Shape optimization in three-dimensional space (Q2706052)

The authors of this interesting paper study the sufficient conditions of optimality for a class of shape optimization problems in the two-dimensional space. A shape optimization problem is formulated as below. Find the shape of an open set \(\Omega \in \)\(\mathcal{R}\)\(^3\) which attains a maximum (minimum) of \(J(\Omega)\equiv \int_{\Omega }g(u(x)) dx\), where \(g(u)\) is a given function that is smooth enough; \(u(x)\) is the solution of the boundary value problem \(\triangle u(x)=-1\) \((x\in\Omega)\), \(u(x)=0\) \((x\in\Gamma\equiv \partial\Omega)\). Here it is shown that there is a class of problems in which disks do not attain any optimum in spite of the fact that the problems have full symmetry. Also a shape optimization problem in three-dimensional space is considered and it is shown when spheres attain minimum or maximum. Other interesting fact is that there is a class of problems having full symmetry in which spheres do not attain any optimum in the three-dimensional space. An interesting example of electrostatic capacity problem is treated as well.