The eigenvalue problem for the $$\infty $$-Bilaplacian

Abstract

Abstract. We consider the problem of finding and describing minimisers of the Rayleigh quotient Λ∞ := infwhere Ω ⊆ R n is a bounded C 1,1 domain and W 2,∞ (Ω) is a class of weakly twice differentiable functions satisfying either u = 0 on ∂Ω or u = |Du| = 0 on ∂Ω. Our first main result, obtained through approximation by L p -problems as p → ∞, is the existence of a minimiser u∞ ∈ W 2,∞ (Ω) satisfying ∆u∞ ∈ Λ∞Sgn(f∞) a.e. in Ω,for some f∞ ∈ L 1 (Ω) ∩ BV loc (Ω) and a measure µ∞ ∈ M(Ω), for either choice of boundary conditions. Here Sgn is the multi-valued sign function. We also study the dependence of the eigenvalue Λ∞ on the domain, establishing the validity of a Faber-Krahn type inequality: among all C 1,1 domains with fixed measure, the ball is a strict minimiser of Ω → Λ∞(Ω). This result is shown to hold true for either choice of boundary conditions and in every dimension.

Existence, Uniqueness and Structure of Second Order Absolute Minimisers

“…Preliminary investigations on the second order 1-dimensional case H(·, u, u , u ) had previously been performed via different methods by Aronsson and Aronsson-Barron in [6,7]. Very recently, arguments inspired by the present paper have been applied in [34] to eigenvalue problems for the ∞-Bilaplacian. It is remarkable that for our specific variational problem, global minimisers are unique and are automatically absolute minimisers.…”

Weak vs. 𝒟-solutions to linear hyperbolic first-order systems with constant coefficients

“…Diffuse derivatives can be seen as measure-theoretic disintegrations whose barycentres are the distributional derivatives (see [29]). For further results relevant to D-solutions and their applications, see [30]- [32], [34,10,35,13], [36]- [38].…”

A minimisation problem in L∞with PDE and unilateral constraints

“…In this paper, motivated by the problem of optical tomography and by the recent developments in Calculus of Variations in L ∞ appearing in the papers [38,39,40], we consider the problem of minimising over the class of all admissible parameters ξ a certain cost functional which measures the deviation of the solution v on the boundary ∂Ω from some predictionṽ of its values. Given the high complexity of the optical tomography problem, in this work which is the companion paper of [37] we will make the simplifying assumption that the diffusion coefficients A, B and the optical terms K, L do not depend explicitly on the dye distribution.…”