Discontinuous solutions in L ∞ for Hamilton-Jacobi equations

Journal article

An approach is introduced to construct global discontinuous solutions in L ∞ for Hamilton-Jacobi equations. This approach allows the initial data only in L ∞ and applies to the equations with nonconvex Hamiltonians. The profit functions are introduced to formulate the notion of discontinuous solutions in L ∞. The existence of global discontinuous solutions in L ∞ is established. These solutions in L ∞ coincide with the viscosity solutions and the minimax solutions, provided that the initial data are continuous. A prototypical equation is analyzed to examine the L ∞ stability of our L ∞ solutions. The analysis also shows that global discontinuous solutions are determined by the topology in which the initial data are approximated. © 1983 Shanghai Scientific and Technological Literature Publishing House.

Authors: Guiqiang, C; Bo, S

• Publisher: Kluwer Academic Publishers
• Journal: Chinese Annals of Mathematics
• Volume: 21
• Issue: 2
• Pages: 165-186
• Publication date: 2000-04-01
• DOI: 10.1007/BF02484190
• EISSN: 1860-6261
• ISSN: 0252-9599
• Language: English
• Keywords: O175.29; Viscosity solutions; O175.26; Profit functions; Hamilton-Jacobi equations; 1991 MR Subject Classification: 35F25, 35D05, 35B35, 49L99, 90D25; Discontinuous solutions; Minimax solutions; Stability