scholarly journals Sobolev capacity on the spaceW1, p(⋅)(ℝn)

2003 ◽  
Vol 1 (1) ◽  
pp. 17-33 ◽  
Author(s):  
Petteri Harjulehto ◽  
Peter Hästö ◽  
Mika Koskenoja ◽  
Susanna Varonen
Keyword(s):  
Hausdorff Dimension ◽  
Sobolev Space ◽  
Variable Exponent ◽  
Choquet Capacity ◽  
Sobolev Capacity

We define Sobolev capacity on the generalized Sobolev spaceW1, p(⋅)(ℝn). It is a Choquet capacity provided that the variable exponentp:ℝn→[1,∞)is bounded away from 1 and∞. We discuss the relation between the Hausdorff dimension and the Sobolev capacity. As another application we study quasicontinuous representatives in the spaceW1, p(⋅)(ℝn).

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).

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 The variable exponent BV-Sobolev capacity

2012 ◽  
Vol 27 (1) ◽  
pp. 13-40 ◽  
Author(s):  
Heikki Hakkarainen ◽  
Matti Nuortio
Keyword(s):  
Variable Exponent ◽  
Sobolev Capacity

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 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.

FRÉCHET DIFFERENTIABILITY OF THE NORM IN A SOBOLEV SPACE WITH A VARIABLE EXPONENT

2013 ◽  
Vol 11 (04) ◽  
pp. 1350012 ◽  
Author(s):  
PHILIPPE G. CIARLET ◽  
GEORGE DINCA ◽  
PAVEL MATEI
Keyword(s):  
Sobolev Space ◽  
Banach Spaces ◽  
Explicit Form ◽  
Variable Exponent ◽  
Measurable Subset ◽  
Uniformly Convex ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Differentiable Norm ◽  
Fréchet Differentiable

Let Ω be a domain in ℝN, let [Formula: see text] be such that p(x) > 1 for all [Formula: see text], let W1,p(⋅) (Ω) be the Sobolev space with variable exponent p(⋅), let Γ0 be a dΓ-measurable subset of Γ = ∂Ω that satisfies dΓ-meas Γ0 > 0, and let UΓ0 = {u ∈ W1,p(⋅)(Ω); tr u = 0 on Γ0}. It is shown that the map u ∈ UΓ0 ↦ ‖u‖0,p(⋅), ∇ = ‖|∇u|‖0,p(⋅) is a Fréchet-differentiable norm on UΓ0, and a formula expressing the Fréchet derivative of this norm at any nonzero u ∈ UΓ0 is given. We also show that, if p(x) ≥ 2 for all [Formula: see text], (UΓ0, ‖u‖0,p(⋅), ∇) is uniformly convex. Using properties of duality mappings defined on Banach spaces having a Fréchet-differentiable norm, we give the explicit form of continuous linear functionals on (UΓ0, ‖u‖0,p(⋅), ∇). It is also shown that the space UΓ0 and its dual have the same Krein–Krasnoselski–Milman dimension.

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