A New Characterisation of ∞-Harmonic and p-Harmonic Maps via Affine Variations in L^∞

2015/09/06 by Nikos Katzourakis, Katzourakis, Nikos · 3 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1509.01811

openalex publication_date 2015/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let u: Ω⊆ ℝⁿ \longrightarrow ℝ^N be a smooth map and n,N ∈ ℕ. The ∞-Laplacian is the PDE system Δ_∞ u := (Du ⊗ Du + |Du|²[Du]^\bot ⊗ I) :D²u = 0, where [Du]^\bot := Proj_R(Du)^\bot. \eqref1 constitutes the fundamental equation of vectorial Calculus of Variations in L^∞, associated to the model functional E_∞ (u,Ω') = ‖ |Du|²‖_L^∞(Ω') , Ω' \Subset Ω. We show that generalised solutions to \eqref1 can be characterised in terms of \eqref2 via a set of designated affine variations. For the scalar case N=1, we utilise the theory of viscosity solutions of Crandall-Ishii-Lions. For the vectorial case N≥ 2, we utilise the recently proposed by the author theory of D-solutions. Moreover, we extend the result described above to the p-Laplacian, 1

Citations

• Absolutely Minimising Generalised Solutions to the Equations of Vectorial Calculus of Variations in L^∞

Cited by

• A Remark on Global W^1,p Bounds for Harmonic Functions with\n Lipschitz Boundary Values
• Second Order L^∞ Variational Problems and the ∞-Polylaplacian
• On the Numerical Approximation of ∞-Harmonic Mappings

Related

• On the Numerical Approximation of ∞-Harmonic Mappings
• On the structure of ∞-Harmonic maps
• A Pointwise Characterisation of the PDE System of Vectorial Calculus of Variations in L^∞
• Counterexamples in Calculus of Variations in L^∞ through the vectorial Eikonal equation
• Explicit Infinity-Harmonic Maps whose Interfaces have Junctions and Corners