Generalized external cone condition for domains in Riemannian manifolds

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of domains in Riemaniann manifolds giving a geometric and intuitive proof of this fact.Comment: This paper has been withdrawn by the author. This problem has already been solved, and in a much more general way than I did. Moreover, in the field it appears to be a standard resul

On the equivalence of stochastic completeness, Liouville and Khas'minskii condition in linear and nonlinear setting

Set in Riemannian enviroment, the aim of this paper is to present and discuss some equivalent characterizations of the Liouville property relative to special operators, in some sense modeled after the p-Laplacian with potential. In particular, we discuss the equivalence between the Lioville property and the Khas'minskii condition, i.e. the existence of an exhaustion functions which is also a supersolution for the operator outside a compact set. This generalizes a previous result obtained by one of the authors and answers to a question in "Aspects of potential theory, linear and nonlinear" by Pigola, Rigoli and Setti.Comment: 34 pages. The pasting lemma has been improved to fix a technical problem in the main theorem. Final version, to appear on Trans. Amer. Math. So

Stokes' theorem, volume growth and parabolicity

We present some new Stokes' type theorems on complete non-compact manifolds that extend, in different directions, previous work by Gaffney and Karp and also the so called Kelvin-Nevanlinna-Royden criterion for (p-)parabolicity. Applications to comparison and uniqueness results involving the p-Laplacian are deduced.Comment: 15 pages. Corrected typos. Accepted for publication in Tohoku Mathematical Journa