Stable solutions of symmetric systems involving hypoelliptic operators

Abstract

Let X and Y be two noncommuting vector fields in an open set Ω in a manifold M equipped with a sub-Riemannian structure. We examine stable solutions of the following symmetric systemΔXYui=Hi(u1,⋯,um)in Ω for 1≤i≤m, when the operator ΔXY is the Hörmander's operator given by ΔXY(⋅):=X(X⋅)+Y(Y⋅) and Hi∈C1(Rm). We prove the following identity for any w∈C2(Ω)|∇XYXw|2+|∇XYYw|2−|X|∇XYw||2−|Y|∇XYw||2={|∇XYw|2[A2+B2]in {|∇XYw|>0}∩Ω,0a.e.in {|∇XYw|=0}∩Ω, where A is the intrinsic curvature of the level sets of w and B is connected with the intrinsic normal and the intrinsic tangent direction to the level sets and also with the Lie bracket [X,Y]. We then apply this to establish a geometric Poincaré inequality for stable solutions of the above system for general vector fields X and Y. This inequality enables us to analyze the level sets of stable solutions. In addition, we provide certain reduction of dimensions results which can be regarded as counterparts of the classical De Giorgi type results. This is remarkable since the classical one-dimensional symmetry results do not hold for general vector fields. Our approaches can be applied, but not limited, to the Grushin vector fields X=(1,0) and Y=(0,x) in R2 and the Heisenberg vector fields X=(1,0,−y2) and Y=(0,1,x2) in R3 and their multidimensional extensions. These specific vector fields generate nonelliptic operators which are hypoelliptic.