Mathematical analysis/Partial differential equations
A note on estimates for elliptic systems with L1 data
Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 851-857.

In this paper, we give necessary and sufficient conditions on the compatibility of a kth-order homogeneous linear elliptic differential operator A and differential constraint C for solutions to

Au=fsubject toCf=0 in Rn

to satisfy the estimates

DkjuLnnj(Rn)cfL1(Rn)

for j{1,,min{k,n1}} and

DknuL(Rn)cfL1(Rn)

when kn.

Dans cet article, nous donnons des conditions nécessaires et suffisantes sur la compatibilité d'un opérateur différentiel elliptique linéaire homogène A d'ordre k et d'une contrainte différentielle C pour que les solutions de

Au=fsujet àCf=0 dans Rn

vérifient les inégalités

DkjuLnnj(Rn)cfL1(Rn)
pour j{1,,min{k,n1}} et
DknuL(Rn)cfL1(Rn)
si kn.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2019.11.007
Raita, Bogdan 1; Spector, Daniel 2, 3

1 Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany
2 Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan
3 Nonlinear Analysis Unit, Okinawa Institute of Science and Technology Graduate University, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, Japan
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     author = {Raita, Bogdan and Spector, Daniel},
     title = {A note on estimates for elliptic systems with {\protect\emph{L}\protect\textsuperscript{1}} data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {851--857},
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Raita, Bogdan; Spector, Daniel. A note on estimates for elliptic systems with L1 data. Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 851-857. doi : 10.1016/j.crma.2019.11.007. http://www.numdam.org/articles/10.1016/j.crma.2019.11.007/

[1] Bourgain, J.; Brezis, H. On the equation divY=f and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003) no. 2, pp. 393-426

[2] Bourgain, J.; Brezis, H. New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc., Volume 9 (2007) no. 2, pp. 277-315

[3] Bousquet, P.; Mironescu, P. An elementary proof of an inequality of Maz'ya involving L1-vector fields, Contemp. Math., Volume 540 (2011), pp. 59-63

[4] Bousquet, P.; Van Schaftingen, J. Hardy–Sobolev inequalities for vector fields and canceling linear differential operators, Indiana Univ. Math. J. (2014), pp. 1419-1445

[5] Guerra, A.; Raita, B. Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints, 2019 (arXiv preprint) | arXiv

[6] Hörmander, L. The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Springer, 2015

[7] Maz'ya, V. Estimates for differential operators of vector analysis involving L1-norm, J. Eur. Math. Soc., Volume 12 (2009) no. 1, pp. 221-240

[8] Raita, B. Critical Lp-differentiability of BVA-maps and canceling operators, Trans. Amer. Math. Soc., Volume 372 (2019), pp. 7297-7326 | DOI

[9] Smith, K.T. Formulas to represent functions by their derivatives, Math. Ann., Volume 188 (1970) no. 1, pp. 53-77

[10] Van Schaftingen, J. Estimates for L1-vector fields, C. R. Acad. Sci. Paris, Ser. I, Volume 339 (2004) no. 3, pp. 181-186

[11] Van Schaftingen, J. Function spaces between BMO and critical Sobolev spaces, J. Funct. Anal., Volume 236 (2006) no. 2, pp. 490-516

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

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