The Dirichlet problem for a family of totally degenerate differential operators

TL;DR

This paper establishes existence and regularity criteria for solutions to the Dirichlet problem involving a broad class of totally degenerate differential operators, extending potential theory methods to these complex cases.

Contribution

It proves existence and regularity results for the Dirichlet problem for a wide family of strongly degenerate operators, including new Wiener-type criteria and boundary conditions.

Findings

Existence of Perron-Weiner-Brelot solutions for degenerate operators

Wiener-type criterion for boundary regularity

Exterior cone condition for boundary point regularity

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 $\mathcal{L} = t^2\Delta_x + \langle x, \nabla_y \rangle -\partial_t$, for $(x,y,t) \in \mathbb{R}^N \times \mathbb{R}^{N} \times \mathbb{R}$.