We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in ${\mathbb{R}}^3$, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.

中文翻译：

---

最优绝缘中出现的特征值问题的最优凸和旋转对称形状的数值逼近

我们对 PDE 约束下的凸域优化感兴趣。由于近似凸域的困难${\mathbb{R}}^3$，对旋转对称域的限制用于将形状优化问题简化为二维设置。对于最优绝缘问题中出现的特征值的优化，证明了最优域的存在。提出了一种适用于凸度和旋转对称几何约束下的一般形状优化问题的算法。讨论了最优绝缘中特征值问题的近似最优域。