We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F (x, u, du, d 2 u) = 0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d 2 u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.

We show that for every Lipschitz function f defined on a separable Riemannian manifold M (possibly of infinite dimension), for every continuous ε : M → (0, +∞), and for every positive number r > 0, there exists a C ∞ smooth Lipschitz function g : M → R such that |f (p) − g(p)| ε(p) for every p ∈ M and Lip(g) Lip(f ) + r. Consequently, every separable Riemannian manifold is uniformly bumpable. We also present some applications of this result, such as a general version for separable Riemannian manifolds of Deville-Godefroy-Zizler's smooth variational principle.

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M . We give some applications of this theory, concerning, for instance, a Borwein-Preiss type variational principle on a Riemannian manifold M , as well as differentiability and geometrical properties of the distance function to a closed subset C of M . (2000). 49J52, 58E30, 58C30, 47H10. Mathematics Subject Classification

We show how an operation of inf-convolution can be used to approximate convex functions with C 1 smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima and sets of minimizers), and we give some applications.