A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity
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Let u : Ω ⊆ ℝn → ℝN be a smooth map and n, N ∈ ℕ. The ∞-Laplacian is the PDE system
∆∞u ≔ (Du ⊗ Du + |Du|2 [Du]⊥ ⊗ I) : D2u = 0,
where [Du]⊥ ≔ ProjR(Du)⊥. This system constitutes the fundamental equation of vectorial Calculus of Variations in L∞, associated with the model functional
E∞(u, Ω′) = ∥|Du|2∥L∞(Ω′), Ω′ ⋐ Ω.
We show that generalised solutions to the system can be characterised in terms of the functional via a set of designated affine variations. For the scalar case N = 1, we utilise the theory of viscosity solutions by 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 < p < ∞.
CitationKatzourakis, N. (2017). A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity. Electronic Journal of Differential Equations, 2017(29), pp. 1-19.
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