Variable Sobolev Spaces Research Articles

Wavelet Characterization of Local Muckenhoupt Weighted Sobolev Spaces with Variable Exponents

The goal of this paper is to define local weighted variable Sobolev spaces of fractional and negative order and their characterization by wavelets. We first consider local weighted variable Sobolev spaces by means of weak derivatives and obtain a wavelet characterization for these spaces. Using the Bessel potentials, we next define local weighted variable Sobolev spaces of fractional order. We show that Sobolev spaces obtained by weak derivatives and those by the Bessel potentials coincide. Finally, using duality, we define local weighted variable Sobolev spaces with negative order. We also show that local weighted variable Sobolev spaces are closed under complex interpolation. Some examples are given including the applications to weighted uniformly local Lebesgue spaces with variable exponents and periodic function spaces as a by-product, although the exponent is constant.

A unified divergent approach to Hardy–Poincaré inequalities in classical and variable Sobolev spaces

We present a unified strategy to derive Hardy–Poincaré inequalities on bounded and unbounded domains. The approach allows proving a general Hardy–Poincaré inequality from which the classical Poincaré and Hardy inequalities immediately follow. We extend the idea to the more general context of variable exponent Sobolev spaces. Surprisingly, despite the well-known counterexamples of Fan et al. (2005) [28], we show that a modular form of the Poincaré inequality is actually possible provided one restricts to the class of functions in u∈Cc∞(Ω) such that |u|⩽1. The argument, concise and constructive, does not require a priori knowledge of compactness results and retrieves geometric information on the best constants.

Characterizations of even-order Musielak–Orlicz–Sobolev spaces via ball averages and their derivatives

In this paper, the authors present some new characterizations of the Musielak–Orlicz–Sobolev spaces with even smoothness order via ball averages and their derivatives on the radius. Consequently, as special examples of the Musielak–Orlicz–Sobolev spaces studied in this paper, the corresponding characterizations for some weighted Sobolev spaces, Orlicz–Sobolev spaces and variable Sobolev spaces are also obtained. Since these characterizations depend only on ball averages and their derivatives on the radius, they provide some possible ways to introduce the corresponding function spaces on any metric measure space.

On the Convergence of Solutions of Variational Problems with Implicit Constraints Defined by Rapidly Oscillating Functions

For functionals defined on variable Sobolev spaces, we establish a series of results on the convergence of minimizers and minimum values on sets of functions subject to implicit constraints by means of periodic rapidly oscillating functions. In connection with the formulation and justification of these results, we introduce the definition of Γ-convergence of functionals corresponding to the given sets of constraints. The specificity of the introduced definition is that it refers to the convergence of a sequence of functionals defined on variable Sobolev spaces to a function on the real line. The considered minimization problems have the feature that, to justify the convergence of a sequence of their solutions, the strong connectedness of the domains of the corresponding functionals is not required, while this connectedness is essential, for instance, in the study of the convergence of solutions of the Neumann variational problems and variational problems with explicit unilateral and bilateral constraints in variable domains. In addition to the mentioned results, we establish theorems on the Γ-compactness of sequences of functionals with respect to the given sets of constraints.

Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

Asymptotic behavior for a class of parabolic equations in weighted variable Sobolev spaces

The Cauchy problem for a class of parabolic equations in weighted variable Sobolev spaces: Existence and asymptotic behavior

The paper addresses the questions of existence and asymptotic behavior of solutions to the Cauchy problem for the equationut−div(D(x)|∇u|p(x)−2∇u)+A(x)|u|q(x)−2u=f(x,t,u). The coefficients D, A are nonnegative functions which may vanish on a set of zero measure in Rn, and A(x)→∞ as |x|→∞, f(x,t,u) is globally Lipschitz with respect to u. The exponents p,q:Rn↦(1,∞) are given measurable functions. We prove that the problem admits at least one weak solution in a weighted Sobolev space with variable exponents, provided that p−=essinfRn⁡p(x)>max⁡{2nn+2,1}, q−=essinfRn⁡q(x)>2, A−2q(x)−2∈L1(Rn) and D−sp(x)−s∈L1(BR1(0)) with constants max⁡{1,2nn+2}<s<min⁡{p−,q−} and R1>0. In the case p−>2, q(x)=p(x) a.e. in Rn, and f≡f(u), there exists a unique strong solution and the problem has a global attractor in L2(Rn).

Blow-up properties in the parabolic problems with anisotropic nonstandard growth conditions

In this paper, we study the parabolic problems with anisotropic nonstandard growth nonlinearities. We first give the existence and uniqueness of weak solutions in variable Sobolev spaces. Second, we use the energy methods to show the existence of blow-up solutions with negative or positive initial energy, respectively. Both the variable exponents and the coefficients make important roles in Fujita blow-up phenomena. Moreover, asymptotic properties of the blow-up solutions are determined.

Fourier multiplier theorems for Besov and Triebel-Lizorkin spaces with variable exponents

In this paper, we will prove Fourier multiplier theorems on Besov and Triebel–Lizorkin spaces with variable exponents. It was shown by many authors that variable Triebel–Lizorkin spaces coincide with variable Bessel potential spaces, variable Sobolev spaces and variable Lebesgue spaces when appropriate indices are chosen. In consequence of the results, we also have Fourier multiplier theorems on these variable function spaces. Mathematics subject classification (2010): 42B15.

Obstacle problems in variable domains

A series of results related to the study of variational problems with unilateral and bilateral obstacles for integral functionals defined on nonweighted and weighted variable Sobolev spaces is given.

Pointwise Inequalities in Variable Sobolev Spaces and Applications

Pointwise estimates for variable exponent Sobolev functions are derived to obtain several results on Sobolev spaces with variable exponent. Hypersingular operators acting in these spaces are considered and the corresponding boundedness and pointwise statements are given over bounded open sets with Lipschitz boundary. Moreover, classical Sobolev embeddings into Hölder spaces are generalized to the variable exponent setting.

Approximate identities in variable Lp spaces

AbstractWe give conditions for the convergence of approximate identities, both pointwise and in norm, in variable L p spaces. We unify and extend results due to Diening [8], Samko [18] and Sharapudinov [19]. As applications, we give criteria for smooth functions to be dense in the variable Sobolev spaces, and we give solutions of the Laplace equation and the heat equation with boundary values in the variable L p spaces. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Characterization of Riesz and Bessel potentials on variable Lebesgue spaces

Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.

Two‐scale convergence with respect to measures and homogenization of monotone operators

In 1989 Nguetseng introduced two‐scale convergence, which now is a frequently used tool in homogenization of partial differential operators. In this paper we discuss the notion of two‐scale convergence with respect to measures. We make an exposition of the basic facts of this theory and develope it in various ways. In particular, we consider both variable Lp spaces and variable Sobolev spaces. Moreover, we apply the results to a homogenization problem connected to a class of monotone operators.