We prove the existence and regularity of optimal shapes for the problem min{P(Ω)+G(Ω):Ω⊂D,|Ω|=m}, where P denotes the perimeter, |⋅| is the volume, and the functional G is either one of the following: • the Dirichlet energy Ef, with respect to a (possibly sign-changing) function f∈Lp;• a spectral functional of the form F(λ1,…,λk), where λk is the kth eigenvalue of the Dirichlet Laplacian and F:Rk→R is locally Lipschitz continuous and increasing in each variable.The domain D is the whole space Rd or a bounded domain. We also give general assumptions on the functional G so that the result remains valid.
Regularity of minimizers of shape optimization problems involving perimeter / De Philippis, G.; Lamboley, J.; Pierre, M.; Velichkov, B.. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - 109:(2018), pp. 147-181. [10.1016/j.matpur.2017.05.021]
Regularity of minimizers of shape optimization problems involving perimeter
De Philippis G.;Velichkov B.
2018
Abstract
We prove the existence and regularity of optimal shapes for the problem min{P(Ω)+G(Ω):Ω⊂D,|Ω|=m}, where P denotes the perimeter, |⋅| is the volume, and the functional G is either one of the following: • the Dirichlet energy Ef, with respect to a (possibly sign-changing) function f∈Lp;• a spectral functional of the form F(λ1,…,λk), where λk is the kth eigenvalue of the Dirichlet Laplacian and F:Rk→R is locally Lipschitz continuous and increasing in each variable.The domain D is the whole space Rd or a bounded domain. We also give general assumptions on the functional G so that the result remains valid.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
