The Aronsson equation for absolute minimizers of L∞-functionals associated with vector fields satisfying Hörmander's condition

الأصل · EN

Given a Carnot-Carathéodory metric space (Rⁿ, dcc) generated by vector fields {Xᵢ}ᵢ₌₁ᵐ satisfying Hörmander's condition, we prove in theorem A that any absolute minimizer u∈ W¹,∞cc() to F(v,)=∈(x,Xv(x)) is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on f. In particular, any AMLE is a viscosity solution to the subelliptic ∞-Laplacian equation (1.7). If the Carnot-Carathédory space is a Carnot group G and f is independent of x-variable, we establish in theorem C the uniquness of viscosity solutions to the Aronsson equation (1.13) under suitable conditions on f. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic ∞-Laplacian equation is established in G