2021/03/29 by Nikos Katzourakis, Katzourakis, Nikos
Computer Science · Mathematics · #35D30 #35D40 #35J47 #35J70 #35J92 #35J99 #35P30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:35D30 #msc:35D40 #msc:35J47 #msc:35J70 #msc:35J92 #msc:35J99 #msc:35P30
paper · pdf · doi:10.48550/arxiv.2103.15911
30 pages, Journal: Nonlinear Analysis (in press)
openalex publication_date 2021/03/29 · arxiv created 2022/02/04 · arxiv updated 2022/02/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Let Ω\Subset \mathbb Rn, f ∈ C1(\mathbb RN× n) and g∈ C1(\mathbb RN), where N,n ∈ \mathbb N. We study the minimisation problem of finding u ∈ W1,∞0(Ω;\mathbb RN) that satisfies ‖ f(\mathrm D u) ‖L^∞(Ω) = inf \‖ f(\mathrm D v) ‖L^∞(Ω) : v ∈ W1,∞0(Ω;\mathbb RN), ‖ g(v) ‖L^∞(Ω) =1\, under natural assumptions on f,g. This includes the ∞-eigenvalue problem as a special case. Herein we prove existence of a minimiser u_∞ with extra properties, derived as the limit of minimisers of approximating constrained Lp problems as p→ ∞. A central contribution and novelty of this work is that u_∞ is shown to solve a divergence PDE with measure coefficients, whose leading term is a divergence counterpart equation of the non-divergence ∞-Laplacian. Our results are new even in the scalar case of the ∞-eigenvalue problem.