Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations


Author: Ali Taheri
Journal: Proc. Amer. Math. Soc. 131 (2003), 3101-3107
MSC (2000): Primary 49J10, 49J45
Published electronically: January 28, 2003
MathSciNet review: 1993219
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega \subset {\mathbb R}^n$ be a bounded starshaped domain. In this note we consider critical points $\bar{u} \in \bar{\xi} y + W_0 ^{1,p} (\Omega ; {\mathbb R}^m)$ of the functional

\begin{displaymath}{\mathcal F}(u, \Omega) := \int_{\Omega} f( \nabla u(y)) \, dy, \end{displaymath}

where $f: {\mathbb R}^{m \times n} \to {\mathbb R}$ of class ${C}^1$ satisfies the natural growth

\begin{displaymath}\vert f (\xi)\vert \le c (1 + \vert \xi\vert^p) \end{displaymath}

for some $1 \le p < \infty$ and $c>0$, is suitably rank-one convex and in addition is strictly quasiconvex at $\bar{\xi} \in {\mathbb R}^{m \times n}$. We establish uniqueness results under the extra assumption that ${\mathcal F}$ is stationary at $\bar{u}$ with respect to variations of the domain. These statements should be compared to the uniqueness result of Knops & Stuart (1984) in the smooth case and recent counterexamples to regularity produced by Müller & Sverák (2003).


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

  • 1. Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
  • 2. H. Federer, Geometric measure theory, Graduate Texts in Mathematics 153, Springer-Verlag, 1969.
  • 3. A. E. Green, On some general formulae in finite elastostatics, Arch. Rational Mech. Anal. 50 (1973), 73–80. MR 0441065
  • 4. Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 1, 15–34. MR 1397098, 10.1090/S0273-0979-97-00692-7
  • 5. Fritz John, Uniqueness of non-linear elastic equilibrium for prescribed boundary displacements and sufficiently small strains, Comm. Pure Appl. Math. 25 (1972), 617–634. MR 0315308
  • 6. R. J. Knops and C. A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rational Mech. Anal. 86 (1984), no. 3, 233–249. MR 751508, 10.1007/BF00281557
  • 7. J. Kristensen, A. Taheri, Partial regularity of strong local minimizers in the multi-dimensional calculus of variations, Max-Planck-Institute MIS (Leipzig) Preprint-Nr. 59, 2001.
  • 8. S. Müller, V. Sverák, Convex integration for Lipschitz mappings and counterexamples to regularity, To appear in Ann. of Math., 2003.
  • 9. K. D. E. Post and J. Sivaloganathan, On homotopy conditions and the existence of multiple equilibria in finite elasticity, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 3, 595–614. MR 1453283, 10.1017/S0308210500029929
  • 10. Ali Taheri, Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), no. 1, 155–184. MR 1820298, 10.1017/S0308210500000822
  • 11. A. Taheri, On critical points of functionals with polyconvex integrands, J. Convex Anal., Vol. 9, 2002, pp. 55-72.
  • 12. A. Taheri, On Artin's braid group and polyconvexity in the calculus of variations, To appear in J. Lond. Math. Soc., 2002.
  • 13. A. Taheri, Local minimizers and quasiconvexity - the impact of Topology, Max-Planck-Institute MIS (Leipzig) Preprint-Nr. 27, 2002.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 49J10, 49J45

Retrieve articles in all journals with MSC (2000): 49J10, 49J45


Additional Information

Ali Taheri
Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
Email: taheri@mis.mpg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06852-7
Received by editor(s): July 31, 2001
Received by editor(s) in revised form: April 24, 2002
Published electronically: January 28, 2003
Communicated by: Bennett Chow
Article copyright: © Copyright 2003 American Mathematical Society