An elementary approach to the homological properties of constant-rank operators

Adolfo Arroyo-Rabasa; José Simental

doi:10.5802/crmath.388

Analyse fonctionnelle, Équations aux dérivées partielles

An elementary approach to the homological properties of constant-rank operators

Adolfo Arroyo-Rabasa ¹ ; José Simental ²

¹ Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
² Instituto de Matemáticas, Universidad Nacional Autónoma de México. Ciudad Universitaria, Mexico City, Mexico

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 45-63.

Résumé

We give a simple and constructive extension of Raiță’s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement over the order of the operators constructed by Raiță and the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over $ℝ^{d}$ , and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.

Reçu le : 2021-07-14
Révisé le : 2022-05-06
Accepté le : 2022-06-02
Publié le : 2023-01-12

DOI : 10.5802/crmath.388

Classification : 35E20, 47F10, 13D02

Affiliations des auteurs :

Adolfo Arroyo-Rabasa ¹ ; José Simental ²

¹ Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium
² Instituto de Matemáticas, Universidad Nacional Autónoma de México. Ciudad Universitaria, Mexico City, Mexico

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Adolfo Arroyo-Rabasa and Jos\'e Simental},
     title = {An elementary approach to the homological properties of constant-rank operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {45--63},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.388},
     language = {en},
}

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Adolfo Arroyo-Rabasa; José Simental. An elementary approach to the homological properties of constant-rank operators. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 45-63. doi : 10.5802/crmath.388. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.388/

Bibliographie
Cité par

[1] Shmuel Agmon; Avron Douglis; Louis Nirenberg Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., Volume 12 (1959), pp. 623-727 | DOI | MR | Zbl

[2] Shmuel Agmon; Avron Douglis; Louis Nirenberg Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Commun. Pure Appl. Math., Volume 17 (1964), pp. 35-92 | DOI | MR | Zbl

[3] Adolfo Arroyo-Rabasa Slicing and fine properties for functions with bounded $𝒜$ -variation (2020) (https://arxiv.org/abs/2009.13513) | DOI

[4] Adolfo Arroyo-Rabasa Characterization of generalized young measures generated by $𝒜$ -free measures, Arch. Ration. Mech. Anal., Volume 242 (2021) no. 1, pp. 235-325 | DOI | MR | Zbl

[5] Adolfo Arroyo-Rabasa; Guido De Philippis; Filip Rindler Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints, Adv. Calc. Var., Volume 13 (2020) no. 3, pp. 219-255 | DOI | MR | Zbl

[6] Adolfo Arroyo-Rabasa; Anna Skorobogatova A look into some of the fine properties of functions with bounded $𝒜$ -variation (2019) (https://arxiv.org/abs/1911.08474) | DOI

[7] Armand Borel; Pierre-Paul Grivel; Burchard Kaup; André Haefliger; Bernard Malgrange; F. Ehless Algebraic $D$ -modules, Perspectives in Mathematics, 2, Academic Press Inc., 1987 | MR | Zbl

[8] Dominic Breit; Lars Diening; Franz Gmeineder On the trace operator for functions of bounded $𝔸$ -variation, Anal. PDE, Volume 13 (2020) no. 2, pp. 559-594 | DOI | MR | Zbl

[9] Rida Ait El Manssour; Marc Härkönen; Bernd Sturmfels Linear PDE with constant coefficients, Glasg. Math. J. (2021), pp. 1-26 | DOI

[10] 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

[11] Franz Gmeineder; Stefan Schiffer Natural annihilators and operators of constant rank over $ℂ$ (2022) (https://arxiv.org/abs/2203.10355) | DOI

[12] Loukas Grafakos Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, 2014 | DOI | MR | Zbl

[13] André Guerra; Bogdan Raiţă On the necessity of the constant rank condition for $L^{p}$ estimates, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 9-10, pp. 1091-1095 | DOI | MR | Zbl

[14] André Guerra; Bogdan Raiţă Quasiconvexity, null Lagrangians, and Hardy space integrability under constant rank constraints, Arch. Ration. Mech. Anal., Volume 245 (2022) no. 1, pp. 279-320 | DOI | MR | Zbl

[15] André Guerra; Bogdan Raiţă; Matthew R. I. Schrecker Compensated compactness: continuity in optimal weak topologies, J. Funct. Anal., Volume 283 (2022) no. 7, 109596 | DOI | MR | Zbl

[16] Derek Gustafson A generalized Poincaré inequality for a class of constant coefficient differential operators, Proc. Am. Math. Soc., Volume 139 (2011) no. 8, pp. 2721-2728 | DOI | MR | Zbl

[17] Marc Härkönen; Jonas Hirsch; Bernd Sturmfels Making Waves (2021) (https://arxiv.org/abs/2111.14045) | DOI

[18] Marc Härkönen; Lisa Nicklasson; Bogdan Raiţă Syzygies, constant rank, and beyond (2021) (https://arxiv.org/abs/2112.12663) | DOI

[19] Ryoshi Hotta; Kiyoshi Takeuchi; Toshiyuki Tanisaki $D$ -modules, perverse sheaves, and representation theory, Progress in Mathematics, 236, Birkhäuser, 2008 (Translated from the 1995 Japanese edition by Takeuchi) | DOI | MR | Zbl

[20] Tosio Kato On a coerciveness theorem by Schulenberger and Wilcox, Indiana Univ. Math. J., Volume 24 (1974/75), pp. 979-985 | DOI | MR | Zbl

[21] Jan Kristensen; Bogdan Raiţă Oscillation and concentration in sequences of PDE constrained measures (2019) (https://arxiv.org/abs/1912.09190) | DOI

[22] Laurent Manivel Symmetric functions, Schubert polynomials and degeneracy loci, SMF/AMS Texts and Monographs, 6, American Mathematical Society; Société Mathématique de France, 2001 (Translated from the 1998 French original by John R. Swallow, Cours Spécialisés [Specialized Courses], 3) | MR | Zbl

[23] François Murat Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 8 (1981) no. 1, pp. 69-102 | MR | Zbl

[24] Julius Plücker On a new geometry of space, Royal Society, 1865

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

[26] Leonard Sarason Remarks on an inequality of Schulenberger and Wilcox, Ann. Mat. Pura Appl., Volume 92 (1972), pp. 23-28 | DOI | MR | Zbl

[27] John R. Schulenberger; Calvin H. Wilcox A coerciveness inequality for a class of nonelliptic operators of constant deficit, Ann. Mat. Pura Appl., Volume 92 (1972), pp. 77-84 | DOI | MR | Zbl

[28] Kennan T. Smith Inequalities for formally positive integro-differential forms, Bull. Am. Math. Soc., Volume 67 (1961), pp. 368-370 | DOI | MR | Zbl

[29] Kennan T. Smith Formulas to represent functions by their derivatives, Math. Ann., Volume 188 (1970), pp. 53-77 | DOI | MR | Zbl

[30] Hans Triebel Theory of function spaces, Monographs in Mathematics, 78, Birkhäuser, 1983, 284 pages | DOI | MR | Zbl

[31] Hans Triebel Tempered homogeneous function spaces, EMS Series of Lectures in Mathematics, European Mathematical Society, 2015 | DOI | MR | Zbl

[32] 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

[33] Calvin H. Wilcox A coerciveness inequality for a class of nonelliptic operators and its applications, Séminaire Goulaouic–Schwartz 1970–1971: Équations aux dérivées partielles et analyse fonctionnelle, Exp. No. 30, Centre de Math., École Polytech., Paris, 1971, 30 | MR | Zbl

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