What's the shape optimization

Let $\Omega $ be a bounded domain with Lipschitz boundary in $\mathbb{R}^d, d=2,3$ and $u(\Omega )$ a minimizer of an energy functional $\mathcal{E}$ over a subset $V(\Omega )$ in the function space $W^{1,p}(\Omega ;\mathbb{R}^m),m\ge 1$, Consider a cost functional $F(u(\Omega ))$. The shape optimization of boundary is a problem of finding a method of deforming a given set $\Omega $ so as to lower the cost function, and optimized shape $\Omega ^O$ is given by $$ F(u(\Omega ^O)) \lt F(u(\Omega ))$$

A mathematician asked me whether $\Omega ^O$ gives the minimum or a local minimum? In practical problems, it is not important to find the minimizer of the cost functions. Consider windows as an example. The minimizer of energy functional of Poisson problem $-\Delta u=1$ with Dirichlet boudary is the circle if the area is same. This indicates that the influence of external forces is uniform in round windows. However, many windows are square shaped because they provide a wider field of view, are easier to make and have fewer losses to cut out. On the other hand, windows on ships that require strength are round windows.

In practical problems it is based on multiple evaluations and is not created by a single evaluation. Here, the optimisation problem is the problem to find lowering the cost functional from mathematical view. With advances in computers, the study of shape optimization has become interesting. Recent mathematical research is dominated by topological optimisation, level set methods, etc., but here the traditional methods based on shape sensitivity analysis are used as a basis. It differs from traditional research that we obtain interesting results due to the singularity of the solution.

Definitions

$\Omega $:: Bounded domain with Lipschitz boundaryu,
Variational problems:: "$u\in \arg \min _{v\in V(\Omega )}\int _{\Omega }\{\widehat{W}(v)-fv\} -\int _{\Gamma _N}gv$" means that $u$ is the minimizer of the functional $\mathcal{E}(v;f,g,\Omega ) :=\int _{\Omega }\{\widehat{W}(v)-fv\} -\int _{\Gamma _N}gv$ over the set $V(\Omega )$ of functions, where $\Gamma _N$ is the part of $\partial \Omega $, $f,g$ are given functions.
Shape sensitivity:: Let $[t\mapsto \varphi _t ]\in C^2([0,\epsilon _0 );C^{0,1}(\mathbb{R}^d;\mathbb{R}^d))$, such that $\varphi _0(x)=x$.

Under construction

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