scholarly journals Characterization of the variable exponent Sobolev norm without derivatives

2017 ◽  
Vol 19 (03) ◽  
pp. 1650022 ◽  
Author(s):  
Peter Hästö ◽  
Ana Margarida Ribeiro
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Sobolev Norm ◽  
Difference Quotient ◽  
Variable Exponent Sobolev Space ◽  
The Difference ◽  
Fractional Smoothness

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Since the difference quotient is based on shifting the function, it cannot be generalized to the variable exponent case. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the variable exponent Sobolev space.

scholarly journals Characterization of generalized Orlicz spaces

2018 ◽  
Vol 22 (02) ◽  
pp. 1850079 ◽  
Author(s):  
Rita Ferreira ◽  
Peter Hästö ◽  
Ana Margarida Ribeiro
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Orlicz Spaces ◽  
Difference Quotient ◽  
Variable Exponent Spaces ◽  
The Difference ◽  
Fractional Smoothness

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

scholarly journals Generalization of Poincar ´e inequality in a Sobolev Space with exponent constant to the case of Sobolev space with a variable exponent

2021 ◽  
Vol 10 (2) ◽  
pp. 31-37
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Ibrahim Dahi
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Equivalent Norms

In this work, we study the Poincare inequality in Sobolev spaces with variable exponent. As a consequence of this ´ result we show the equivalent norms over such cones. The approach we adopt in this work avoids the difficulty arising from the possible lack of density of the space C∞ 0 (Ω).

GEOMETRY OF SOBOLEV SPACES WITH VARIABLE EXPONENT AND A GENERALIZATION OF THE p-LAPLACIAN

2009 ◽  
Vol 07 (04) ◽  
pp. 373-390 ◽  
Author(s):  
GEORGE DINCA ◽  
PAVEL MATEI
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Uniformly Convex ◽  
Smooth Bounded Domain ◽  
Generalized Lebesgue Space

Let Ω ⊂ ℝN, N ≥ 2, be a smooth bounded domain. It is shown that: (a) if [Formula: see text] and ess inf x ∈ y p(x) > 1, then the generalized Lebesgue space (Lp (·)(Ω), ‖·‖p(·)) is smooth; (b) if [Formula: see text] and p(x) > 1, [Formula: see text], then the generalized Sobolev space [Formula: see text] is smooth. In both situations, the formulae for the Gâteaux gradient of the norm corresponding to each of the above spaces are given; (c) if [Formula: see text] and p(x) ≥ 2, [Formula: see text], then [Formula: see text] is uniformly convex and smooth.

scholarly journals On compact and bounded embedding in variable exponent Sobolev spaces and its applications

2019 ◽  
Vol 9 (2) ◽  
pp. 401-414
Author(s):  
Farman Mamedov ◽  
Sayali Mammadli ◽  
Yashar Shukurov
Keyword(s):  
Sobolev Space ◽  
Growth Condition ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Lebesgue Spaces ◽  
Nonstandard Growth ◽  
Variable Exponent Sobolev Spaces ◽  
Nonstandard Growth Condition ◽  
Nonlinear Ode ◽  
Weighted Variable

Abstract For a weighted variable exponent Sobolev space, the compact and bounded embedding results are proved. For that, new boundedness and compact action properties are established for Hardy’s operator and its conjugate in weighted variable exponent Lebesgue spaces. Furthermore, the obtained results are applied to the existence of positive eigenfunctions for a concrete class of nonlinear ode with nonstandard growth condition.

scholarly journals A Characterization of Multi-Wavelet Dual Frames in Sobolev Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Jianping Zhang ◽  
Jiajia Li
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Wavelet Frames ◽  
Dual Frames ◽  
The Past

For the past few years, wavelet and multi-wavelet frames have attracted interest from researchers. In this paper, we address some of these problems in the setting of the Sobolev space, and characterize of multi-wavelet dual frames in these spaces by using a pair of equations.

scholarly journals Characterization of Riesz and Bessel potentials on variable Lebesgue spaces

2006 ◽  
Vol 4 (2) ◽  
pp. 113-144 ◽  
Author(s):  
Alexandre Almeida ◽  
Stefan Samko
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Regularity Conditions ◽  
Bessel Potential ◽  
Lebesgue Spaces ◽  
Hypersingular Integrals ◽  
Variable Lebesgue Spaces ◽  
Bessel Potentials ◽  
Bessel Potential 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.

scholarly journals The Existence of Solutions to the NonhomogeneousA-Harmonic Equations with Variable Exponent

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Haiyu Wen
Keyword(s):  
Weak Solution ◽  
Sobolev Space ◽  
Existence And Uniqueness ◽  
Obstacle Problem ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Variable Exponent Sobolev Space ◽  
Harmonic Equation ◽  
Weighted Variable

We first discuss the existence and uniqueness of weak solution for the obstacle problem of the nonhomogeneousA-harmonic equation with variable exponent, and then we obtain the existence of the solutions of the equationd⋆A(x,dω)=B(x,dω)in the weighted variable exponent Sobolev spaceWdp(x)(Ω,Λl,μ).

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