Main result

Theorem 2.1 If there exists a linear form $L_\rho$ defined on $\mathcal{V}_\rho$ , a function $f:\mathbb{R}^+\to\mathbb{R}^+$ , and four real numbers $\delta J_1$ , $\delta J_2$ , $\delta a$ and $\delta l$ such that

$\bullet$: $\displaystyle \lim_{\rho\to 0} f(\rho) = 0$ ,
$\bullet$: $J(\Omega_\rho,u_\rho) - J(\Omega_\rho,u_0) = L_\rho(u_\rho-u_0)+f(\rho)\delta J_1+o(f(\rho)),$
$\bullet$: $J(\Omega_\rho,u_0) - J(\Omega,u_0) = f(\rho)\delta J_2+o(f(\rho)),$
$\bullet$: $(a_\rho-a_0)(u_0,p_\rho) = f(\rho)\delta a + o(f(\rho)),$
$\bullet$: $(l_\rho-l_0)(p_\rho) = f(\rho)\delta l + o(f(\rho))$ ,

where the adjoint state $p_\rho$ is the solution of the adjoint equation

$\displaystyle a_\rho(w,p_\rho)=-L_\rho(w),\forall w\in\mathcal{V}_\rho,$

(2.5)

and $u_\rho$ is the solution of the direct equation (2.2), then the cost function has the asymptotic expansion (2.4), where the topological gradient is given by

$\displaystyle G(x) = \delta J_1 + \delta J_2 + \delta a - \delta l.$

(2.6)