A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 818-838.

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order $\frac{1}{\sqrt{\epsilon }}$ concentrated on an $\epsilon$-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

DOI : https://doi.org/10.1051/cocv:2008048
Classification : 49Q20,  28A33
Mots clés : gradient Young measures, concentration measures, minimization problems, quasiconvexity
@article{COCV_2009__15_4_818_0,
author = {Croce, Gisella and Lacour, Catherine and Michaille, G\'erard},
title = {A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {818--838},
publisher = {EDP-Sciences},
volume = {15},
number = {4},
year = {2009},
doi = {10.1051/cocv:2008048},
zbl = {1175.49039},
mrnumber = {2567247},
language = {en},
url = {http://www.numdam.org/item/COCV_2009__15_4_818_0/}
}
Croce, Gisella; Lacour, Catherine; Michaille, Gérard. A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 4, pp. 818-838. doi : 10.1051/cocv:2008048. http://www.numdam.org/item/COCV_2009__15_4_818_0/

[1] H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series. Pitman Advanced Publishing Program (1985). | MR 773850 | Zbl 0561.49012

[2] H. Federer, Geometric Measure Theory, Classic in Mathematics. Springer-Verlag (1969). | MR 257325 | Zbl 0176.00801

[3] I. Fonseca, S. Müller and P. Pedregal, Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal. 29 (1998) 736-756. | MR 1617712 | Zbl 0920.49009

[4] D. Kinderlehrer and P. Pedregal, Characterization of Young measures generated by gradients. Arch. Rational Mech. Anal. 119 (1991) 329-365. | MR 1120852 | Zbl 0754.49020

[5] C. Licht and G. Michaille, A modelling of elastic adhesive bonded joints. Adv. Math. Sci. Appl. 7 (1997) 711-740. | MR 1476274 | Zbl 0892.73007

[6] C. Licht, G. Michaille and S. Pagano, A model of elastic adhesive bonded joints through oscillation-concentration measures. J. Math. Pures Appl. 87 (2007) 343-365. | MR 2317338 | Zbl 1118.49009