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

Donatella Danielli, Nicola Garofalo, Duy Minh Nhieu

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

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 uε ∈ ℒ2,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 Lp norm of Lu when p < Q. Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as supΩ 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 languageEnglish (US)
Pages (from-to)3487-3498
Number of pages12
JournalProceedings of the American Mathematical Society
Volume131
Issue number11
DOIs
StatePublished - Nov 2003
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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