Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds

By D. Azagra, J. Ferrera, Fernando López-Mesas. 2003

176 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8b15070c4a20523567250a3345f5167d32d2f5db

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Prescribing symmetric functions of the eigenvalues of the Ricci tensor (2004, background): “on |∇wdw|w = 1, away from the base point x 0. Proof. First we observe that each dw i = dw i (x 0,·) is a viscosity solution of the Hamilton-Jacobi equation |∇idw i |w i = 1 away from x 0 (see [MM03], [AFLM05]). Using the convergence of the di from Lemma 7.5, and convergence of the metrics in 46 MATT”
Global and fine approximation of convex functions (2012, background): “To see that the reverse inequality holds on B, take x ∈ B and a subdifferential ζ ∈ D−f(x) (we refer to [3], [2] for the definitions and some properties of the Fréchet subdifferential and inf convolution on Riemannian manifolds).”
Smooth approximation of Lipschitz functions on Riemannian manifolds (2006, methodology): “In [2] we were able to prove such a variational principle under the assumption that the manifold was uniformly bumpable (see Definition 2), but the question whether or not every Riemannian manifold is uniformly bumpable remained open.”
Invex sets and preinvex functions on Riemannian manifolds (2007, background): “Recall that a subset S of a Riemannian manifold is called geodesic convex if any two points x, y ∈ S can be joined by exactly one geodesic of length d(x, y) which belongs entirely to S (see [2,11]).”
Proximal point method for a special class of nonconvex functions on Hadamard manifolds (2008, methodology): “Rþþ with initial conditions x(0)¼ x0 and x0(0)¼ v is given by xðtÞ ¼ x0eðv=x0Þt: From the above equation it can be easily seen that any interval I Rþþ is a convex set of the manifold (Rþþ, h, i).”
Monotone and Accretive Vector Fields on Riemannian Manifolds (2010, background): “On the other hand, various derivative-like and subdifferential constructions for nondifferentiable functions on spaces with nonlinear structure have been developed and applied to the study of constrained optimization problems, nonclassical problems of the calculus of variations and optimal control; ”