A $$\texttt {p}(\cdot )$$-Poincaré-type inequality for variable exponent Sobolev spaces with zero boundary values in Carnot groups

2018 ◽  
Vol 8 (2) ◽  
pp. 289-308 ◽  
Author(s):  
Thomas Bieske ◽  
Robert D. Freeman
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Type Inequality ◽  
Carnot Groups ◽  
Boundary Values ◽  
Variable Exponent Sobolev Spaces

The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

2006 ◽  
Vol 25 (3) ◽  
pp. 205-222 ◽  
Author(s):  
Petteri Harjulehto ◽  
Peter Hästö ◽  
Mika Koskenoja ◽  
Susanna Varonen
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Dirichlet Energy ◽  
Energy Integral ◽  
Boundary Values ◽  
Variable Exponent Sobolev Spaces

scholarly journals Nonlocal Variational Principles with Variable Growth

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Yongqiang Fu ◽  
Miaomiao Yang
Keyword(s):  
Sobolev Spaces ◽  
Lower Semicontinuity ◽  
Variable Exponent ◽  
Variational Principles ◽  
Young Measures ◽  
Bounded Set ◽  
Weak Lower Semicontinuity ◽  
Variable Exponent Sobolev Spaces ◽  
Regular Open

This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.

scholarly journals Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent

2021 ◽  
Vol 7 (1) ◽  
pp. 50-65
Author(s):  
Mustapha Ait Hammou ◽  
Elhoussine Azroul
Keyword(s):  
Sobolev Spaces ◽  
Elliptic Problem ◽  
Topological Degree ◽  
Variable Exponent ◽  
Existence Of Solutions ◽  
Nonlinear Elliptic Problem ◽  
Technical Approach ◽  
Nonlinear Elliptic ◽  
Variable Exponent Sobolev Spaces ◽  
Topological Degree Method

AbstractThe aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form\left\{ {\matrix{{A\left( u \right) = f} \hfill & {in} \hfill & \Omega \hfill \cr {u = 0} \hfill & {on} \hfill & {\partial \Omega } \hfill \cr } } \right.where A(u) = −diva(x, u, ∇u) is a Leray-Lions operator and f ∈ W−1,p′(.)(Ω) with p(x) ∈ (1, ∞). Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.

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