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 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 establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u t + F (t, d x u) = 0, u(0, x) = u 0 (x), where u 0 : M → R is a bounded uniformly continuous function, M is a Riemannian manifold, and F : [0, ∞) × T * M → R. This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations.First order Hamilton-Jacobi equations are partial differential equations of the form F x, u(x), du(x) = 0 in the stationary case, and of the form, F t, x, u(x, t), du(t, x) = 0 in the evolution case. Such equations arise, for instance, in optimal control theory and differential games. It is very well known that, even in the simplest case where u : R → R, there are examples
AbstraetWe establish approximate Rolle's theorems for the proximal subgradient and for the generalized gradient We also show that an exact Rolle's theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions),
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