Shape sensitivity of energy



Interfaces are (a) $\Gamma _2$, (b) $\Gamma _{12}$ and (c) $\Gamma _{12}, \Gamma _{23}, \Gamma _{34}, \Gamma _{14}$
Assume that the region $\Omega $ is divided into partial regions $\Omega _{\kappa },\kappa =1,…,K$ such that $\overline{\Omega }=\sum _{\kappa =1}^K\overline{\Omega _{\kappa }}$, $\Omega _{\kappa }\cap \Omega _{l}=\emptyset $ if $\kappa \neq l$ and there is $[(\xi ,z,\zeta )\mapsto \widehat{W}_{\kappa }(\xi ,z,\xi )]\in C^1(\mathbb{R}^d\times \mathbb{R}^m\times \mathbb{R}^{m\times d})$ for each $1\le \kappa \le K$. Now we define $\widehat{W}(v)$ as follows for $v\in H^1(\Omega ;\mathbb{R}^m)$: \begin{eqnarray} \widehat{W}(x,v(x),\nabla v(x))&:=&\widehat{W}_{\kappa }(x,v(x),\nabla v(x)) \textrm{if }x\in \Omega _{\kappa } \tag{2.50} \end{eqnarray} \begin{eqnarray} \mathcal{E}(v;f,g,\cup \Omega _{\kappa })&:=&\sum _{\kappa =1}^K \int _{\Omega _{\kappa }}\{\widehat{W}_{\kappa }(v)-fv\} dx-\int _{\Gamma _N}gv\, ds \tag{2.51} \end{eqnarray} Here we denote $\mathcal{E}(v;f,g,\cup \Omega _{\kappa })$ the left-hand side of (2.51) to distinguish it from $\mathcal{E}(v;f,g,\Omega )$ in (2.2) .
Corollary 2.18
If , for $\varphi $ near $\varphi _0$ in $C^{0,1}(\Omega _0;\mathbb{R}^d)$, $\varphi _0(x)=x, \overline{\Omega }\subset \Omega _0$, \begin{eqnarray*} \tilde{\mathcal{E}}(v;f,g,\varphi )&:=& \sum _{\kappa =1}^K \int _{\Omega _{\kappa }}\left \lbrace \widehat{W}_{\kappa }(\varphi (x),v,(\nabla \varphi ^T)^{-1}\nabla v)-(f\circ \varphi )v\right \} \textrm{det}\nabla \varphi dx\\ &&-\int _{\Gamma _N}gv\,ds\notag \end{eqnarray*} satisfies the conditions (F1)--(F7), then the following holds when $\varphi _t=\varphi _0+t\mu _{\varphi }$ and $\varphi _t(\textrm{supp}_{\Gamma _N}g)=\textrm{supp}_{\Gamma _N}g$, $0\le t \lt \epsilon $ with sufficiently small $\epsilon \gt 0$: \begin{eqnarray} \frac{d}{dt}\mathcal{E}(u^{\varphi _t};f,g,\cup \Omega _{\kappa }(t))|_{t=0}&=& -\sum _{\kappa =1}^KR_{\Omega _{\kappa }}(u,\mu _{\varphi })-\int _{\partial \Omega }fu(\mu _{\varphi }\cdot n)ds \tag{2.52} \end{eqnarray}
If $u |_{\Omega _{\kappa }}$ are regular inside $\Omega _{\kappa }$ for all $1\le \kappa \le K$, then \begin{eqnarray*} \frac{d}{dt}\mathcal{E}(u^{\varphi _t};f,g,\cup \Omega _{\kappa }(t))|_{t=0}&=& \sum _{\kappa =1}^KP_{\Omega _{\kappa }}(u,\mu _{\varphi })-\int _{\partial \Omega }fu(\mu _{\varphi }\cdot n)ds \end{eqnarray*}

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