Generalized external cone condition for domains in Riemannian manifolds

TL;DR

This paper provides a new geometric proof for the regularity of domains in Riemannian manifolds, focusing on the Dirichlet problem and p-regularity, using a generalized external cone condition.

Contribution

It introduces an alternative, geometric proof for the regularity of domains, including those with submanifolds of variable codimension, in Riemannian manifolds.

Findings

Established a generalized external cone condition for domain regularity

Proved p-regularity for domains of the form O-K in Riemannian manifolds

Provided an intuitive geometric proof for regularity criteria

Abstract

The aim of this note is to present an alternative proof for an already known result relative to the solvability of the Dirichlet problem in Riemannian manifolds (see remark 0.1). In particular, we discuss the p-regularity (regularity relative to the p-laplacian) of domains of the form I = O-K, where O is a regular domain and K is a regular submanifold of variable codimension (see theorem 4.4). In theorem 5.1 we prove a sort of generalized external cone condition for the regularity of domains in Riemaniann manifolds giving a geometric and intuitive proof of this fact.