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Some new results on differential inclusions for differential forms


Authors: Saugata Bandyopadhyay, Bernard Dacorogna and Olivier Kneuss
Journal: Trans. Amer. Math. Soc. 367 (2015), 3119-3138
MSC (2010): Primary 35F60
DOI: https://doi.org/10.1090/S0002-9947-2014-06014-5
Published electronically: December 3, 2014
MathSciNet review: 3314803
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Abstract | References | Similar Articles | Additional Information

Abstract: In this article we study some necessary and sufficient conditions for the existence of solutions in $ W_{0}^{1,\infty }(\Omega ;\Lambda ^{k})$ of the differential inclusion

$\displaystyle d\omega \in E$$\displaystyle \quad \text {a.e. in }\Omega $

where $ E\subset \Lambda ^{k+1}$ is a prescribed set.

References [Enhancements On Off] (What's this?)

  • [1] Saugata Bandyopadhyay, Ana Cristina Barroso, Bernard Dacorogna, and José Matias, Differential inclusions for differential forms, Calc. Var. Partial Differential Equations 28 (2007), no. 4, 449-469. MR 2293981 (2007m:35297), https://doi.org/10.1007/s00526-006-0049-6
  • [2] Ana Cristina Barroso and José Matias, Necessary and sufficient conditions for existence of solutions of a variational problem involving the curl, Discrete Contin. Dyn. Syst. 12 (2005), no. 1, 97-114. MR 2121251 (2005k:49004)
  • [3] Alberto Bressan and Fabián Flores, On total differential inclusions, Rend. Sem. Mat. Univ. Padova 92 (1994), 9-16. MR 1320474 (96b:35244)
  • [4] Arrigo Cellina, On minima of a functional of the gradient: necessary conditions, Nonlinear Anal. 20 (1993), no. 4, 337-341. MR 1206422 (94b:49036a), https://doi.org/10.1016/0362-546X(93)90137-H
  • [5] Arrigo Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlinear Anal. 20 (1993), no. 4, 343-347. MR 1206423 (94b:49036b), https://doi.org/10.1016/0362-546X(93)90138-I
  • [6] Gyula Csató, Bernard Dacorogna, and Olivier Kneuss, The pullback equation for differential forms, Progress in Nonlinear Differential Equations and their Applications, 83, Birkhäuser/Springer, New York, 2012. MR 2883631
  • [7] Bernard Dacorogna, Direct methods in the calculus of variations, second edition, Springer-Verlag, Berlin, 2007.
  • [8] Bernard Dacorogna and Irene Fonseca, A-B quasiconvexity and implicit partial differential equations, Calc. Var. Partial Differential Equations 14 (2002), no. 2, 115-149. MR 1890397 (2003e:35037), https://doi.org/10.1007/s005260100092
  • [9] Bernard Dacorogna and Paolo Marcellini, General existence theorems for Hamilton-Jacobi equations in the scalar and vectorial cases, Acta Math. 178 (1997), no. 1, 1-37. MR 1448710 (98d:35029), https://doi.org/10.1007/BF02392708
  • [10] Bernard Dacorogna and Paolo Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications, 37, Birkhäuser Boston Inc., Boston, MA, 1999. MR 1702252 (2000f:35005)
  • [11] Gero Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 3, 437-471. MR 1286914 (96g:49001), https://doi.org/10.1017/S0308210500028730
  • [12] R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn. 2 (1990), no. 3, 215-239. MR 1069400 (92a:82132), https://doi.org/10.1007/BF01129598
  • [13] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683 (43 #445)

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Additional Information

Saugata Bandyopadhyay
Affiliation: Department of Mathematics & Statistics, Indian Institutes of Science Education and Research, Kolkata, India
Email: saugata.bandyopadhyay@gmail.com

Bernard Dacorogna
Affiliation: Section de Mathématiques, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
Email: bernard.dacorogna@epfl.ch

Olivier Kneuss
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Address at time of publication: Department of Mathematics, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
Email: olivier.kneuss@gmail.com

DOI: https://doi.org/10.1090/S0002-9947-2014-06014-5
Received by editor(s): July 24, 2012
Received by editor(s) in revised form: November 5, 2012
Published electronically: December 3, 2014
Additional Notes: Part of the present work was done while the first and third authors were visiting EPFL, whose hospitality is gratefully acknowledged
Article copyright: © Copyright 2014 American Mathematical Society

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