On the necessity of the constant rank condition for $L^p$ estimates

André Guerra; Bogdan Raiţă

doi:10.5802/crmath.105

Théorie spectrale, Théorie des opérateurs

On the necessity of the constant rank condition for

L^{p}

estimates

André Guerra ¹ ; Bogdan Raiţă ²

¹ University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, United Kingdom
² Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1091-1095.

Résumé

We consider a generalization of the elliptic $L^{p}$ -estimate suited for linear operators with non-trivial kernels. A classical result of Schulenberger and Wilcox (Ann. Mat. Pura Appl. 88 (1971), no. 1, p. 229-305) shows that if the operator has constant rank then the estimate holds. We prove necessity of the constant rank condition for such an estimate.

Reçu le : 2020-07-05
Accepté le : 2020-08-10
Publié le : 2021-01-05

DOI : 10.5802/crmath.105

Classification : 26D10, 42B20

Affiliations des auteurs :

André Guerra ¹ ; Bogdan Raiţă ²

¹ University of Oxford, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, United Kingdom
² Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2020__358_9-10_1091_0,
     author = {Andr\'e Guerra and Bogdan Rai\c{t}\u{a}},
     title = {On the necessity of the constant rank condition for $L^p$ estimates},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1091--1095},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {9-10},
     year = {2020},
     doi = {10.5802/crmath.105},
     language = {en},
}

TY  - JOUR
AU  - André Guerra
AU  - Bogdan Raiţă
TI  - On the necessity of the constant rank condition for $L^p$ estimates
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 1091
EP  - 1095
VL  - 358
IS  - 9-10
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.105
LA  - en
ID  - CRMATH_2020__358_9-10_1091_0
ER  -

%0 Journal Article
%A André Guerra
%A Bogdan Raiţă
%T On the necessity of the constant rank condition for $L^p$ estimates
%J Comptes Rendus. Mathématique
%D 2020
%P 1091-1095
%V 358
%N 9-10
%I Académie des sciences, Paris
%R 10.5802/crmath.105
%G en
%F CRMATH_2020__358_9-10_1091_0

André Guerra; Bogdan Raiţă. On the necessity of the constant rank condition for $L^p$ estimates. Comptes Rendus. Mathématique, Volume 358 (2020) no. 9-10, pp. 1091-1095. doi : 10.5802/crmath.105. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.105/

Bibliographie
Cité par

[1] Jan Boman Supremum norm estimates for partial derivatives of functions of several real variables, Ill. J. Math., Volume 16 (1972) no. 2, pp. 203-216 | DOI | MR | Zbl

[2] Haim Brezis Functional analysis, Sobolev spaces and partial differential equations, Springer, 2010 | Zbl

[3] Alberto P. Calderón; Antoni Zygmund On the existence of certain singular integrals, Acta Math., Volume 88 (1952) no. 1, p. 85 | DOI | MR | Zbl

[4] Stephen L. Campbell; Carl D. Meyer Generalized inverses of linear transformations, Classics in Applied Mathematics, 56, Society for Industrial and Applied Mathematics, 2009 | MR | Zbl

[5] Henry P. Decell An application of the Cayley–Hamilton theorem to generalized matrix inversion, SIAM Rev., Volume 7 (1965) no. 4, pp. 526-528 | DOI | MR | Zbl

[6] Daniel Faraco; André Guerra A short proof of Ornstein’s non-inequality in $ℝ^{2 \times 2}$ (2020) (preprint, https://arxiv.org/abs/2006.09060)

[7] Irene Fonseca; Stefan Müller $𝒜$ -Quasiconvexity, Lower Semicontinuity, and Young Measures, SIAM J. Math. Anal., Volume 30 (1999) no. 6, pp. 1355-1390 | DOI | MR | Zbl

[8] André Guerra; Bogdan Raiţă Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints (2019) (preprint, https://arxiv.org/abs/1909.03923)

[9] Tosio Kato On a coerciveness theorem by Schulenberger and Wilcox, Indiana Univ. Math. J., Volume 24 (1975) no. 10, pp. 979-985 | DOI | MR | Zbl

[10] Bernd Kirchheim; Jan Kristensen On rank one convex functions that are homogeneous of degree one, Arch. Ration. Mech. Anal., Volume 221 (2016) no. 1, pp. 527-558 | DOI | MR | Zbl

[11] Karel de Leeuw; Hazleton Mirkil A priori estimates for differential operators in $L_{\infty}$ norm, Ill. J. Math., Volume 8 (1964) no. 1, pp. 112-114 | DOI | MR | Zbl

[12] Chun Li; Alan McIntosh; Kewei Zhang; Zhijian Wu Compensated compactness, paracommutators, and Hardy spaces, J. Funct. Anal., Volume 150 (1997) no. 2, pp. 289-306 | MR

[13] Boris Samuilovich Mityagin On second mixed derivative, Dokl. Akad. Nauk SSSR, Volume 123 (1958) no. 4, pp. 606-609 | MR

[14] Stefan Müller Rank-one convexity implies quasiconvexity on diagonal matrices, Int. Math. Res. Not., Volume 1999 (1999) no. 20, pp. 1087-1095 | DOI | MR | Zbl

[15] François Murat Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothese de rang constant, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, Volume 8 (1981) no. 1, pp. 69-102 | Numdam | Zbl

[16] Donald Ornstein A non-inequality for differential operators in the L 1 norm, Arch. Ration. Mech. Anal., Volume 11 (1962) no. 1, pp. 40-49 | DOI | MR | Zbl

[17] Bogdan Raiţă $L^{1}$ -estimates for constant rank operators (2018) (preprint, https://arxiv.org/abs/1811.10057)

[18] Bogdan Raiţă Potentials for $𝒜$ -quasiconvexity, Calc. Var. Partial Differ. Equ., Volume 58 (2019) no. 3, p. 105 | DOI | MR | Zbl

[19] John R. Schulenberger; Calvin H. Wilcox Coerciveness inequalities for nonelliptic systems of partial differential equations, Ann. Mat. Pura Appl., Volume 88 (1971) no. 1, pp. 229-305 | DOI | MR | Zbl

[20] Luc Tartar Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot–Watt symposium. Vol. IV (Research Notes in Mathematics), Volume 39 (1979), pp. 136-212 | MR | Zbl

[21] Jean Van Schaftingen Limiting Sobolev inequalities for vector fields and canceling linear differential operators, J. Eur. Math. Soc., Volume 15 (2013) no. 3, pp. 877-921 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique