Higher order functional inequalities related to the clamped 1-biharmonic operator

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.