### The Dirichlet problem for a class of ultraparabolic equations

Maria Manfredini

#### Abstract

In this paper we study the Dirichlet problem for a class of ultraparabolic equations. More precisely, we prove the existence of a generalized Perron-Wiener solution and we provide a geometric condition for the regularity of the boundary points which extends the classical Zaremba exterior cone criterion to our setting. The main steps for deriving our results are: i) the introduction in $\mathbf{R}^{N+1}$ of a homogeneous structure; ii) the proof of some interior estimates in a suitable space of Hölder-continuous functions; iii) the construction of a basis of open subsets of $\mathbf{R}^{N+1}$ for which the Dirichlet problem is univocally solvable.

#### Article information

Source
Adv. Differential Equations Volume 2, Number 5 (1997), 831-866.

Dates
First available in Project Euclid: 22 April 2013