Ultra-parabolic H-measures and compensated compactness
Panov, E.Yu.
Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011), p. 47-62 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Nous présentons ici une généralisation de la théorie de la « compacité par compensation ». Le cas d'une forme quadratique et de contraintes différentielles avec coefficients variables, éventuellement discontinus en espace, est considéré. Ces contraintes différentielles peuvent être d'ordre un, mais aussi d'ordre deux. Notre outil principal est le principe de localisation pour les H-mesures ultra-paraboliques associées à des suites de fonctions faiblement convergentes.

We present a generalization of compensated compactness theory to the case of variable and generally discontinuous coefficients, both in the quadratic form and in the linear, up to the second order, constraints. The main tool is the localization properties for ultra-parabolic H-measures corresponding to weakly convergent sequences.

Publié le : 2011-01-01
DOI : https://doi.org/10.1016/j.anihpc.2010.10.002
Classification:  35A27,  35K55,  46G10,  42B15,  42B30 
@article{AIHPC_2011__28_1_47_0,
     author = {Panov, E.Yu.},
     title = {Ultra-parabolic H-measures and compensated compactness},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {28},
     year = {2011},
     pages = {47-62},
     doi = {10.1016/j.anihpc.2010.10.002},
     mrnumber = {2765509},
     zbl = {1211.35013},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2011__28_1_47_0}
}

Panov, E.Yu. Ultra-parabolic H-measures and compensated compactness. Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) pp. 47-62. doi : 10.1016/j.anihpc.2010.10.002. http://gdmltest.u-ga.fr/item/AIHPC_2011__28_1_47_0/

[1] N. Antonić, M. Lazar, H-measures and variants applied to parabolic equations, J. Math. Anal. Appl. 343 (2008), 207-225 | MR 2412122 | Zbl 1139.35310

[2] J. Bergh, J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin (1980) | MR 482275

[3] G.A. Francfort, An introduction to H-measures and their applications, G. Dal Maso, A. Desimone, Antonio F. Tomarelli (ed.), Variational Problems in Materials Science, Progr. Nonlinear Differential Equations Appl. vol. 68, Birkhäuser Verlag, Basel (2006), 85-110

[4] P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), 1761-1794 | MR 1135919 | Zbl 0770.35001

[5] F. Murat, Compacité par compensation, Ann. Sc. Norm. Super. Pisa 5 (1978), 489-507 | Numdam | MR 506997 | Zbl 0399.46022

[6] E.Yu. Panov, Ultra-parabolic equations with rough coefficients. Entropy solutions and strong precompactness property, J. Math. Sci. 159 no. 2 (2009), 180-228 | MR 2544036 | Zbl 1207.35199

[7] E.Yu. Panov, On the strong pre-compactness property for entropy solutions of a degenerate elliptic equation with discontinuous flux, J. Differential Equations 247 no. 10 (2009), 2821-2870 | MR 2568159 | Zbl 1185.35100

[8] P. Pedregal, Parametrized Measures and Variational Principles, Progr. Nonlinear Differential Equations Appl. vol. 30, Birkhäuser Verlag, Basel (1997) | MR 1452107 | Zbl 0879.49017

[9] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. vol. 30, Princeton University Press, Princeton, NJ (1970) | MR 290095 | Zbl 0207.13501

[10] L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear Analysis and Mechanics: Heriot. Watt Symposium, vol. 4, Edinburgh, 1979, Res. Notes Math. vol. 39 (1979), 136-212 | MR 584398 | Zbl 0437.35004

[11] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 no. 3–4 (1990), 193-230 | MR 1069518 | Zbl 0774.35008