Neohookean deformations of annuli, existence, uniqueness and radial symmetry

Abstract

Let ${\mathbb {X} = \{x\in \mathbb {R}^2 ; r < |x|<R\}}$ and ${\mathbb {Y} = \{y\in \mathbb {R}^2 ; r_\ast < |y|<R_\ast\}}$ be annuli in the plane. We consider the class ${\fancyscript {F}(\mathbb {X}, \mathbb {Y})}$ of all orientation preserving homeomorphisms ${h :\mathbb {X}\overset{\textnormal{\tiny{onto}}}{\longrightarrow}\mathbb {Y}}$ in the Sobolev space ${{\fancyscript {W}^{1,2}(\mathbb {X}, \mathbb {Y})}}$ which keep the boundary circles in the same order. This means that ${\lim_{|x| \searrow r} |h(x)| =r_\ast}$ and ${\lim_{|x| \nearrow R} |h(x)| =R_\ast}$ . We study the Neohookean energy integral

$$\mathcal {E}[h]= \int\limits_\mathbb {X} |Dh|^2 + \Phi ({\rm det}\, Dh) \quad {\rm for}\, h \in \fancyscript{F} (\mathbb {X} , \mathbb {Y})\quad\quad\quad (1)$$

where ${\Phi \in \fancyscript {C}^\infty (0, \infty)}$ is positive and strictly convex. We assume in addition that the function ${\Psi (z)= {1}/{\ddot{\Phi}(z)}}$ and its derivative extend continuously to [0, ∞), with Ψ(0) = 0. Then we prove:

Theorem 1 The minimum of energy within the class${\fancyscript F (\mathbb X , \mathbb Y)}$is attained for a radial map${h(x)=H(|x|) \frac{x}{|x|}}$. The minimizer is${\fancyscript {C}^\infty}$ -smooth and is unique up to a rotation of the annuli.

We believe that not only the result but also some novelties in the computation might gain a particular interest.