  The necessity of a Maximum Principle arises naturally when one is interested
in the study of qualitative properties of solutions to partial differential
equations. In general, to ensure the validity of these kind of principles one
has to consider some additional assumptions on the ambient manifold or on the
differential operator. The present work aims to address, using both of these
approaches, the problem of proving Maximum Principles for second order,
elliptic operators acting on unbounded Riemannian domains under Dirichlet
boundary conditions. Hence there is a natural division of this article in two
distinct and standalone sections.
