2016/11/18 by Ayanbayev, Birzhan, Katzourakis, Nikos
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1611.05936
Let n,N∈ ℕ with Ω⊆ ℝn open. Given H ∈ C2(Ω× ℝN× ℝNn), we consider the functional E_∞ (u,O) := \undersetOess sup H (⋅,u,D u) , u∈ W1,∞loc(Ω,ℝN), O \Subset Ω. The associated PDE system which plays the role of Euler-Lagrange equations in L^∞ is \ HP(⋅, u, Du) D (H(⋅, u, D u)) = 0, H(⋅, u, D u) [ [HP(⋅, u, D u)] ]^\bot (Div(HP(⋅, u, D u))- Hη(⋅, u, D u)) = 0, . where [ [A] ]^\bot := ProjR(A)^\bot. Herein we establish that generalised solutions to \eqref2 can be characterised as local minimisers of \eqref1 for appropriate classes of affine variations of the energy. Generalised solutions to \eqref2 are understood as D-solutions, a general framework recently introduced by one of the authors.