## Extending the Knops-Stuart-Taheri technique to $C^{1}$ weak local minimizers in nonlinear elasticity

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## Abstract:

We prove that any $C^{1}$ weak local minimizer of a certain class of elastic stored-energy functionals $I(u) = \int _{\Omega } f(\nabla u) dx$ subject to a linear boundary displacement $u_{0}(x)=\xi x$ on a star-shaped domain $\Omega$ with $C^{1}$ boundary is necessarily affine provided $f$ is strictly quasiconvex at $\xi$. This is done without assuming that the local minimizer satisfies the Euler-Lagrange equations, and therefore extends in a certain sense the results of Knops and Stuart, and those of Taheri, to a class of functionals whose integrands take the value $+\infty$ in an essential way.## References

- John M. Ball,
*Convexity conditions and existence theorems in nonlinear elasticity*, Arch. Rational Mech. Anal.**63**(1976/77), no. 4, 337–403. MR**475169**, DOI 10.1007/BF00279992 - J. M. Ball,
*Discontinuous equilibrium solutions and cavitation in nonlinear elasticity*, Philos. Trans. Roy. Soc. London Ser. A**306**(1982), no. 1496, 557–611. MR**703623**, DOI 10.1098/rsta.1982.0095 - John M. Ball,
*Some open problems in elasticity*, Geometry, mechanics, and dynamics, Springer, New York, 2002, pp. 3–59. MR**1919825**, DOI 10.1007/0-387-21791-6_{1} - J. M. Ball,
*Minimizers and the Euler-Lagrange equations*, Trends and applications of pure mathematics to mechanics (Palaiseau, 1983) Lecture Notes in Phys., vol. 195, Springer, Berlin, 1984, pp. 1–4. MR**755716**, DOI 10.1007/3-540-12916-2_{4}7 - J. M. Ball and F. Murat,
*$W^{1,p}$-quasiconvexity and variational problems for multiple integrals*, J. Funct. Anal.**58**(1984), no. 3, 225–253. MR**759098**, DOI 10.1016/0022-1236(84)90041-7 - Patricia Bauman, Daniel Phillips, and Nicholas C. Owen,
*Maximal smoothness of solutions to certain Euler-Lagrange equations from nonlinear elasticity*, Proc. Roy. Soc. Edinburgh Sect. A**119**(1991), no. 3-4, 241–263. MR**1135972**, DOI 10.1017/S0308210500014815 - Patricia Bauman, Nicholas C. Owen, and Daniel Phillips,
*Maximum principles and a priori estimates for a class of problems from nonlinear elasticity*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**8**(1991), no. 2, 119–157 (English, with French summary). MR**1096601**, DOI 10.1016/S0294-1449(16)30269-4 - J. J. Bevan. On one-homogeneous solutions to elliptic systems with spatial variable dependence in two dimensions. Proc. Roy. Soc. Edinburgh 140A (2010), 447-475.
- Bernard Dacorogna,
*Direct methods in the calculus of variations*, 2nd ed., Applied Mathematical Sciences, vol. 78, Springer, New York, 2008. MR**2361288** - Lawrence C. Evans and Ronald F. Gariepy,
*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660** - J. L. Ericksen. Special topics in elastostatics. In: Advances in Applied Mechanics, 17, 189-224. Academic Press, 1977.
- I. M. Gelfand and S. V. Fomin,
*Calculus of variations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. Revised English edition translated and edited by Richard A. Silverman. MR**0160139** - 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 - Jan Kristensen and Ali Taheri,
*Partial regularity of strong local minimizers in the multi-dimensional calculus of variations*, Arch. Ration. Mech. Anal.**170**(2003), no. 1, 63–89. MR**2012647**, DOI 10.1007/s00205-003-0275-4 - J. E. Littlewood,
*Lectures on the Theory of Functions*, Oxford University Press, 1944. MR**0012121** - Stefan Müller, Scott J. Spector, and Qi Tang,
*Invertibility and a topological property of Sobolev maps*, SIAM J. Math. Anal.**27**(1996), no. 4, 959–976. MR**1393418**, DOI 10.1137/S0036141094263767 - S. Müller and V. Šverák,
*Convex integration for Lipschitz mappings and counterexamples to regularity*, Ann. of Math. (2)**157**(2003), no. 3, 715–742. MR**1983780**, DOI 10.4007/annals.2003.157.715 - S. Müller, Tang Qi, and B. S. Yan,
*On a new class of elastic deformations not allowing for cavitation*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**11**(1994), no. 2, 217–243 (English, with English and French summaries). MR**1267368**, DOI 10.1016/S0294-1449(16)30193-7 - Peter J. Olver,
*Applications of Lie groups to differential equations*, Graduate Texts in Mathematics, vol. 107, Springer-Verlag, New York, 1986. MR**836734**, DOI 10.1007/978-1-4684-0274-2 - Jeyabal Sivaloganathan,
*Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity*, Arch. Rational Mech. Anal.**96**(1986), no. 2, 97–136. MR**853969**, DOI 10.1007/BF00251407 - J. Sivaloganathan,
*Implications of rank one convexity*, Ann. Inst. H. Poincaré Anal. Non Linéaire**5**(1988), no. 2, 99–118 (English, with French summary). MR**954467** - Ali Taheri,
*Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations*, Proc. Amer. Math. Soc.**131**(2003), no. 10, 3101–3107. MR**1993219**, DOI 10.1090/S0002-9939-03-06852-7

## Additional Information

**J. J. Bevan**- Affiliation: Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom
- Email: j.bevan@surrey.ac.uk
- Received by editor(s): September 15, 2009
- Received by editor(s) in revised form: May 18, 2010
- Published electronically: October 8, 2010
- Additional Notes: The author gratefully acknowledges the support of an RCUK Academic Fellowship
- Communicated by: Matthew J. Gursky
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**139**(2011), 1667-1679 - MSC (2010): Primary 49J40; Secondary 49N60, 74G30
- DOI: https://doi.org/10.1090/S0002-9939-2010-10637-8
- MathSciNet review: 2763756