A note about weak * lower semicontinuity for functionals with linear growth in W 1,1 × L 1

2017 ◽  
Vol 3 (1-2) ◽  
pp. 93-103
Author(s):  
Elvira Zappale ◽  
Hamdi Zorgati
Keyword(s):  
Lower Semicontinuity ◽  
Linear Growth ◽  
Weak Lower Semicontinuity

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.

Weak lower semicontinuity for polyconvex integrals in the limit case

2013 ◽  
Vol 51 (1-2) ◽  
pp. 171-193 ◽  
Author(s):  
M. Focardi ◽  
N. Fusco ◽  
C. Leone ◽  
P. Marcellini ◽  
E. Mascolo ◽  
...  
Keyword(s):  
Lower Semicontinuity ◽  
Limit Case ◽  
Weak Lower Semicontinuity

scholarly journals Weak lower semicontinuity for non coercive polyconvex integrals

2008 ◽  
Vol 1 (2) ◽  
Author(s):  
Micol Amar ◽  
Virginia De Cicco ◽  
Paolo Marcellini ◽  
Elvira Mascolo
Keyword(s):  
Lower Semicontinuity ◽  
Weak Lower Semicontinuity

scholarly journals Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

2020 ◽  
Vol 13 (3) ◽  
pp. 219-255 ◽  
Author(s):  
Adolfo Arroyo-Rabasa ◽  
Guido De Philippis ◽  
Filip Rindler
Keyword(s):  
Lower Semicontinuity ◽  
Generalized Convexity ◽  
Linear Growth ◽  
Arbitrary Order ◽  
Integral Functionals ◽  
Vector Measures ◽  
First Order ◽  
Linear Pde ◽  
Side Constraints ◽  
Linear Pdes

AbstractWe show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

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