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

DOI:
https://doi.org/10.1090/S0002-9939-03-06852-7

Published electronically:
January 28, 2003

MathSciNet review:
1993219

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Abstract: Let be a bounded starshaped domain. In this note we consider critical points of the functional

where of class satisfies the natural growth

for some and , is suitably rank-one convex and in addition is strictly quasiconvex at . We establish uniqueness results under the extra assumption that is stationary at 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).

**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.

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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:
https://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