A Shape Optimization Problem on Planar Sets with Prescribed Topology

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