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.