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 Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

scholarly journals Sobolev spaces and hyperbolic fillings

2018 ◽  
Vol 2018 (737) ◽  
pp. 161-187 ◽  
Author(s):  
Mario Bonk ◽  
Eero Saksman
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Weak Type ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Fractional Sobolev Space ◽  
Gradient Condition

AbstractLetZbe an AhlforsQ-regular compact metric measure space, where{Q>0}. For{p>1}we introduce a new (fractional) Sobolev space{A^{p}(Z)}consisting of functions whose extensions to the hyperbolic filling ofZsatisfy a weak-type gradient condition. IfZsupports aQ-Poincaré inequality with{Q>1}, then{A^{Q}(Z)}coincides with the familiar (homogeneous) Hajłasz–Sobolev space.

scholarly journals Poincaré inequality and Hajłasz–Sobolev spaces on nested fractals

2013 ◽  
Vol 218 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Katarzyna Pietruska-Pałuba ◽  
Andrzej Stós
Keyword(s):  
Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality

Extension Theorems on Weighted Sobolev Spaces and Some Applications

2006 ◽  
Vol 58 (3) ◽  
pp. 492-528 ◽  
Author(s):  
Seng-Kee Chua
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Extension Theorems ◽  
Doubling Weight ◽  
Interpolation Inequalities

AbstractWe extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

scholarly journals Poincaré Inequality for Weighted First Order Sobolev Spaces on Loop Spaces

2001 ◽  
Vol 185 (2) ◽  
pp. 527-563 ◽  
Author(s):  
Fuzhou Gong ◽  
Michael Röckner ◽  
Wu Liming
Keyword(s):  
Sobolev Spaces ◽  
Poincaré Inequality ◽  
Loop Spaces ◽  
Poincare Inequality ◽  
First Order

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.

On Weighted Sobolev Spaces

1996 ◽  
Vol 48 (3) ◽  
pp. 527-541 ◽  
Author(s):  
Seng-Kee Chua
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Extension Theorems ◽  
Doubling Weight ◽  
Weighted Poincaré Inequality

AbstractWe study density and extension problems for weighted Sobolev spaces on bounded (ε, δ) domains𝓓when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of 𝓓 and when it is in the MuckenhouptApclass locally in 𝓓. Moreover, when the weightswi(x) are of the form dist(x,Mi)αi,αi∈ ℝ,Mi⊂ 𝓓that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.

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.

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