Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations
HTML articles powered by AMS MathViewer
- by Ali Taheri
- Proc. Amer. Math. Soc. 131 (2003), 3101-3107
- DOI: https://doi.org/10.1090/S0002-9939-03-06852-7
- Published electronically: January 28, 2003
- PDF | Request permission
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 \[ {\mathcal F}(u, \Omega ) := \int _{\Omega } f( \nabla u(y)) dy, \] where $f: {\mathbb R}^{m \times n} \to {\mathbb R}$ of class $\mathrm {C}^1$ satisfies the natural growth \[ |f (\xi )| \le c (1 + | \xi |^p) \] 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 & Šverák (2003).References
- Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890, DOI 10.1007/978-3-642-51440-1
- H. Federer, Geometric measure theory, Graduate Texts in Mathematics 153, Springer-Verlag, 1969.
- A. E. Green, On some general formulae in finite elastostatics, Arch. Rational Mech. Anal. 50 (1973), 73–80. MR 441065, DOI 10.1007/BF00251294
- Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 1, 15–34. MR 1397098, DOI 10.1090/S0273-0979-97-00692-7
- 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 315308, DOI 10.1002/cpa.3160250505
- 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, DOI 10.1007/BF00281557
- 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.
- S. Müller, V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity, To appear in Ann. of Math., 2003.
- 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, DOI 10.1017/S0308210500029929
- 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, DOI 10.1017/S0308210500000822
- A. Taheri, On critical points of functionals with polyconvex integrands, J. Convex Anal., Vol. 9, 2002, pp. 55-72.
- A. Taheri, On Artin’s braid group and polyconvexity in the calculus of variations, To appear in J. Lond. Math. Soc., 2002.
- A. Taheri, Local minimizers and quasiconvexity - the impact of Topology, Max-Planck-Institute MIS (Leipzig) Preprint-Nr. 27, 2002.
Bibliographic Information
- Ali Taheri
- Affiliation: Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, D-04103 Leipzig, Germany
- Email: taheri@mis.mpg.de
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 3101-3107
- MSC (2000): Primary 49J10, 49J45
- DOI: https://doi.org/10.1090/S0002-9939-03-06852-7
- MathSciNet review: 1993219