2013/04/18 by Nikos Katzourakis, Katzourakis, Nikos
Mathematics · #30C75 #Analysis of PDEs (math.AP) #FOS: Mathematics #Primary 30C70 #Secondary 35J47 #math.AP #msc:30C70 #msc:30C75 #msc:35J47
paper · pdf · doi:10.48550/arxiv.1304.5273
16 pages, 3 figures, to appear in Comm. on Pure Appl. Anal
arxiv created 2014/04/15 · arxiv updated 2014/04/16
For a Hamiltonian H ∈ C2(ℝN × n) and a map u:Ω⊆ ℝn /!\longrightarrow ℝN, we consider the supremal functional E_∞ (u,Ω) := ‖H(Du)‖L^∞(Ω) . The "Euler-Lagrange" PDE associated to \eqref1 is the quasilinear system A_∞ u := (HP ⊗ HP + H[HP]^\bot /! HPP)(Du):D2 u = 0. \eqref1 and \eqref2 are the fundamental objects of vector-valued Calculus of Variations in L^∞ and first arose in recent work of the author [K1]. Herein we show that the Dirichlet problem for \eqref2 admits for all n=N≥ 2 infinitely-many smooth solutions on the punctured ball, in the case of H(P)=|P|2 for the ∞-Laplacian and of H(P)= |P|2det(P^\top /! P)-1/n for optimised Quasiconformal maps. Nonuniqueness for the linear degenerate elliptic system A(x):D2u =0 follows as a corollary. Hence, the celebrated L^∞ scalar uniqueness theory of Jensen [J] has no counterpart when N≥ 2. The key idea in the proofs is to recast \eqref2 as a first order differential inclusion Du(x) ∈ K ⊆ ℝn× n, x∈ Ω.