On the best possible character of the $L^Q$ norm in some a priori estimates for non-divergence form equations in Carnot groups

Abstract:

Let $\boldsymbol {G}$ be a group of Heisenberg type with homogeneous dimension $Q$. For every $0<\epsilon <Q$ we construct a non-divergence form operator $L^\epsilon$ and a non-trivial solution $u^\epsilon \in \mathcal {L}^{2,Q-\epsilon }(\Omega )\cap C(\overline {\Omega })$ to the Dirichlet problem: $Lu=0$ in $\Omega$, $u=0$ on $\partial \Omega$. This non-uniqueness result shows the impossibility of controlling the maximum of $u$ with an $L^p$ norm of $Lu$ when $p<Q$. Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as \[ \sup _\Omega u\le C\left (\int _{\Omega }|\operatorname {det}(u_{,ij})| dg\right ) ^{1/m},\] where $m$ is the dimension of the horizontal layer of the Lie algebra and $(u_{,ij})$ is the symmetrized horizontal Hessian of $u$.