On the role of lower bounds in characterizations of weak lower semicontinuity of multiple integrals

2010 ◽  
Vol 3 (4) ◽  
Author(s):  
Stefan Krömer
Keyword(s):  
Lower Bounds ◽  
Lower Semicontinuity ◽  
Weak Lower Semicontinuity ◽  
Multiple Integrals

Weak lower semicontinuity of polyconvex integrals

1993 ◽  
Vol 123 (4) ◽  
pp. 681-691 ◽  
Author(s):  
Jan Malý
Keyword(s):  
Open Subset ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Weak Lower Semicontinuity ◽  
Multiple Integrals

SynopsisMultiple integrals with polyconvex integrands are studied on the class of all sense-preserving diffeomorphisms from W1,p(Ω, Rn) where Ω is an open subset of Rn. They are proved to be sequentially weakly lower semicontinuous if 1 < p = n –1. An example is presented showing that a similar result is not valid if p <n –1.

Lower semicontinuity of multiple integrals and the Biting Lemma

1990 ◽  
Vol 114 (3-4) ◽  
pp. 367-379 ◽  
Author(s):  
J. M. Ball ◽  
K.-W. Zhang
Keyword(s):  
Integral Representation ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Young Measure ◽  
Weak Limit ◽  
Periodic Case ◽  
Quasiconvex Functions ◽  
General Sequence ◽  
Weak Lower Semicontinuity ◽  
Multiple Integrals

SynopsisWeak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

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 On lower semicontinuity of multiple integrals

1997 ◽  
Vol 74 (1) ◽  
pp. 71-78 ◽  
Author(s):  
Agnieszka Kałamajska
Keyword(s):  
Lower Semicontinuity ◽  
Multiple Integrals

A note on relaxation with constraints on the determinant

2019 ◽  
Vol 25 ◽  
pp. 41 ◽  
Author(s):  
Marco Cicalese ◽  
Nicola Fusco
Keyword(s):  
Calculus Of Variations ◽  
Upper Bound ◽  
Lower Semicontinuity ◽  
Carathéodory Function ◽  
Incompressibility Constraint ◽  
Multiple Integrals ◽  
Quasiconvex Envelope

We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413–437] relative to the case where W depends only on the gradient variable.

Sign in / Sign up

Export Citation Format

Share Document