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.

Keywords: nonlinear elasticity, singular minimizer, stability

@article{COCV_2008__14_1_192_0, author = {Yan, Xiaodong and Bevan, Jonathan}, title = {Minimizers with topological singularities in two dimensional elasticity}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {192--209}, publisher = {EDP-Sciences}, volume = {14}, number = {1}, year = {2008}, doi = {10.1051/cocv:2007043}, mrnumber = {2375756}, zbl = {1140.49014}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv:2007043/} }

TY - JOUR AU - Yan, Xiaodong AU - Bevan, Jonathan TI - Minimizers with topological singularities in two dimensional elasticity JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 192 EP - 209 VL - 14 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv:2007043/ DO - 10.1051/cocv:2007043 LA - en ID - COCV_2008__14_1_192_0 ER -

%0 Journal Article %A Yan, Xiaodong %A Bevan, Jonathan %T Minimizers with topological singularities in two dimensional elasticity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 192-209 %V 14 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv:2007043/ %R 10.1051/cocv:2007043 %G en %F COCV_2008__14_1_192_0

Yan, Xiaodong; Bevan, Jonathan. Minimizers with topological singularities in two dimensional elasticity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 192-209. doi : 10.1051/cocv:2007043. http://www.numdam.org/articles/10.1051/cocv:2007043/

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

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

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

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

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

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

and ,[7] 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

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

, and ,[9] Compensated compactness and hardy spaces. J. Math. Pures. Appl. 72 (1993) 247-286. | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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