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.

ON MINIMIZATION OF A NON-CONVEX FUNCTIONAL IN VARIABLE EXPONENT SPACES

2014 ◽  
Vol 25 (01) ◽  
pp. 1450011 ◽  
Author(s):  
GERARDO R. CHACÓN ◽  
RENATO COLUCCI ◽  
HUMBERTO RAFEIRO ◽  
ANDRÉS VARGAS
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Direct Method ◽  
Convex Functional ◽  
Lebesgue Spaces ◽  
Variable Exponent Spaces ◽  
Existence Of Minimizers ◽  
Variable Exponent Sobolev Spaces

We study the existence of minimizers of a regularized non-convex functional in the context of variable exponent Sobolev spaces by application of the direct method in the calculus of variations. The results are new even in the framework of classical Lebesgue spaces.

The Dirichlet Problem for Harmonic Functions in the Smirnov Class with Variable Exponent

2007 ◽  
Vol 14 (2) ◽  
pp. 289-299
Author(s):  
Vakhtang Kokilashvili ◽  
Vakhtang Paatashvili
Keyword(s):  
Dirichlet Problem ◽  
Explicit Form ◽  
Growth Condition ◽  
Harmonic Functions ◽  
Variable Exponent ◽  
Domain Boundary ◽  
Problem Solution ◽  
Smirnov Class ◽  
Nonstandard Growth ◽  
Nonstandard Growth Condition

Abstract A solution of the Dirichlet problem for harmonic functions from the Smirnov class is obtained in the framework of functional spaces with a nonstandard growth condition. It is found that the domain boundary geometry influences the character of a problem solution. In the case of solvability, solutions are constructed in explicit form.

Polynomial approximation of functions in weighted Lebesgue and Smirnov spaces with nonstandard growth

2011 ◽  
Vol 18 (2) ◽  
pp. 203-235
Author(s):  
Ramazan Akgün
Keyword(s):  
Variable Exponent ◽  
Trigonometric Approximation ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Approximation Of Functions ◽  
Nonstandard Growth ◽  
Approximation Problems ◽  
Finite Domains ◽  
Weighted Variable ◽  
Polynomial Approximation Of Functions

Abstract This work deals with basic approximation problems such as direct, inverse and simultaneous theorems of trigonometric approximation of functions of weighted Lebesgue spaces with a variable exponent on weights satisfying a variable Muckenhoupt A p(·) type condition. Several applications of these results help us transfer the approximation results for weighted variable Smirnov spaces of functions defined on sufficiently smooth finite domains of complex plane ℂ.

scholarly journals Nontrivial Solution for a Nonlocal Elliptic Transmission Problem in Variable Exponent Sobolev Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Bilal Cekic ◽  
Rabil A. Mashiyev
Keyword(s):  
Nontrivial Solution ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Variable Exponent ◽  
Nonlinear Elliptic Equations ◽  
Transmission Problem ◽  
Variational Techniques ◽  
Nonlinear Elliptic ◽  
Variable Exponent Sobolev Spaces ◽  
Nonstandard Growth Condition

In this paper, by means of adequate variational techniques and the theory of the variable exponent Sobolev spaces, we show the existence of nontrivial solution for a transmission problem given by a system of two nonlinear elliptic equations ofp(x)-Kirchhoff type with nonstandard growth condition.

scholarly journals Weighted Variable Exponent Sobolev spaces on metric measure spaces

2018 ◽  
Vol 4 (2) ◽  
pp. 62-76
Author(s):  
Moulay Cherif Hassib ◽  
Youssef Akdim
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Metric Spaces ◽  
Maximal Inequality ◽  
Outer Measure ◽  
Sobolev Function ◽  
Convergence Results ◽  
Variable Exponent Sobolev Spaces ◽  
Measure Spaces ◽  
Weighted Variable

AbstractIn this article we define the weighted variable exponent-Sobolev spaces on arbitrary metric spaces, with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a point wise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds and that lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each weighted variable exponent-Sobolev function has a quasi-continuous representative, we study different definitions of the first order weighted variable exponent-Sobolev spaces with zero boundary values, we define the Dirichlet energy and we prove that it has a minimizer in the weighted variable exponent -Sobolev spaces case.

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