A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity
Date
2017-01-26
Authors
Katzourakis, Nikos
Journal Title
Journal ISSN
Volume Title
Publisher
Texas State University, Department of Mathematics
Abstract
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 < ∞.
Description
Keywords
Infinity-Laplacian, p-Laplacian, Generalised solutions, Viscosity solutions, Calculus of variations in L-infinity, Young measures, Fully nonlinear systems
Citation
Katzourakis, N. (2017). A characterisation of infinity-harmonic and p-harmonic maps via affine variations in L-infinity. <i>Electronic Journal of Differential Equations, 2017</i>(29), pp. 1-19.
Rights
Attribution 4.0 International