The Aronsson equation for absolute minimizers of ${L^\infty}$-functionals associated with vector fields satisfying Hörmander's condition

By Changyou Wang. 2003

50 citations recorded by the source corpus.

Given a Carnot-Caratheodory metric space (R", d X ) generated by vector fields {X i } m i=1 satisfying Hormander's condition, we prove in Theorem A that any absolute minimizer u ∈ W 1,∞ X (Ω) to F(v, Q) = ess sup x∈Ω f(x, Xv(x)) is a viscosity solution to the Aronsson equation - Σ X i (f(x, X u (x)))f pi (x, X u (x)) = 0, in Ω, under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic oo-Laplacian equation Δ (X) ∞ u:= - Σ X i uX j uX i X j u = 0, in Ω. If the Carnot-Caratheodory space is a Carnot group G and f is independent of the x-variable, we establish in Theorem C the uniqueness of viscosity solutions to the Aronsson equation A(Xu,(D 2 u)*):= - Σ f pi (Xu)f pj (Xu)X i X j u = 0, in Ω, u = Φ, on ∂Ω, under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic oo-Laplacian equation is established on any Carnot group G.

Subfields: Applied Mathematics, Genetics

View Paper: https://www.semanticscholar.org/paper/8f9c7c4fc090862f93b9499996447f62b62c1d67

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

  1. G. Lu
  2. Juan J. Manfredi
  3. Petri Juutinen
  4. Bianca Stroffolini
  5. Nguyen Huong Lam
  6. Valentino Magnani
  7. Caifeng Zhang
  8. Changyou Wang

In Genetics

  1. G. Lu

Citations in the corpus