*Submitted Paper*

**Inserted:** 17 jan 2021

**Last Updated:** 18 jan 2021

**Year:** 2021

**Abstract:**

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 $k=1$.

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