Almost continuous solutions of geometric Hamilton–Jacobi equations

✍ Scribed by Antonio Siconolfi

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 197 KB
• Volume: 20
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(02)00010-0

✦ Synopsis

We study the Hamilton-Jacobi equation

where H is a continuous positively homogeneous Hamiltonian with constant sign and verifying suitable assumptions but no convexity properties. We look for discontinuous (viscosity) solutions verifying certain initial conditions with discontinuous data. Our aim is to give representation formulae as well as uniqueness and stability results.

We find that the condition

where u # (u # ) denotes the upper (lower) semicontinuous envelope of u, can be used as a uniqueness criterion and determines a class of solutions defined and continuous on certain dense subsets of R N × ]0, +∞[ that we call almost continuous. Such solutions can be represented by a formula which is a generalization of the Lax-Hopf one for the eikonal equation.