We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form \(P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}\), and the class of admissible domains consists of two-dimensional open sets \(\Omega \) satisfying the topological constraints of having a prescribed number k of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when \(q<1/2\) an optimal relaxed domain exists. When \(q>1/2\), the problem is ill-posed, and for \(q=1/2\), the explicit value of the infimum is provided in the cases \(k=0\) and \(k=1\).

我们根据周长、扭转刚度和 Lebesgue 度量考虑涉及泛函的形状优化问题。无标度成本泛函的形式为\(P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}\)，并且可接受域的类别由二维组成开集\(\Omega \)满足具有规定数量k的互补集的有界连通分量的拓扑约束。需要一个宽松的程序来解决一个适定的问题，我们证明当\(q<1/2\) 时存在一个最优的宽松域。当\(q>1/2\)，问题是不适定的，对于\(q=1/2\)，在这些情况下提供了下界的显式值\(k=0\)和\(k=1\)。