlicz–Sobolev space with variable exponents depending on time and space, then the uniqueness and continuity of solutions are also obtained. Finally, by verifying the pullback ‐asymptotic compactness, the existence of pullback ‐attractors is proved. In particular, the upper semicontinuity of pullback ‐attractors of the corresponding equation with respect to the disturbance parameter λ is also proved.

On Coupled Two-Level Variational Problem in Sobolev-Orlicz Space

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.

A two-level variational algorithm in the Sobolev–Orlicz space to predict daily surface reflectance at LANDSAT high spatial resolution and MODIS temporal frequency

We propose a new two-level variational model in Sobolev–Orlicz spaces with nonstandard growth conditions of the objective functional and discuss its applications to the spatiotemporal interpolation of multispectral satellite images. At the first level, we deal with the temporal interpolation problem that can be cast as a state constrained optimal control problem for anisotropic convection–diffusion equation, whereas at the second level we solve a constrained minimization problem with a nonstandard growth energy functional that lives in variable Sobolev–Orlicz spaces. The characteristic feature of the proposed model is the fact that the variable exponent, which is associated with nonstandard growth in spatial interpolation problem, is unknown a priori and it depends on the solution of the first-level optimal control problem. It makes this spatiotemporal interpolation problem rather challenging. In view of this, we discuss the consistency of the proposed model, study the existence of optimal solutions, and derive the corresponding optimality systems. In particular, we apply this approach to the well-known prediction problem of the Daily MODIS Surface Reflectance at the Landsat-Like Resolution.

Pullback exponential attractors for second-order lattice system with nonstandard growth condition

In this paper, we study the existence of pullback attractors and pullback exponential attractors for lattice dynamical system in time-dependent sequence space. First, we introduce a new sequence space with time-dependent variable exponents. Second, two abstract criteria (or sufficient conditions) about the existence of pullback attractors and pullback exponential attractors are established for infinite dimensional lattice dynamical systems on time-dependent spaces of infinite sequences. Finally, for making full use of the above-mentioned abstract criteria, we consider a second order lattice system with nonstandard growth nonlinearity, and then the existence of bi-space pullback attractors and pullback exponential attractors on a time-dependent Musielak–Orlicz space is obtained. In particular, we point out that these criteria and analytical skills can be utilized to deal with other lattice systems satisfying nonstandard growth conditions.

Local boundedness under nonstandard growth conditions

This paper deals with local boundedness under nonstandard growth conditions by using De Giorgi's iteration method. We first consider quasilinear elliptic systems, and assume that the coefficients satisfying some boundedness and coercivity conditions, then we derive local boundedness of solutions. Secondly, we consider elliptic equations with a drift term, under some nonstandard growth conditions, we derive that solutions are locally bounded. Finally, we consider variational integrals with the integrand satisfy nonstandard growth conditions, local boundedness of minimizers is derived.

On Hölder continuity to a function class with nonstandard growth condition

With the help of De Giorgi's method, the embedding of the class into is proved. This result can be applied to the continuity to some elliptic equations with variable exponent.

Existence result for a Neumann boundary value problem governed by a class of p ( x )-Laplacian-like equation

In this article, we consider a Neumann boundary value problem driven by p ( x )-Laplacian-like operator with a reaction term depending also on the gradient (convection) and on three real parameters, originated from a capillary phenomena, of the following form: − Δ p ( x ) l u + δ | u | ζ ( x ) − 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) in Ω , ∂ u ∂ η = 0 on ∂ Ω , where Δ p ( x ) l u is the p ( x )-Laplacian-like operator, Ω is a smooth bounded domain in R N , δ, μ and λ are three real parameters, p ( x ) , ζ ( x ) ∈ C + ( Ω ‾ ), η is the outer unit normal to ∂ Ω and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized ( S + ) type and the theory of variable exponent Sobolev spaces, we establish the existence of weak solution for the above problem.

A nonlinear anisotropic diffusion model with non-standard growth for image segmentation

The anisotropic diffusion equation with non-standard growth embodies the physical characteristics of “point-by-point anisotropy” as well as has important potential value in computer vision. In this paper, a general anisotropic diffusion framework of the level set function is proposed for image segmentation in scalar-value and vector-value images. Specifically, we develop a new regularization term that uses a diffusion coefficient with non-standard growth conditions and diffusion tensors. The existence and uniqueness of the model are obtained by the Galerkin method. We establish the numerical algorithms for obtaining the texture feature and evolving the level set function of images. Some numerical tests on medical and natural images confirm the accuracy of the proposed method and the improvement in segmenting small features.

Weak mean random attractors for nonautonomous stochastic parabolic equation with variable exponents

In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term.

Singular solutions for the fourth‐order parabolic equation with nonstandard growth conditions and absorption

This paper deals with the fourth‐order parabolic equation in a bounded domain, subject to homogeneous Navier boundary conditions. Under some conditions on the variable exponents, we give a complete and optimal classification on the singularity of solutions, characterized by the signs of the Nehari energy and the difference between the initial energy and the potential depth.

On a Variational Problem with a Nonstandard Growth Functional and Its Applications to Image Processing

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.

Upper and lower bounds of convergence rates for strong solutions of the generalized Newtonian fluids with non-standard growth conditions

We consider the motion of an incompressible shear-thickening power-law-like non-Newtonian fluid in $R^3$ with a variable power-law index. This system of nonlinear partial differential equations arises in mathematical models of electrorheological fluids. The aim of this paper is to investigate the large-time behaviour of the difference $u-\tilde{u}$ where $u$ is a strong solution of the given equations with the initial data $u_0$ and $\tilde{u}$ is the strong solution of the same equations with perturbed initial data $u_0+w_0$. The initial perturbation $w_0$ is not required to be small, but is assumed to satisfy certain decay condition. In particular, we can show that $(1+t)^{-\frac{\gamma}{2}}\lesssim \|u(t)-\tilde{u}(t)\|_2\lesssim(1+t)^{-\frac{\gamma}{2}}$, for sufficiently large $t>0$, where $\gamma\in(2,\frac{5}{2})$. The proof is based on the observation that the solution of the linear heat equation describes the asymptotic behaviour of the solutions of the electrorheological fluids well for sufficiently large time $t>0$, and the generalized Fourier splitting method with an iterative argument. Furthermore, it will also be discussed that the argument used in the present paper can improve the previous results for the generalized Newtonian fluids with a constant power-law index.

Regularity Results for Bounded Solutions to Obstacle Problems with Non-standard Growth Conditions

In this paper, we consider a class of obstacle problems of the type min∫Ωf(x,Dv)dx:v∈Kψ(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\min \\left\\{ \\int _{\\Omega }f(x, Dv)\\, {\\mathrm d}x\\,:\\, v\\in {\\mathcal {K}}_\\psi (\\Omega )\\right\\} \\end{aligned}$$\\end{document}where psi is the obstacle, {mathcal {K}}_psi (Omega )={vin u_0+W^{1, p}_{0}(Omega , {mathbb {R}}): vge psi text { a.e. in }Omega }, with u_0 in W^{1,p}(Omega ) a fixed boundary datum, the class of the admissible functions and the integrand f(x, Dv) satisfies non standard (p, q)-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map xmapsto A(x, xi ) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one W^{1,n}.

A Local Boundedness Result for a Class of Obstacle Problems with Non-Standard Growth Conditions

We prove the local boundedness for solutions to a class of obstacle problems with non-standard growth conditions. The novelty here is that we are able to establish the local boundedness under a sharp bound on the gap between the growth exponents.

Gradient estimates of very weak solutions to general quasilinear elliptic equations

We establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the p-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems.