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Recent Advances in Partial Differential Equations and Applications

2016; American Mathematical Society; Linguagem: Inglês

10.1090/conm/666

1098-3627

VicenÅ£iu Rădulescu, Adélia Sequeira, V. A. Solonnikov,

Differential Equations and Numerical Methods

We consider spectral optimization problems of the form min λ1(Ω; D) : Ω ⊂ D, |Ω| = 1 , where D is a given subset of the Euclidean space R d .Here λ1(Ω; D) is the first eigenvalue of the Laplace operator -∆ with Dirichlet conditions on ∂Ω ∩ D and Neumann or Robin conditions on ∂Ω ∩ ∂D.The equivalent variational formulation λ1(Ω; D) = minreminds the classical drop problems, where the first eigenvalue replaces the total variation functional.We prove an existence result for general shape cost functionals and we show some qualitative properties of the optimal domains.