2012/04/24 by Nicholas Katzourakis, Katzourakis, Nicholas
Computer Science · Mathematics · #35J62 #53C24 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Primary 35J47 #Secondary 49J99 #math.AP #math.CA #msc:35J47 #msc:35J62 #msc:49J99 #msc:53C24
paper · pdf · doi:10.48550/arxiv.1204.5374
30 pages, 10 figures, revised including referees' comments, (Communications in PDE)
openalex publication_date 2012/04/24 · arxiv created 2014/01/07 · arxiv updated 2014/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Let H ∈ C2(ℝN × n), H≥ 0. The PDE system A_∞ u := (HP ⊗ HP + H [HP]^\bot HPP )(Du) : D2 u = 0 arises as the ``Euler-Lagrange PDE" of vectorial variational problems for the functional E∞(u,Ω) = ‖ H(Du) ‖L^∞(Ω) defined on maps u : Ω⊆ ℝn \longrightarrow ℝN. \eqref1 first appeared in the author's recent work \citeK3. The scalar case though has a long history initiated by Aronsson in \citeA1. Herein we study the solutions of \eqref1 with emphasis on the case of n=2≤ N with H the Euclidean norm on ℝN × n, which we call the ``∞-Laplacian". By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non-existence of zeros of |Du| and prove a Maximum Principle. We further characterise all H for which \eqref1 is elliptic and also study the initial value problem for the ODE system arising for n=1 but with H(⋅,u,u') depending on all the arguments.