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

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. B. Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences 78, Springer-Verlag, 1988. MR 90e:49001
  • 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., Vol. 50, 1973, pp. 73-80. MR 55:13931
  • 4. R.M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc., Vol. 34, No. 1, 1997, pp. 15-34. MR 98b:58046
  • 5. F. John, Uniqueness of nonlinear equilibrium for prescribed boundary displacement and sufficiently small strains, Comm. Pure Appl. Math., Vol. 25, 1972, pp. 617-634. MR 47:3857
  • 6. R.J. Knops, C.A. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in non-linear elasticity, Arch. Rational Mech. Anal., Vol 86, No. 3, 1984, pp. 233-249. MR 85j:73012
  • 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. Post, J. Sivaloganathan, On homotopy conditions and the existence of multiple equilibria in finite elasticity, Proc. Roy. Soc. Edin. A, Vol. 127, 1997, pp. 595-614. MR 98h:73030a
  • 10. A. Taheri, Sufficiency theorems for local minimizers of the multiple integrals of the calculus of variations, Proc. Roy. Soc. Edin. A, Vol. 131, 2001, pp. 155-184. MR 2002e:49036
  • 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

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

American Mathematical Society