# Regularity and Variationality of Solutions to Hamilton—Jacobi Equations. Part II: Variationality, Existence, Uniqueness

Regularity and Variationality of Solutions to Hamilton—Jacobi Equations. Part II:... We formulate an Hamilton–Jacobi partial differential equation \$\$H(x,Du(x))=0\$\$ on a n dimensional manifold M , with assumptions of convexity of the sets \$\{p:H(x,p)\le 0\}\subset T^{*}_{x}M\$ , for all x . We reduce the above problem to a simpler problem; this shows that u may be built using an asymmetric distance (this is a generalization of the “distance function” in Finsler geometry); this brings forth a ‘completeness’ condition, and a Hopf–Rinow theorem adapted to Hamilton–Jacobi problems. The ‘completeness’ condition implies that u is the unique viscosity solution to the above problem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Regularity and Variationality of Solutions to Hamilton—Jacobi Equations. Part II: Variationality, Existence, Uniqueness

, Volume 63 (2) – Apr 1, 2011
26 pages

Publisher
Springer-Verlag
Subject
Mathematics; Numerical and Computational Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-010-9116-7
Publisher site
See Article on Publisher Site

### Abstract

We formulate an Hamilton–Jacobi partial differential equation \$\$H(x,Du(x))=0\$\$ on a n dimensional manifold M , with assumptions of convexity of the sets \$\{p:H(x,p)\le 0\}\subset T^{*}_{x}M\$ , for all x . We reduce the above problem to a simpler problem; this shows that u may be built using an asymmetric distance (this is a generalization of the “distance function” in Finsler geometry); this brings forth a ‘completeness’ condition, and a Hopf–Rinow theorem adapted to Hamilton–Jacobi problems. The ‘completeness’ condition implies that u is the unique viscosity solution to the above problem.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2011

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