Liouville theorem for X-elliptic operators

Abstract

We prove a Liouville-type theorem for a class of degenerate elliptic operators of the form L u ≔ ∑ i , j = 1 N ∂ x i ( a i j ∂ x j u ) + ∑ i = 1 N b i ∂ x i u . L is supposed to be X -elliptic, with respect to a family X = ( X 1 , … , X m ) of locally Lipschitz continuous vector fields, in the sense introduced in [E. Lanconelli, A.E. Kogoj, X -elliptic operators and X -control distances, Contributions in Honor of the Memory of Ennio De Giorgi, Ricerche di Matematica 49 (Suppl.) (2000) 223–243].