具有给定拓扑的平面集上的形状最优化问题
A shape optimization problem on planar sets with prescribed topology
摘要：
我们考虑形状最优化问题涉及泛函取决于周长，扭转刚度和勒贝格措施。无标度代价泛函的形式是$P(Ω)T^q(Ω)|Ω|^{-2q-1/2}$和类的容许域由二维开集$Ω$满足拓扑约束，有一个规定数量的$k$有界连通组件的互补集。一个放松的程序需要有一个适定性的问题，我们表明，当$q<1/2$最佳放松域存在。当$q>1/2$的问题是不适定的和$q=1/2$的明确值的下确界提供的情况下$k=0$和$k=1$。
英文摘要：
We consider shape optimization problems involving functionals depending on perimeter, torsional rigidity and Lebesgue measure. The scaling free cost functionals are of the form $P(Ω)T^q(Ω)|Ω|^{-2q-1/2}$ and the class of admissible domains consists of two-dimensional open sets $Ω$ 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 $k=1$.
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