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of domains in Riemaniann manifolds giving a geometric and intuitive proof of this fact.Comment: This paper has been withdrawn by the author.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

Reverse Khas'minskii condition

The aim of this paper is to present and discuss some equivalent characterizations of p-parabolicity in terms of existence of special exhaustion functions. In particular, Khas'minskii in [K] proved that if there exists a 2-superharmonic function k defined outside a compact set such that $\lim_{x\to \infty} k(x)=\infty$, then R is 2-parabolic, and Sario and Nakai in [SN] were able to improve this result by showing that R is 2-parabolic if and only if there exists an Evans potential, i.e. a 2-harmonic function $E:R\setminus K \to \R^+$ with \lim_{x\to \infty} \E(x)=\infty. In this paper, we will prove a reverse Khas'minskii condition valid for any p>1 and discuss the existence of Evans potentials in the nonlinear case.Comment: final version of the article available at http://www.springer.co

Sharp estimates on the first eigenvalue of the pp p -Laplacian with negative Ricci lower bound

We complete the picture of sharp eigenvalue estimates for the $$p$$ p -Laplacian on a compact manifold by providing sharp estimates on the first nonzero eigenvalue of the nonlinear operator $$\Delta _p$$ Δ p when the Ricci curvature is bounded from below by a negative constant. We assume that the boundary of the manifold is convex, and put Neumann boundary conditions on it. The proof is based on a refined gradient comparison technique and a careful analysis of the underlying model spaces