The comparsion principle for viscosity solutions of fully nonlinear subelliptic equations in Carnot groups

Abstract

For any Carnot group $\mathbf{G}$ and a bounded domain $\Omega \subset \mathbf{G}$, we prove that viscosity solutions in $C(\bar\Om)$ of the fully nonlinear subelliptic equation $F(u,{\nabla }_{h}u,{\nabla }_h^{2}u)=0$ are unique when $F\in C(R\times R^m\times {\Cal S}(m))$ satisfies (i) $F$ is degenerate subelliptic and decreasing in $u$ or (ii) $F$ is uniformly subelliptic and nonincreasing in $u$. This extends Jensen's uniqueness theorem from the Euclidean space to the sub-Riemannian setting of the Carnot group.

15 pages

References in corpus (1)

• Viscosity convex functions on Carnot groups