Affordable Access

Publisher Website

Viscosity solutions for partial differential equations with Neumann type boundary conditions and some aspects of Aubry–Mather theory

Authors
Journal
Journal of Mathematical Analysis and Applications
0022-247X
Publisher
Elsevier
Publication Date
Volume
336
Issue
1
Identifiers
DOI: 10.1016/j.jmaa.2007.02.079
Keywords
  • Reflected Control Problems
  • Boundary Reflection
  • Viscosity Solutions
  • Hamilton–Jacobi Inequalities
  • Aubry–Mather Set

Abstract

Abstract We study partial differential inequalities (PDI) of the type c + H ( x , ∂ v ∂ x ) − N K ( x ) ∋ 0 where N K ( ⋅ ) is the normal cone to the set K. We prove existence of a constant c : = c ¯ such that the PDI of Hamilton–Jacobi type has a unique (global) Lipschitz viscosity solution. We provide a formula to calculate this constant. Moreover, we define a subset K ˜ of K such that any two solutions of the previous PDI which coincide on K ˜ will coincide on K. Our paper generalizes results of the case without boundary conditions for convex Hamiltonians obtained by L.C. Evans and A. Fathi.

There are no comments yet on this publication. Be the first to share your thoughts.