Contents

Shape optimization

Eigenvalue problems

We can also define the GJ-integral $J_{\omega }^E(u_{\lambda },\mu )$ of an eigenfunction $u_{\lambda }$ with an eigenvalue $\lambda $, that is, \begin{eqnarray} a(u_{\lambda }(t),v)=\lambda (t) \int _{\Omega }u_{\lambda }(t)v\, dx  ∀v\in V^{\varphi _t}(\Omega (t)) \tag{3.10} \end{eqnarray} Putting $u_{\lambda }=u_{\lambda }(0)$, we define $J_{\omega }^E(u_{\lambda },\mu )=P_{\omega }(u_{\lambda },\mu )+R_{\omega }^E(u_{\lambda },\mu )$ by \begin{eqnarray} R_{\omega }^E(u_{\lambda },\mu )& =&-\int _{\omega \cap \Omega }\left \lbrace \nabla _{\xi }\widehat{W}(u_{\lambda })\cdot \mu +\lambda u_{\lambda }(\nabla u_{\lambda }\cdot \mu )\right \} dx \tag{3.11}\\ && +\int _{\omega \cap \Omega }\left \{ \nabla _{\zeta }\widehat{W}(u_{\lambda }):[\nabla u_{\lambda }\nabla \mu ]-\widehat{W}(u_{\lambda })\,\textrm{div}\mu \right \} \,dx, \notag \end{eqnarray} where $P_{\omega }(u_{\lambda },\mu )$ is the surface integral substituting $u_{\lambda }$ into $u$ in (2.5b) . We have that $J_{\omega }^E(u_{\lambda },\mu )=0$ for all $\mu \in W^{1,\infty }(\Omega ;\mathbb{R}^d)$ if $u_{\lambda }$ is regular by a proof similar to that of Theorem 2.2 . From \cite [Theorem 3.5.2]{Hau86}, we have the shape sensitivity of eigenvalue.
Theorem 3.3
Under the same assumption of Theorem 2.8 , if $\lambda $ is single, then \begin{equation} \left .\frac{d}{dt}\lambda (t)\right |_{t=0} \int _{\Omega } u_{\lambda }^2\, dx=-2R_{\Omega }^E(u_{\lambda },\mu _{\varphi }) -\lambda \int _{\partial \Omega }u_{\lambda }^2(\mu _{\varphi }\cdot n)ds \tag{3.12} \end{equation}

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