On a necessary condition in the calculus of variations in Sobolev spaces with variable exponent

2014 ◽  
Vol 41 (2) ◽  
pp. 165-174
Author(s):  
Elhoussine Azroul ◽  
Meryem El Lekhlifi ◽  
Badr Lahmi ◽  
Abdelfattah Touzani
Keyword(s):  
Calculus Of Variations ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Necessary 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.

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.

The general double integral problem in the calculus of variations

1933 ◽  
Vol 29 (2) ◽  
pp. 207-211
Author(s):  
R. P. Gillespie
Keyword(s):  
Calculus Of Variations ◽  
General Problem ◽  
Necessary Condition ◽  
Double Integral ◽  
Integral Problem ◽  
Image Position

In a previous paper in these Proceedings the problem of the double integralwas discussed when the function F had the formwhereIt is proposed in the present paper to extend the method to the general problem, where F may have any form provided only that it satisfies the necessary condition of being homogeneous of the first degree in A, B, C.

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