Diffusion-type operators,Liouville theorems and gradient estimates on complete manifolds

We study Liouville theorems and gradient estimates for solutions of Eq. (1.1) with the help of a diffusion operator and the related geometry.     

On the compact support principle on complete manifolds

In this paper we study the compact support principle for singular elliptic inequalities on complete manifolds with a significant dependence on the spatial variable. In the process we underline the influence in our results both of the spatial variable and of the geometry of the ambient space.     

A remark on the maximum principle and stochastic completeness

We prove that the stochastic completeness of a Riemannian manifold is equivalent to the validity of a weak form of the Omori-Yau maximum principle. Some geometric applications of this result are also presented.

     

Finite element approximation of a model for phase separation of a multi-component alloy with a concentration-dependent mobility matrix

We consider a model for phase separation of a multi-componentalloy with a concentration-dependent mobility matrix and logarithmicfree energy. In particular we prove that there exists a uniquesolution for sufficiently smooth initial data. Further, we provean error bound for a fully practical piecewise linear finiteelement approximation in one and two space dimensions. Finallynumerical experiments with three components in one space dimensionare presented.     

Spherical-type hypersurfaces in a Riemannian manifold

We extend some rigidity results of Aleksandrov and Ros on compact hypersurfaces inR ⁿ to more general ambient spaces with the aid of the notion of almost conformal vector fields. These latter, at least locally, always exist and allow us to find interesting integral formulas fitting our purposes.     

A Liouville theorem for a class of superlinear elliptic equations on cones

We give sufficient conditions for non-existence of positive solutions of the equation on a cone of We further analyze the existence of positive solutions in the radial, subcritical case, and show that under suitable conditions on the coefficients, every radial solution whose value in 0 is sufficiently large must vanish. Received April 2000