On the best possible character of the <IMG BORDER="0" SRC="/proc/2003-131-11/S0002-9939-03-07105-3/gif-title0/img1.gif" > norm in some a priori estimates for non-divergence form equations in Carnot groups期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Donatella Danielli Nicola Garofalo Duy-Minh Nhieu

Institution:	Department of Mathematics, Purdue University, West Lafayette, Indiana 47907 ; 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 ; Department of Mathematics, Georgetown University, Washington, DC 20057-1233

Abstract:	Let $\boldsymbol{G}$ be a group of Heisenberg type with homogeneous dimension . 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: in $\Omega$ , on $\partial\Omega$ . This non-uniqueness result shows the impossibility of controlling the maximum of with an norm of when . 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 is the dimension of the horizontal layer of the Lie algebra and $(u_{,ij})$ is the symmetrized horizontal Hessian of .

Keywords:	Alexandrov-Bakelman-Pucci estimate geometric maximum principle horizontal Monge-Amp\`ere equation $\infty$-horizontal Laplacian

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