Generalized external cone condition for domains in Riemannian manifolds

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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.