For a given function $f\in L^2(\mathbb{R}^2)$, consider the Poisson equation $-\Delta u=f$ defined on a polygonal domain $\Omega $ with the boundary $\partial \Omega =\cup _{j=1}^{j_M}\overline{\Gamma }_j$ such that $\overline{\Gamma }_j$ is a segment and $\Gamma _j$ is open for each $j$, $\Gamma _i\cap \Gamma _j=\emptyset $ if $i\neq j$, and the segments are numbered in such way that $\Gamma _{j+1}$ follows $\Gamma _j$ according to the positive orientation except $\Gamma _{j_M+1}:=\Gamma _1$.We denote by $\gamma _j$ the endpoint of $\Gamma _j$ and by $\Theta _j$ the measure of the interior angle at $\gamma _j$. We now put $\Gamma _D=\cup _{j=1}^{j_D}\overline{\Gamma }_j\setminus \{\gamma _{j_D},\gamma _{j_M}\}$, $\Gamma _N=\cup _{j=j_D+1}^{j_M}\overline{\Gamma }_j\setminus \{\gamma _{j_D},\gamma _{j_M}\}$ and assume that $\Theta _{j_D}=\Theta _{j_M}=\pi $.

Domain removed from the neighbourhood of singular points $\Gamma _{DN}:=\{\gamma _{j_D},\gamma _{j_M}\}$

The minimizer $u\in V(\Omega )$ of (2.1) over $V(\Omega )=\{v\in H^1(\Omega ):\, v=0\textrm{ on }\Gamma _D\}$ belong to $u\in H^s(\Omega _{\epsilon })\, s \gt 3/2, \Omega _{\epsilon }=\Omega \setminus \overline{B_{\epsilon }(\Gamma _{DN})}, B_{\epsilon }(\Gamma _{DN}):=B_{\epsilon }(\gamma _{j_D}) \cup B_{\epsilon }(\gamma _{j_M})$ where $B_{\epsilon }(\gamma )=\{x:\, |x-\gamma | \lt \epsilon \}$. By a perturbation $\{\varphi _t\} _{t\in [0,T_0]}$, we have \begin{eqnarray} &&\frac{d}{dt}\mathcal{E}(u^{\varphi _t};f,0,\Omega (t))|_{t=0} =-R_{\Omega }(u;\mu _{\varphi })-\int _{\Gamma _N}fu(\mu _{\varphi }\cdot n)ds\tag{2.45}\\ && =P_{\Omega _{\epsilon }}(u,\mu _{\varphi }|\Gamma _D) +P_{\Omega _{\epsilon }}(u,\mu _{\varphi }|\Gamma _N)\notag \\ && -\sum _{\gamma \in \{\gamma _{j_D},\gamma _{j_M}\} }J_{B_{\epsilon }(\gamma )}(u,\mu _{\varphi }|\partial B_{\epsilon }(\gamma )) -\int _{\Gamma _N}fu(\mu _{\varphi }\cdot n)ds\notag \end{eqnarray} \begin{eqnarray} &&P_{\Omega _{\epsilon }}(u,\mu _{\varphi }|\Gamma _D) =-\frac{1}{2}\int _{\Gamma _D\setminus B_{\epsilon }(\Gamma _{DN})}\left (\frac{\partial u}{\partial n}\right )^2ds \tag{2.46} \end{eqnarray} \begin{eqnarray} &&P_{\Omega _{\epsilon }}(u,\mu _{\varphi }|\Gamma _N) =\frac{1}{2}\int _{\Gamma _N\setminus B_{\epsilon }(\Gamma _{DN})}|\nabla u |^2ds \tag{2.47} \end{eqnarray} Here we used that $J_{\Omega _{\epsilon }}(u;\mu _{\varphi })=0$ and $u=0$ on $\Gamma _j, 1\le j\le j_D$ and $\partial u/\partial n=0$ on $\Gamma _j, j_D \lt j\le j_M$. The solution $u$ has the structure $u |_{B_{\epsilon }(\gamma _{j_M})\cap \Omega }=K(\gamma _{j_M})S+u_R, u_R\in H^2(B_{\epsilon }(j_M)\cap \Omega )$ with a constant $K(\gamma _{j_M})$ and $S=\sqrt{r_{j_M}}\sin (\theta _{j_M}/2)$, where $(r_{j_M},\theta _{j_M})$ is the local polar coordinate with origin at $\gamma _{j_M}$; $r_{j_M}=|x-\gamma _{j_M}|$ (see e.g.[Gr92]), $\Gamma _1$ is on the half-axis $\theta _{j_M}=0$ and $\Gamma _{j_M}$ is on the half-axis $\theta _{j_M}=\pi $.

Local polar coordinate system $(r_{j_M},\theta _{j_M})$ with origin at $\gamma _{j_M}$

So then we have \begin{equation} \lim _{\epsilon →0}J_{B_{\epsilon }(\gamma _{j_M})}(u,\mu _{\varphi }| \partial B_{\epsilon }(\gamma _{j_M}))=\frac{\pi }{8} K(\gamma _{j_M})^2 (\mu _{\varphi }\cdot \tau )(\gamma _{j_M}), \tag{2.48} \end{equation} where $\tau (x)$ is the tangential vector at $x\in \partial \Omega $ with positive direction.

Theorem Poisson with mixed boundary

The shape sensitivity of energy $\frac{d}{dt}\mathcal{E}(u^{\varphi _t};f,0,\Omega (t))|_{t=0}$ with the mixed boundary condition stated just above is \begin{eqnarray} &&\lim _{\epsilon →0}\left \{P_{\Omega _{\epsilon }}(u,\mu _{\varphi }|\Gamma _D) +P_{\Omega _{\epsilon }}(u,\mu _{\varphi }|\Gamma _N)\right \} \tag{2.49} \\ &&-\frac{\pi }{8} \left \{K(\gamma _{j_M})^2 (\mu _{\varphi }\cdot \tau )(\gamma _{j_M}) -K(\gamma _{j_D})^2 (\mu _{\varphi }\cdot \tau )(\gamma _{j_D}) \right \} \notag \\ &&-\int _{\Gamma _N}fu(\mu _{\varphi }\cdot n)ds\notag \end{eqnarray} In particular in the case of $\varphi _t(\Omega )=\Omega $, the energy decreases as $\Gamma _D(t)\subset \Gamma _D$.

If $\Theta _{j_M},\Theta _{j_D} \lt \pi $, then the second term in (2.49) disappear. If $\Theta _{j_M} \gt \pi $, (2.45) is true but the existence of the limit as in (2.49) is not clear. The method used here is applicable to various boundary value problems with mixed boundary conditions.

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