Minimizers with topological singularities in two dimensional elasticity

Yan, Xiaodong; Bevan, Jonathan

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 192-209.

For a class of 2-D elastic energies we show that a radial equilibrium solution is the unique global minimizer in a subclass of all admissible maps. The boundary constraint is a double cover of $S^1$; the minimizer $u$ is $C^1$ and is such that $det\nabla u$ vanishes at one point.

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | MR | Zbl

[2] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337-403. | MR | Zbl

[3] J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Roy. Soc. London A 306 (1982) 557-611. | MR | Zbl

[4] J.M. Ball, Some open questions in elasticity. Geometry, mechanics, and dynamics. Springer, New York (2002) 3-59. | MR | Zbl

[5] J.M. Ball, J. Currie and P. Olver, Null Lagrangians, weak continuity and variational problems of arbitrary order. J. Func. Anal. 41 (1981) 135-174. | MR | Zbl

[6] J.M. Ball and F. Murat, $W^{1,p}$-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR | Zbl

[7] P. Bauman, N. Owen and D. Phillips, Maximum Principles and a priori estimates for a class of problems in nonlinear elasticity. Ann. Inst. H. Poincaré Anal. non Linéaire 8 (1991) 119-157. | Numdam | MR | Zbl

[8] P. Bauman, N. Owen and D. Phillips, Maximal smoothness of solutions to certain Euler-Lagrange equations from noninear elasticity. Proc. Roy. Soc. Edinburgh 119A (1991) 241-263. | MR | Zbl

[9] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and hardy spaces. J. Math. Pures. Appl. 72 (1993) 247-286. | MR | Zbl

[10] C. Horgan and D. Polignone, Cavitation in nonlinearly elastic solids; A review. Appl. Mech. Rev. 48 (1995) 471-485.

[11] R. Knops and C. Stuart, Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity. Arch. Rational Mech. Anal. 86 (1984) 233-249. | MR | Zbl

[12] S. Müller, Higher integrability of determinants and weak convergence in $L^1.$ J. reine angew. Math. 412 (1990) 20-34. | MR | Zbl

[13] R. Ogden, Large-deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. Edinburgh 328A (1972) 567-583. | Zbl

[14] P. Quintela-Estevez, Critical point in the energy of hyperelastic materials. RAIRO: Math. Modél. Numér. Anal. 25 (1990) 103-132. | Numdam | MR | Zbl

[15] J. Sivaloganathan, The generalized Hamilton-Jacobi inequality and the stability of equilibria in nonlinear elasticity. Arch. Rational Mech. Anal. 107 (1989) 347-369. | MR | Zbl

[16] J. Sivaloganathan, Singular minimizers in the calculus of variations: a degenerate form of cavitation. Ann. Inst. H. Poincaré Anal. non linéaire 9 (1992) 657-681. | Numdam | MR | Zbl

[17] J. Sivaloganathan, On the stability of cavitating equilibria. Q. Appl. Math. 53 (1995) 301-313. | MR | Zbl

[18] J. Spector, Linear deformations as global minimizers in nonlinear elasticity. Q. Appl. Math. 52 (1994) 59-64. | MR | Zbl

[19] V. Šverák, Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988) 105-127. | MR | Zbl

[20] A. Taheri, Quasiconvexity and uniqueness of stationary points in the multi-dimensional calculus of variations. Proc. Amer. Math. Soc. 131 (2003) 3101-3107. | MR | Zbl

[21] K. Zhang, Polyconvexity and stability of equilibria in nonlinear elasticity. Quart. J. Mech. appl. Math. 43 (1990) 215-221. | MR | Zbl

[22] K. Zhang, Energy minimizers in nonlinear elastostatics and the implicit function theorem. Arch. Rational Mech. Anal. 114 (1991) 95-117. | MR | Zbl