On the lower semicontinuity of supremal functional under differential constraints

Ansini, Nadia; Prinari, Francesca

doi:10.1051/cocv/2014058

Ansini, Nadia ^1,² ; Prinari, Francesca ³

¹ Dip. di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy.
² Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
³ Dip. di Matematica e Informatica, Università di Ferrara, Via Machiavelli 35, 44121 Ferrara, Italy.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1053-1075.

Résumé

We study the weak* lower semicontinuity of functionals of the form

F (V) =_{x \in} f (x, V (x)),

where $Ω \subset ℝ^{N}$ is a bounded open set, $V \in L^{\infty} (Ω;) \cap Ker$ and �� is a constant-rank partial differential operator. The notion of ��-Young quasiconvexity, which is introduced here, provides a sufficient condition when $f (x, \cdot)$ is only lower semicontinuous. We also establish necessary conditions for weak* lower semicontinuity. Finally, we discuss the divergence and curl-free cases and, as an application, we characterise the strength set in the context of electrical resistivity.

Reçu le : 2014-05-05

MR Zbl

DOI : 10.1051/cocv/2014058

Classification : 49J45, 35E99
Mots clés : Supremal functionals, Γ-convergence, L^p-approximation, lower semicontinuity, 𝒜-quasiconvexity

Affiliations des auteurs :

Ansini, Nadia ^{1, 2} ; Prinari, Francesca ³

@article{COCV_2015__21_4_1053_0,
     author = {Ansini, Nadia and Prinari, Francesca},
     title = {On the lower semicontinuity of supremal functional under differential constraints},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {1053--1075},
     publisher = {EDP-Sciences},
     volume = {21},
     number = {4},
     year = {2015},
     doi = {10.1051/cocv/2014058},
     mrnumber = {3395755},
     zbl = {1336.49015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2014058/}
}

TY  - JOUR
AU  - Ansini, Nadia
AU  - Prinari, Francesca
TI  - On the lower semicontinuity of supremal functional under differential constraints
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2015
SP  - 1053
EP  - 1075
VL  - 21
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2014058/
DO  - 10.1051/cocv/2014058
LA  - en
ID  - COCV_2015__21_4_1053_0
ER  -

%0 Journal Article
%A Ansini, Nadia
%A Prinari, Francesca
%T On the lower semicontinuity of supremal functional under differential constraints
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2015
%P 1053-1075
%V 21
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2014058/
%R 10.1051/cocv/2014058
%G en
%F COCV_2015__21_4_1053_0

Ansini, Nadia; Prinari, Francesca. On the lower semicontinuity of supremal functional under differential constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 4, pp. 1053-1075. doi : 10.1051/cocv/2014058. http://www.numdam.org/articles/10.1051/cocv/2014058/

Bibliographie
Cité par

E. Acerbi, G. Buttazzo and F. Prinari, The class of functionals which can be represented by a supremum. J. Convex Anal. 9 (2002) 225–236. | MR | Zbl

N. Ansini and A. Garroni, $Γ$ -convergence of Functionals on Divergence Free Fields. ESAIM: COCV 13 (2007) 809–828. | Numdam | MR | Zbl

N. Ansini and F. Prinari, Power-law approximation under differential constraint. SIAM J. Math. Anal. 46 (2014) 1085–1115. | DOI | MR | Zbl

J.M. Ball, A version of the fundamental theorem for Young measures. PDE’s and Continuum Models of Phase Transitions. Edited by M. Rascle, D. Serre and M. Slemrod. In vol. 344 of Lect. Notes Phys. Springer-Verlag, Berlin (1989) 207–215. | MR | Zbl

E.N. Barron and W. Liu, Calculus of Variation in $L^{\infty}$ . Appl. Math. Optim. 35 (1997) 237–263. | MR | Zbl

E.N. Barron, R. Jensen and W. Liu, Hopf-Lax type formula for $u_{t} + H (u, d u) = 0$ . J. Differ. Eq. 126 (1996) 48–61. | DOI | MR | Zbl

E.N. Barron, R. Jensen and C.Y. Wang, Lower semicontinuity of $L^{\infty}$ functionals. Ann. Inst. Henri Poincaré 4 (2001) 495–517. | DOI | Numdam | MR | Zbl

M. Bocea and V. Nesi, $Γ$ -convergence of power-law functionals, variational principles in $L^{\infty}$ and applications. SIAM J. Math. Anal. 39 (2008) 1550–1576. | DOI | MR | Zbl

H. Berliocchi and J.M. Lasry, Intégral normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France 101 (1973) 129–184. | DOI | Numdam | MR | Zbl

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998). | MR | Zbl

A. Braides, I. Fonseca and G. Leoni, ��-Quasiconvexity: Relaxation and Homogenization. ESAIM: COCV 5 (2000) 539–577. | Numdam | MR | Zbl

P. Cardaliaguet and F. Prinari, Supremal representation of $L^{\infty}$ functionals. Appl. Math. Optim. 52 (2005) 129–141. | DOI | MR | Zbl

T. Champion, L. De Pascale and F. Prinari, $Γ$ -convergence and absolute minimizers for supremal functionals. ESAIM: COCV 10 (2004) 14–27. | Numdam | MR | Zbl

G. Dal Maso, An Introduction to $Γ$ -convergence. Birkhäuser, Boston (1993). | MR | Zbl

I. Fonseca and S. Müller, ��-Quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. | DOI | MR | Zbl

A. Garroni, V. Nesi and M. Ponsiglione, Dielectric breakdown: optimal bounds. Proc. Roy. Soc. London A 457 (2001) 2317–2335. | DOI | MR | Zbl

F. Murat, Compacité par compensation: condition necessaire et suffisante de continuité faible sous une hypothése de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8 (1981) 68–102. | Numdam | MR | Zbl

F. Prinari, Semicontinuity and supremal representation in the Calculus of Variations. Appl. Math. Optim. 54 (2008) 111–145. | DOI | MR | Zbl

F. Prinari, Semicontinuity and relaxation of $L^{\infty}$ -functionals. Adv. Calc. Var. 2 (2009), 43–71. | DOI | MR | Zbl

F. Prinari, On the necessary condition for the lower semicontinuity of supremal functionals. In preparation.

A. Ribeiro and E. Zappale, Existence of minimizers for non-level convex supremal functionals. SIAM J. Control Optim. 52 (2014) 3341–3370. | DOI | MR | Zbl

L. Tartar, Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot-Watt Symposium. Edited by R. Knops. Longman, Harlow. Pitman Res. Notes Math. Ser. 39 (1979) 136–212. | MR | Zbl

Cité par Sources :