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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 |
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| Authors: | Donatella Danielli Nicola Garofalo Duy-Minh Nhieu |
| Affiliation: | 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  be a group of Heisenberg type with homogeneous dimension  . For every  we construct a non-divergence form operator  and a non-trivial solution  to the Dirichlet problem:  in  ,  on  . 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  where  is the dimension of the horizontal layer of the Lie algebra and  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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