Neohookean deformations of annuli, existence, uniqueness and radial symmetry (Q985689)

The authors investigate the deformations of the annuli \(\mathbb{X}= \{x\in \mathbb{R}^2;\;r<|x|<R\} \) onto \(\mathbb{Y} =\{x\in \mathbb{R}^2;\;r_*<|x|<R_*\}\). They consider the class \(\mathcal{F}(\mathbb{X},\mathbb{Y})\) of all orientation preserving surjective homeomorphisms \(h:\mathbb{X}\to\mathbb{Y}\) in the Sobolev space \(\mathcal{W}^{1,2}(\mathbb{X},\mathbb{Y})\) which keep the boundary circles in the same order, and study the Neohookean energy integral \[ \mathcal{E}[h]=\int_\mathbb{X}|Dh|^2+\Phi(\det Dh),\quad h\in \mathcal{F}(\mathbb{X},\mathbb{Y}), \] where \(\Phi\in \mathcal{C}^\infty(0,\infty)\) is positive and strictly convex. Moreover, they assume that the function \(\Psi(z)=1/\ddot\Phi(z)\) and its derivative extend continuously to \([0,\infty)\), with \(\Psi(0)=0\). Then they prove: Theorem. The minimum of energy within the class \(\mathcal{F}(\mathbb{X},\mathbb{Y})\) is attained for a radial map \[ h(X)=H\big(| x|\big)\frac{x}{| x|}. \] The minimizer is \(\mathcal{C}^\infty\)-smooth and is unique up to a rotation of the annuli.