A Shape Optimization Problem on Planar Sets with Prescribed Topology
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 functionals are of the form $$P(\Omega )T^q(\Omega )|\Omega |^{-2q-1/2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mi>T</mml:mi><mml:mi>q</mml:mi></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>Ω</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:msup><mml:mrow><mml:mo>|</mml:mo><mml:mi>Ω</mml:mi><mml:mo>|</mml:mo></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mn>2</mml:mn><mml:mi>q</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> , and the class of admissible domains consists of two-dimensional open sets $$\Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> satisfying the topological …