## Abstract

Let G be a group of Heisenberg type with homogeneous dimension Q. For every 0 < ε < Q we construct a non-divergence form operator L ^{ε} and a non-trivial solution u^{ε} ∈ ℒ^{2,Q-ε} (Ω) ∩ C (Ω̄) to the Dirichlet problem: Lu = 0 in Ω, u = 0 on ∂Ω. 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_{Ω} u ≤ C (∫_{Ω} | det(u,ij)| dg)^{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.

Original language | English |
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Pages (from-to) | 3487-3498 |

Number of pages | 12 |

Journal | Proceedings of the American Mathematical Society |

Volume | 131 |

Issue number | 11 |

DOIs | |

State | Published - Nov 2003 |

## Keywords

- Alexandrov-Bakelman-Pucci estimate
- Geometric maximum principle
- Horizontal Monge-Ampère equation
- ∞-horizontal Laplacian

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