A note on Meyers’ Theorem in $W^{k,1}$
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- by Irene Fonseca, Giovanni Leoni, Jan Malý and Roberto Paroni
- Trans. Amer. Math. Soc. 354 (2002), 3723-3741
- DOI: https://doi.org/10.1090/S0002-9947-02-03027-1
- Published electronically: April 30, 2002
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Abstract:
Lower semicontinuity properties of multiple integrals \[ u\in W^{k,1}(\Omega ;\mathbb {R}^{d})\mapsto \int _{\Omega }f(x,u(x), \cdots ,\nabla ^{k}u(x)) dx\] are studied when $f$ may grow linearly with respect to the highest-order derivative, $\nabla ^{k}u,$ and admissible $W^{k,1}(\Omega ;\mathbb {R}^{d})$ sequences converge strongly in $W^{k-1,1}(\Omega ;\mathbb {R}^{d}).$ It is shown that under certain continuity assumptions on $f,$ convexity, $1$-quasiconvexity or $k$-polyconvexity of \[ \xi \mapsto f(x_{0},u(x_{0}),\cdots ,\nabla ^{k-1}u(x_{0}),\xi )\] ensures lower semicontinuity. The case where $f(x_{0},u(x_{0}),\cdots ,\nabla ^{k-1}u(x_{0}),\cdot )$ is $k$-quasiconvex remains open except in some very particular cases, such as when $f(x,u(x),\cdots ,\nabla ^{k}u(x))=h(x)g(\nabla ^{k}u(x)).$References
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Bibliographic Information
- Irene Fonseca
- Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
- MR Author ID: 67965
- Email: fonseca@cmu.edu
- Giovanni Leoni
- Affiliation: Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy 15100
- MR Author ID: 321623
- Email: leoni@unipmn.it
- Jan Malý
- Affiliation: Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83,186 75 Praha 8, Czech Republic
- Email: maly@karlin.mff.cuni.cz
- Roberto Paroni
- Affiliation: Dipartimento di Ingegneria Civile, Università degli Studi di Udine, Udine, Italy 33100
- Email: roberto.paroni@dic.uniud.it
- Received by editor(s): April 1, 2001
- Published electronically: April 30, 2002
- Additional Notes: The research of I. Fonseca was partially supported by the National Science Foundation under Grant No. DMS–9731957.
The research of G. Leoni was partially supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”, by the Italian CNR, through the strategic project “Metodi e modelli per la Matematica e l’Ingegneria”, and by GNAFA
The research of J. Malý was supported by CEZ MSM 113200007, grants GA ČR 201/00/0768 and GA UK 170/99.
The authors wish to thank Guy Bouchitté for stimulating discussions on the subject of this work, and the Center for Nonlinear Analysis (NSF Grant No. DMS–9803791) for its support during the preparation of this paper. - © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3723-3741
- MSC (2000): Primary 49J45, 49Q20
- DOI: https://doi.org/10.1090/S0002-9947-02-03027-1
- MathSciNet review: 1911518