A selection lemma for sequences of measurable sets, and lower semicontinuity of multiple integrals

1979 ◽  
Vol 27 (1) ◽  
pp. 73-79 ◽  
Author(s):  
Goswin Eisen
Keyword(s):  
Lower Semicontinuity ◽  
Multiple Integrals ◽  
Selection Lemma

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.

On lower semicontinuity and relaxation

2001 ◽  
Vol 131 (3) ◽  
pp. 519-565 ◽  
Author(s):  
Irene Fonseca ◽  
Giovanni Leoni
Keyword(s):  
Energy Density ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Multiple Integrals ◽  
Image Position

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

On lower semicontinuity and relaxation

2001 ◽  
Vol 131 (3) ◽  
pp. 519-565
Author(s):  
Irene Fonseca ◽  
Giovanni Leoni
Keyword(s):  
Energy Density ◽  
Calculus Of Variations ◽  
Lower Semicontinuity ◽  
Multiple Integrals ◽  
Image Position

Lower semicontinuity and relaxation results in BV are obtained for multiple integrals where the energy density f satisfies lower semicontinuity conditions with respect to (x, u) and is not subjected to coercivity hypotheses. These results call for methods recently developed in the calculus of variations.

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.

Sign in / Sign up

Export Citation Format

Share Document