2017/03/10 by Nikos Katzourakis, Katzourakis, Nikos, Enea Parini +1
Computer Science · Mathematics · #35D40 #35D99 #35G20 #35G30 #35J91 #35P15 #35P30 #49R05 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · doi:10.48550/arxiv.1703.03648
openalex publication_date 2017/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of finding and describing minimisers of the Rayleigh quotient Λ_∞ := inf_u∈ W2,∞(Ω)∖\0\ \frac‖Δu‖L^∞(Ω)‖u‖L^∞(Ω), where Ω⊆ ℝn is a bounded C1,1 domain and W2,∞(Ω) is a class of weakly twice differentiable functions satisfying either u=0 or u=|D u|=0 on ∂ Ω. Our first main result, obtained through approximation by Lp-problems as p→ ∞, is the existence of a minimiser u_∞ ∈ W2,∞(Ω) satisfying \ Δu_∞ ∈ Λ_∞ Sgn(f_∞) · amp; a.e. in Ω,
Δf_∞ = μ_∞ · amp; in D'(Ω), . for some f_∞∈ L1(Ω)∩ BVloc(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 C1,1 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.