The Lax solution to a Hamilton–Jacobi equation and its generalizations: Part 2

✍ Scribed by Ya.V. Mykytiuk; A.K. Prykarpatsky; D. Blackmore

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 232 KB
• Volume: 55
• Category: Article
• ISSN: 0362-546X
• DOI: 10.1016/j.na.2003.08.004

✦ Synopsis

It is proved that the function deÿned by the inÿmum-based Lax formula (for viscosity solutions) provides a solution almost everywhere in x for each ÿxed t ¿ 0 to the Hamilton-Jacobi, Cauchy problem ut + 1 2 ∇u 2 = 0; u(x; 0 + ) = v(x);

where the Cauchy data function v is lower semicontinuous on real n-space. In addition, a generalization of the Lax formula is developed for the more inclusive Hamilton-Jacobi equation ut + 1 2 ( ∇u 2 -ÿu u 2 + Jx; x ) = 0; where J is a diagonal, positive-deÿnite matrix.