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The scaling free cost functionals are of the form$$P(\\Omega )T^q(\\Omega )|\\Omega |^{-2q-1\/2}$$<\/jats:tex-math>P<\/mml:mi>(<\/mml:mo>\u03a9<\/mml:mi>)<\/mml:mo><\/mml:mrow>T<\/mml:mi>q<\/mml:mi><\/mml:msup>(<\/mml:mo>\u03a9<\/mml:mi>)<\/mml:mo><\/mml:mrow>|<\/mml:mo>\u03a9<\/mml:mi>|<\/mml:mo><\/mml:mrow>-<\/mml:mo>2<\/mml:mn>q<\/mml:mi>-<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and the class of admissible domains consists of two-dimensional open sets$$\\Omega $$<\/jats:tex-math>\u03a9<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>satisfying the topological constraints of having a prescribed numberk<\/jats:italic>of bounded connected components of the complementary set. A relaxed procedure is needed to have a well-posed problem, and we show that when$$q<1\/2$$<\/jats:tex-math>q<\/mml:mi><<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>an optimal relaxed domain exists. When$$q>1\/2$$<\/jats:tex-math>q<\/mml:mi>><\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the problem is ill-posed, and for$$q=1\/2$$<\/jats:tex-math>q<\/mml:mi>=<\/mml:mo>1<\/mml:mn>\/<\/mml:mo>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the explicit value of the infimum is provided in the cases$$k=0$$<\/jats:tex-math>k<\/mml:mi>=<\/mml:mo>0<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and$$k=1$$<\/jats:tex-math>k<\/mml:mi>=<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10957-021-01870-7","type":"journal-article","created":{"date-parts":[[2021,6,1]],"date-time":"2021-06-01T00:41:34Z","timestamp":1622508094000},"page":"760-784","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A Shape Optimization Problem on Planar Sets with Prescribed Topology"],"prefix":"10.1007","volume":"193","author":[{"given":"Luca","family":"Briani","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1769-6801","authenticated-orcid":false,"given":"Giuseppe","family":"Buttazzo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesca","family":"Prinari","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,5,31]]},"reference":[{"key":"1870_CR1","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.na.2016.10.012","volume":"153","author":"G Alberti","year":"2017","unstructured":"Alberti, G., Ottolini, M.: On the structure of continua with finite length and Go\u0142ab\u2019s semicontinuity theorem. 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