A note on estimates for elliptic systems with <i>L</i><sup>1</sup> data

Bogdan Raita; Daniel Spector

doi:10.1016/j.crma.2019.11.007

Mathematical analysis/Partial differential equations

A note on estimates for elliptic systems with L¹ data

Presented by: Haïm Brézis

Bogdan Raita ¹; Daniel Spector ^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

Comptes Rendus. Mathématique, Volume 357 (2019) no. 11-12, pp. 851-857.

Abstract
Résumé

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

A u = f subject to C f = 0 in R^{n}

to satisfy the estimates

{‖ D^{k - j} u ‖}_{L^{\frac{n}{n - j}} (R^{n})} ⩽ c {‖ f ‖}_{L^{1} (R^{n})}

for $j \in {1, \dots, \min {k, n - 1}}$ and

{‖ D^{k - n} u ‖}_{L^{\infty} (R^{n})} ⩽ c {‖ f ‖}_{L^{1} (R^{n})}

when $k \geq n$ .

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

A u = f sujet à C f = 0 dans R^{n}

vérifient les inégalités

{‖ D^{k - j} u ‖}_{L^{\frac{n}{n - j}} (R^{n})} ⩽ c {‖ f ‖}_{L^{1} (R^{n})}

pour

j \in {1, \dots, \min {k, n - 1}}

et

{‖ D^{k - n} u ‖}_{L^{\infty} (R^{n})} ⩽ c {‖ f ‖}_{L^{1} (R^{n})}

si

k \geq n

.

Received: 2019-06-27
Accepted: 2019-11-18
Published online: 2019-11-01

DOI: 10.1016/j.crma.2019.11.007

Author's affiliations:

Bogdan Raita ¹; Daniel Spector ^{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

@article{CRMATH_2019__357_11-12_851_0,
     author = {Bogdan Raita and Daniel Spector},
     title = {A note on estimates for elliptic systems with {\protect\emph{L}\protect\textsuperscript{1}} data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {851--857},
     publisher = {Elsevier},
     volume = {357},
     number = {11-12},
     year = {2019},
     doi = {10.1016/j.crma.2019.11.007},
     language = {en},
}

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Bogdan Raita; Daniel Spector. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.11.007/

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