Some new results on differential inclusions for differential forms
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- by Saugata Bandyopadhyay, Bernard Dacorogna and Olivier Kneuss PDF
- Trans. Amer. Math. Soc. 367 (2015), 3119-3138 Request permission
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 \[ d\omega \in E\quad \text {a.e. in }\Omega \] where $E\subset \Lambda ^{k+1}$ is a prescribed set.References
- 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, DOI 10.1007/s00526-006-0049-6
- 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, DOI 10.3934/dcds.2005.12.97
- Alberto Bressan and Fabián Flores, On total differential inclusions, Rend. Sem. Mat. Univ. Padova 92 (1994), 9–16. MR 1320474
- Arrigo Cellina, On minima of a functional of the gradient: necessary conditions, Nonlinear Anal. 20 (1993), no. 4, 337–341. MR 1206422, DOI 10.1016/0362-546X(93)90137-H
- Arrigo Cellina, On minima of a functional of the gradient: sufficient conditions, Nonlinear Anal. 20 (1993), no. 4, 343–347. MR 1206423, DOI 10.1016/0362-546X(93)90138-I
- Gyula Csató, Bernard Dacorogna, and Olivier Kneuss, The pullback equation for differential forms, Progress in Nonlinear Differential Equations and their Applications, vol. 83, Birkhäuser/Springer, New York, 2012. MR 2883631, DOI 10.1007/978-0-8176-8313-9
- Bernard Dacorogna, Direct methods in the calculus of variations, second edition, Springer-Verlag, Berlin, 2007.
- 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, DOI 10.1007/s005260100092
- 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, DOI 10.1007/BF02392708
- Bernard Dacorogna and Paolo Marcellini, Implicit partial differential equations, Progress in Nonlinear Differential Equations and their Applications, vol. 37, Birkhäuser Boston, Inc., Boston, MA, 1999. MR 1702252, DOI 10.1007/978-1-4612-1562-2
- 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, DOI 10.1017/S0308210500028730
- R. D. James and D. Kinderlehrer, Frustration in ferromagnetic materials, Contin. Mech. Thermodyn. 2 (1990), no. 3, 215–239. MR 1069400, DOI 10.1007/BF01129598
- R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR 0274683, DOI 10.1515/9781400873173
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
- 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
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 3119-3138
- MSC (2010): Primary 35F60
- DOI: https://doi.org/10.1090/S0002-9947-2014-06014-5
- MathSciNet review: 3314803