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.
- Alexandrov-Bakelman-Pucci estimate
- Geometric maximum principle
- Horizontal Monge-Ampère equation
- ∞-horizontal Laplacian
ASJC Scopus subject areas
- Applied Mathematics