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

Authors: Donatella Danielli, Nicola Garofalo and Duy-Minh Nhieu
Journal: Proc. Amer. Math. Soc. 131 (2003), 3487-3498
MSC (2000): Primary 35B50, 22E30, 52A30
Published electronically: June 3, 2003
MathSciNet review: 1991760
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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

\begin{displaymath}\sup_\Omega u\le C\left(\int_{\Omega}\vert\operatorname{det}(u_{,ij})\vert\,dg\right) ^{1/m},\end{displaymath}

where $m$ is the dimension of the horizontal layer of the Lie algebra and $(u_{,ij})$ is the symmetrized horizontal Hessian of $u$.

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Additional Information

Donatella Danielli
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Nicola Garofalo
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 – and – Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Padova, 35131 Padova, Italy

Duy-Minh Nhieu
Affiliation: Department of Mathematics, Georgetown University, Washington, DC 20057-1233

Keywords: Alexandrov-Bakelman-Pucci estimate, geometric maximum principle, horizontal Monge-Amp\`ere equation, $\infty$-horizontal Laplacian
Received by editor(s): June 2, 2002
Published electronically: June 3, 2003
Additional Notes: This work was supported in part by NSF Grant No. DMS-0070492
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2003 American Mathematical Society