On-line catalogue and orders (secure paiement, VISA or MASTERCARD only)

Journals available by subscription

Annales scientifiques de l'ENS

Astérisque

Bulletin de la SMF

Mémoires de la SMF

Revue d'Histoire des Mathématiques

Gazette des Mathématiciens

Books

Astérisque

Cours Spécialisés

Documents Mathématiques

Mémoires de la SMF

Panoramas et Synthèses

Séminaires et Congrès

Jean Morlet Chair Series

SMF/AMS Texts and Monographs

La Série T

Volumes "Journée Annuelle"

Other Books

Donald E. Knuth - French translations

Nicolas Bourbaki's seminar new edition

Jean Leray's scientific works new edition

Revue de l'Institut Elie Cartan

Electronic Editions

Annales scientifiques de l'ENS

Bulletin de la SMF

Revue d'Histoire des Mathématiques

Séminaires et Congrès

More information / Subscription

Publications for a general public

L'explosion des mathématiques (smf.emath.fr)

Mathématiques L'explosion continue (smf.emath.fr)

Zoom sur les métiers des maths (smf.emath.fr)

Zoom sur les métiers des mathématiques et de l'informatique (smf.emath.fr)

Où en sont les mathématiques ?

La Série T

For the authors

Submission of manuscripts

Formats and documentation

More info

Electronic distribution list (smf.emath.fr)

Information for bookselers and subscription agencies (smf.emath.fr)

Publications de la SMF
fr en
Your IP number: 54.225.20.73
Access to elec. publ.: SémCong

Annales scientifiques de l'ENS

Presentation of the publication

Titles

Last Titles

Editorial staff committee / Secretary

Serie 4:
Serie 3:
Serie 2:
Serie 1:

Search


Catalogue & orders

Annales scientifiques de l'ENS - Titles - série 4, 48 (2015)

Titles < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 4 (2015)

Pascal Auscher, Andreas Rosén, David Rule
Boundary value problems for degenerate elliptic equations and systems
Annales scientifiques de l'ENS 48, fascicule 4 (2015), 951-1000

Télécharger cet article : Fichier PDF
Buy the book

Résumé :
Problèmes aux limites pour les équations et systèmes elliptiques dégénérés
On étudie les problèmes aux limites pour les équations et systèmes elliptiques dégénérés avec données au bord de carré intégrable. La dégénérescence est contrôlée par un poids A_2. On obtient représentation et trace pour les solutions dans des classes appropriées, des résultats de perturbation pour la résolubilité, et résolubilité pour certaines situations. La méthode de travaux antérieurs par les deux premiers auteurs est adaptée à ces dégénérescences une fois démontrée une estimation quadratique et nous améliorons même certains résultats dans le cas non-dégénéré. La preuve de cette estimation quadratique ne se déduit pas en revanche de résultats antérieurs et est la partie centrale de l'article.

Mots-clefs : Estimées de Littlewood-Paley, calcul fonctionnel, problèmes aux limites, équations et systèmes elliptiques d'ordre deux, inégalités à poids.

Abstract:
We study boundary value problems for degenerate elliptic equations and systems with square integrable boundary data. We can allow for degeneracies in the form of an A_2 weight. We obtain representations and boundary traces for solutions in appropriate classes, perturbation results for solvability and solvability in some situations. The technology of earlier works of the first two authors can be adapted to the weighted setting once the needed quadratic estimate is established and we even improve some results in the unweighted setting. The proof of this quadratic estimate does not follow from earlier results on the topic and is the core of the article.

Keywords: Littlewood-Paley estimates, functional calculus, boundary value problems, second order elliptic equations and systems, weighted norm inequalities.

Class. math. : 35J56, 35J70, 35J25, 42B25, 42B37.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Albrecht, David and Duong, Xuan and McIntosh, Alan
Operator theory and harmonic analysis
in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995)
Proc. Centre Math. Appl. Austral. Nat. Univ. 34 (1996) 77–136
Math Reviews MR1394696 (97e:47001)
2
Auscher, Pascal and Axelsson, Andreas and Hofmann, Steve
Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems
J. Funct. Anal. 255 (2008) 374–448
Math Reviews MR2419965 (2009h:35079)
3
Auscher, Pascal and Axelsson, Andreas
Weighted maximal regularity estimates and solvability of non-smooth elliptic systems I
Invent. Math. 184 (2011) 47–115
Math Reviews MR2782252 (2012c:35111)
4
Auscher, Pascal and Rosén, Andreas
Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems, II
Anal. PDE 5 (2012) 983–1061
Math Reviews MR3022848
5
Auscher, Pascal and Axelsson, Andreas and McIntosh, Alan
Solvability of elliptic systems with square integrable boundary data
Ark. Mat. 48 (2009) 253–287
6
Auscher, Pascal and Axelsson, Andreas and McIntosh, Alan
On a quadratic estimate related to the Kato conjecture and boundary value problems
in Harmonic Analysis and Partial Differential Equations
Contemporary Mathematics 505 (2010) 105–129
7
Auscher, Pascal and Hofmann, Steve and Lacey, Michael and McIntosh, Alan and Tchamitchian, Philippe
The solution of the Kato square root problem for second order elliptic operators on Rn
Ann. of Math. 156 (2002) 633–654
Math Reviews MR1933726 (2004c:47096c)
8
Auscher, Pascal and McIntosh, Alan and Mourgoglou, Mihalis
On L2 solvability of BVPs for elliptic systems
J. Fourier Anal. Appl. 19 (2013) 478–494
Math Reviews MR3048587
9
Auscher, Pascal and Tchamitchian, Philippe
Square root problem for divergence operators and related topics
Soc. Math. France, 1998
Math Reviews MR1651262 (2000c:47092)
10
Axelsson, Andreas and Keith, Stephen and McIntosh, Alan
Quadratic estimates and functional calculi of perturbed Dirac operators
Invent. Math. 163 (2006) 455–497
Math Reviews MR2207232 (2007k:58029)
11
Badr, Nadine
Real interpolation of Sobolev spaces
Math. Scand. 105 (2009) 235–264
Math Reviews MR2573547 (2011c:46156)
12
Bandara, Lashi
Quadratic estimates for perturbed Dirac type operators on doubling measure metric spaces
in Proceedings of the AMSI International Conference on Harmonic Analysis and Applications
(2011) 1–21
13
Biroli, Marco and Mosco, Umberto
Wiener criterion and potential estimates for obstacle problems relative to degenerate elliptic operators
Ann. Mat. Pura Appl. 159 (1991) 255–281
Math Reviews MR1145100 (93a:35056)
14
Caffarelli, Luis and Silvestre, Luis
An extension problem related to the fractional Laplacian
Comm. Partial Differential Equations 32 (2007) 1245–1260
Math Reviews MR2354493 (2009k:35096)
15
Cruz-Uribe, David and Rios, Cristian
The Kato problem for operators with weighted ellipticity
Trans. Amer. Math. Soc. 367 (2015) 4727–4756
Math Reviews MR3335399
16
17
Dindoš, Martin and Rule, David J.
Elliptic equations in the plane satisfying a Carleson measure condition
Rev. Mat. Iberoam. 26 (2010) 1013–1034
Math Reviews MR2789374 (2012d:35057)
18
Duoandikoetxea, Javier and Rubio de Francia, José L.
Maximal and singular integral operators via Fourier transform estimates
Invent. Math. 84 (1986) 541–561
Math Reviews MR837527 (87f:42046)
19
Fabes, Eugene B. and Jerison, D. and Kenig, Carlos E.
The Wiener test for degenerate elliptic equations
Ann. Inst. Fourier (Grenoble) 32 (1982) 151–182
Math Reviews MR688024 (84g:35067)
20
Fabes, Eugene B. and Kenig, Carlos E. and Jerison, D.
Boundary behavior of solutions to degenerate elliptic equations
in Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981)
Wadsworth Math. Ser. (1983) 577–589
Math Reviews MR730093 (85m:35028)
21
Fabes, Eugene B. and Kenig, Carlos E. and Serapioni, Raul P.
The local regularity of solutions of degenerate elliptic equations
Comm. Partial Differential Equations 7 (1982) 77–116
Math Reviews MR643158 (84i:35070)
22
Franchi, B. and Serapioni, Raul P.
Pointwise estimates for a class of strongly degenerate elliptic operators: a geometrical approach
Ann. Scuola Norm. Sup. Pisa Cl. Sci. 14 (1987) 527–568
Math Reviews MR963489 (90e:35076)
23
García-Cuerva, José and Rubio de Francia, José L.
Weighted norm inequalities and related topics
North-Holland Publishing Co., Amsterdam, 1985
Math Reviews MR807149 (87d:42023)
24
Garnett, John B.
Bounded analytic functions
Academic Press, Inc., New York-London, 1981
Math Reviews MR628971 (83g:30037)
25
Grafakos, Loukas
Classical and modern Fourier analysis
Pearson Education, Inc., Upper Saddle River, NJ, 2004
Math Reviews MR2449250
26
Hajłasz, Piotr and Koskela, Pekka
Sobolev met Poincaré
Mem. Amer. Math. Soc. 145 (2000)
Math Reviews MR1683160 (2000j:46063)
27
28
Hytönen, Tuomas P.
The sharp weighted bound for general Calderón-Zygmund operators
Ann. of Math. 175 (2012) 1473–1506
Math Reviews MR2912709
29
Hytönen, Tuomas P. and Rosén, Andreas
On the Carleson duality
Ark. Mat. 51 (2013) 293–313
Math Reviews MR3090198
30
31
Kato, Tosio
Perturbation theory for linear operators
Springer New York, Inc., New York, 1966
Math Reviews MR0203473 (34 \#3324)
32
Kenig, Carlos E. and Pipher, Jill
The Neumann problem for elliptic equations with nonsmooth coefficients
Invent. Math. 113 (1993) 447–509
Math Reviews MR1231834 (95b:35046)
33
Meyer, Yves
Ondelettes et opérateurs. I
Hermann, Paris, 1990
Math Reviews MR1085487 (93i:42002)
34
Rosén, Andreas
Cauchy non-integral formulas
in Harmonic analysis and partial differential equations
Contemp. Math. 612 (2014) 163–178
Math Reviews MR3204863
35
Šneberg, I. Ja.
Spectral properties of linear operators in interpolation families of Banach spaces
Mat. Issled. 9 (1974) 214–229, 254–255
Math Reviews MR0634681 (58 \#30362)