The Dirichlet problem for a family of totally degenerate differential operators

Abstract

In the framework of Potential Theory we prove existence or the Perron-Weiner-Brelot-Bauer solution to the Dirichlet problem related to a family of totally degenerate, in the sense of Bony, differential operators. We also state and prove a Wiener-type criterium and an exterior cone condition for the regularity of a boundary point. Our results apply to a wide family of strongly degenerate operators that includes the following example $\mathcal{L}=t^2{\Delta }_x+⟨x,{\nabla }_y⟩-{\partial }_t$, for $(x,y,t)\in {\mathbb{R}}^N\times {\mathbb{R}}^N\times \mathbb{R}$.

32 pages