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A note on Meyers' Theorem in $W^{k,1}$

Author(s): Irene Fonseca; Giovanni Leoni; Jan Malý; Roberto Paroni
Journal: Trans. Amer. Math. Soc. 354 (2002), 3723-3741.
MSC (2000): Primary 49J45, 49Q20
Posted: April 30, 2002
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Abstract: Lower semicontinuity properties of multiple integrals

\begin{displaymath}u\in W^{k,1}(\Omega;\mathbb{R}^{d})\mapsto\int_{\Omega}f(x,u(x), \cdots,\nabla^{k}u(x))\,dx\end{displaymath}

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

\begin{displaymath}\xi\mapsto f(x_{0},u(x_{0}),\cdots,\nabla^{k-1}u(x_{0}),\xi)\end{displaymath}

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:

1.
S. AGMON, A. DOUGLIS AND L. NIRENBERG, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Applied Math. 12 (1959), 623-727. MR 23:A2610

2.
M. AMAR AND V. DE CICCO, Relaxation of quasi-convex integrals of arbitrary order, Proc. Roy. Soc. Edinburgh 124 (1994), 927-946. MR 95h:49015

3.
L. AMBROSIO, New lower semicontinuity results for integral functionals, Rend. Accad. Naz. Sci. XL, 11 (1987) 1-42. MR 89c:49010
4.
L. AMBROSIO AND G. DAL MASO, On the relaxation in $BV(\Omega;\mathbb{R}^{m})$ of quasi-convex integrals, J. Funct. Anal., 109 (1992) 76-97. MR 93j:49012
5.
L. AMBROSIO, N. FUSCO AND D. PALLARA, Functions of Bounded Variation and Free Discontinuity Problems, Mathematical Monographs, Oxford University Press, 2000.

6.
J. BALL, J. CURRIE AND P. OLVER, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 41 (1981), 315-328. MR 83f:49031

7.
L. BOCCARDO, D. GIACHETTI, J.I. DIAZ AND F. MURAT, Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, J. Diff. Equations 106 (1993), 215-237. MR 94k:35062

8.
A. BRAIDES, I. FONSECA AND G. LEONI, A-quasiconvexity: relaxation and homogenization, ESAIM:COCV 5 (2000), 539-577. MR 2001k:49034

9.
M. CARRIERO, A. LEACI AND F. TOMARELLI, Special bounded hessian and elastic-plastic plate, Rend. Accad. Naz. Sci. XL Mem. Mat. (5) 16 (1992), 223-258. MR 94k:49043

10.
M. CARRIERO, A. LEACI AND F. TOMARELLI, Strong minimizers of Blake & Zisserman functional, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (4) 25 (1997), 257-285. MR 2000a:49019

11.
M. CARRIERO, A. LEACI AND F. TOMARELLI, A second order model in image segmentation: Blake & Zisserman functional, Progr. Nonlinear Differential Equations Appl., 25, Birkhäuser 25 (1996), 57-72. MR 97h:4901b

12.
R. CERNÝ AND J. MALÝ, Counterexample to lower semicontinuity in Calculus of Variations, to appear in Math. Z.

13.
B. DACOROGNA, Direct methods in the calculus of variations, Springer-Verlag, New York, 1989. MR 90e:49001
14.
G. DAL MASO AND C. SBORDONE, Weak lower semicontinuity of polyconvex integrals: a borderline case, Math. Z. 218 (1995), 603-609. MR 96g:49003

15.
E. DE GIORGI AND L. AMBROSIO, Un nuovo tipo di funzionale del calcolo delle variazioni, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82 (1988), 199-210. MR 92j:49043
16.
L.C. EVANS AND R.F. GARIEPY, Measure Theory and Fine Properties of Functions, Studies in Advanced Math., CRC Press, 1992. MR 93f:28001

17.
I. FONSECA AND G. LEONI, On lower semicontinuity and relaxation, Proc. Royal Soc. Edinburgh 131 (2001), 519-565.

18.
I. FONSECA AND G. LEONI, Some remarks on lower semicontinuity, Indiana U. Math. J. 49 (2000), 617-635.

19.
I. FONSECA AND S. MÜLLER, Quasi-convex integrands and lower semicontinuity in $L^{1}$, SIAM J. Math. Anal. 23 (1992), 1081-1098. MR 93i:49020

20.
I. FONSECA AND S. MÜLLER, Relaxation of quasiconvex functionals in $\mathrm{BV}(\Omega,\mathbb{R}^{p})$ for integrands $f(x,u,\nabla u)$, Arch. Rat. Mech. Anal. 123 (1993), 1-49. MR 94h:49023

21.
FONSECA I. AND S. MÜLLER, $\mathcal{A}$-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal., 30 (1999) 1355-1390. MR 2000j:49020

22.
N. FUSCO, Quasiconvessità e semicontinuità per integrali multipli di ordine superiore, Ricerche Mat. 29 (1980), 307-323. MR 83h:49022

23.
N. FUSCO AND J. E. HUTCHINSON, A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math. 85 (1995), 35-50. MR 96f:49020

24.
M. GUIDORZI AND L. POGGIOLINI, Lower semicontinuity for quasiconvex integrals of higher-order, Nonlinear Differential Equations Appl. 6 (1999), 227-246. MR 2000c:49021

25.
P. MARCELLINI, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals quasiconvex integrals, Manuscripta Math. 51 (1985), 1-28. MR 86h:49018

26.
N. MEYERS, Quasi-convexity and lower semi-continuity of multiple variational integrals of any order, Trans. Amer. Math. Soc. 119 (1965), 125-149. MR 32:6270

27.
C. MORREY, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 25-53. MR 14:992a

28.
J. SERRIN, On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc. 161 (1961), 139-167. MR 25:1466

29.
W.P. ZIEMER, Weakly differentiable functions. Sobolev spaces and functions of bounded variation, Springer-Verlag, New York, 1989. MR 91e:46046

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Additional Information:

Irene Fonseca
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
Email: fonseca@cmu.edu

Giovanni Leoni
Affiliation: Dipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy 15100
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

DOI: 10.1090/S0002-9947-02-03027-1
PII: S 0002-9947(02)03027-1
Keywords: $k$-quasiconvexity, higher-order lower semicontinuity, gradient truncation
Received by editor(s): April 1, 2001
Posted: 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 GACR 201/00/0768 and GAUK 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 of article: Copyright 2002, American Mathematical Society

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