×

Relaxation of quasilinear elliptic systems via A-quasiconvex envelopes. (English) Zbl 1037.49011

This paper deals with the weak closure \(WZ\) for the set of solutions of a family of quasilinear elliptic systems in a bounded Lipschitz domain in \(\mathbb R^n\). The main feature of the problem studied in the paper is the presence of a nonlinearity whose derivatives are strictly convex smooth functions with quadratic growth. The main result asserts that \(WZ\) is the zero level set for an integral functional with the integrand \({\mathcal Q}\,{\mathcal F}\) being the A-quasiconvex envelope for a certain function \({\mathcal F}\) and the operator A=(curl,div)\(^m\). Next, is is shown that if the nonlinearity has certain isotropic properties, then the characteristic cone \(\Lambda\) (defined by the operator A) \({\mathcal Q}\,{\mathcal F}\) coincides with the A-polyconvex envelope of \({\mathcal F}\) and it can be computed by means of rank-one laminates.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
49M20 Numerical methods of relaxation type
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML

References:

[1] J. Ball , B. Kirchheim and J. Kristensen , Regularity of quasiconvex envelopes , Preprint No. 72/1999. Max-Planck Institute für Mathematik in der Naturwissenschaften, Leipzig ( 1999 ). MR 1808126 | Zbl 0972.49024 · Zbl 0972.49024
[2] B. Dacorogna , Direct Methods in the Calculus of Variations . Springer: Berlin, Heidelberg, New York ( 1989 ). MR 990890 | Zbl 0703.49001 · Zbl 0703.49001
[3] I. Fonseca and S. Müller , A-quasiconvexity, lower semicontinuity, and Young measures . SIAM J. Math. Anal. 30 ( 1999 ) 1355 - 1390 . MR 1718306 | Zbl 0940.49014 · Zbl 0940.49014
[4] R.V. Kohn and G. Strang , Optimal design and relaxation of variational problems , Parts I-III. Comm. Pure Appl. Math. 39 ( 1986 ) 113 - 137 , 138 - 182 , 353 - 377 . Zbl 0609.49008 · Zbl 0609.49008
[5] K.A. Lurie , A.V. Fedorov and A.V. Cherkaev , Regularization of optimal problems of design of bars and plates , Parts 1 and 2. JOTA 37 ( 1982 ) 499 - 543 . MR 669842 | Zbl 0464.73109 · Zbl 0464.73109
[6] M. Miettinen and U. Raitums , On \(C^1\)-regularity of functions that define G-closure . Z. Anal. Anwendungen 20 ( 2001 ) 203 - 214 . MR 1826327 | Zbl 0984.49018 · Zbl 0984.49018
[7] F. Murat , Compacité par compensation : condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant . Ann. Scuola Norm. Super. Pisa 8 ( 1981 ) 69 - 102 . Numdam | MR 616901 | Zbl 0464.46034 · Zbl 0464.46034
[8] U. Raitums , Properties of optimal control problems for elliptic equations , edited by W. Jäger et al., Partial Differential Equations Theory and Numerical Solutions. Boca Raton: Chapman & Hall/CRC, Res. Notes in Math. 406 ( 2000 ) 290 - 297 . MR 1713894 | Zbl 0942.49006 · Zbl 0942.49006
[9] L. Tartar , An introduction to the homogenization method in optimal design . CIME Summer Course. Troia ( 1998 ). http://www.math.cmu.edu/cna/publications.html Zbl 1040.49022 · Zbl 1040.49022
[10] V.V. Zhikov , S.M. Kozlov and O.A. Oleinik , Homogenization of Differential Operators and Integral Functionals . Springer: Berlin, Hedelberg, New York ( 1994 ). MR 1329546 | Zbl 0838.35001 · Zbl 0838.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.