We consider the problem of finding the optimal constant for the embedding of the space W2,1Δ,0(Ω):={u∈W1,10(Ω)∣∣there exists {uk}⊂C∞c(Ω) s.t. ∥Δuk−Δu∥1→0} into the space L1(Ω), where Ω⊂Rn is a bounded domain with boundary of class C1,1. This is equivalent to find the first eigenvalue Λc1,1(Ω) of the clamped 1-biharmonic operator. In this paper, we identify the correct relaxation of the problem on BL0(Ω), the space of functions whose distributional Laplacian is a finite Radon measure, we obtain the associated Euler–Lagrange equation, and we give lower bounds for Λc1,1(Ω), investigating the validity of an inequality of Faber–Krahn type.
Higher order functional inequalities related to the clamped 1-biharmonic operator / E. Parini, B. Ruf, C. Tarsi. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 194:6(2015 Dec), pp. 1835-1858. [10.1007/s10231-014-0447-x]
Higher order functional inequalities related to the clamped 1-biharmonic operator
B. Ruf
;C. Tarsi
2015
Abstract
We consider the problem of finding the optimal constant for the embedding of the space W2,1Δ,0(Ω):={u∈W1,10(Ω)∣∣there exists {uk}⊂C∞c(Ω) s.t. ∥Δuk−Δu∥1→0} into the space L1(Ω), where Ω⊂Rn is a bounded domain with boundary of class C1,1. This is equivalent to find the first eigenvalue Λc1,1(Ω) of the clamped 1-biharmonic operator. In this paper, we identify the correct relaxation of the problem on BL0(Ω), the space of functions whose distributional Laplacian is a finite Radon measure, we obtain the associated Euler–Lagrange equation, and we give lower bounds for Λc1,1(Ω), investigating the validity of an inequality of Faber–Krahn type.File | Dimensione | Formato | |
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