A Shape Optimization Problem on Planar Sets with Prescribed Topology

Luca Briani (), Giuseppe Buttazzo () and Francesca Prinari ()
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Luca Briani: Università di Pisa
Giuseppe Buttazzo: Università di Pisa
Francesca Prinari: Università di Pisa

Journal of Optimization Theory and Applications, 2022, vol. 193, issue 1, No 32, 760-784

Abstract: 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 $$P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}$$ P ( Ω ) T q ( Ω ) | Ω | - 2 q - 1 / 2 , and the class of admissible domains consists of two-dimensional open sets $$\Omega $$ 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 $$q 1/2$$ q > 1 / 2 , the problem is ill-posed, and for $$q=1/2$$ q = 1 / 2 , the explicit value of the infimum is provided in the cases $$k=0$$ k = 0 and $$k=1$$ k = 1 .

Keywords: Torsional rigidity; Shape optimization; Perimeter; Planar sets; Topological genus; 49Q10; 49J45; 49R05; 35P15; 35J25 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-021-01870-7

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