Neohookean deformations of annuli, existence, uniqueness and radial symmetry
Neohookean deformations of annuli, existence, uniqueness and radial symmetry
复制标题
环的 Neohookean 变形、存在性、唯一性和径向对称性
DOI：
10.1007/s00208-009-0469-7
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发表时间：
2010
影响因子：
1.4
通讯作者：
Jani Onninen
中科院分区：
文献类型：
--
作者：
T. Iwaniec;Jani Onninen
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.
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.