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| DISCONTINUOUS SOLUTIONS IN $L^\infty$ FOR HAMILTON-JACOBI EQUATIONS |
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Citation： |
CHEN Guiqiang,SU Bo.DISCONTINUOUS SOLUTIONS IN $L^\infty$ FOR HAMILTON-JACOBI EQUATIONS[J].Chinese Annals of Mathematics B,2000,21(2):165~186 |
| Page view： 1236
Net amount： 858 |
Authors： |
CHEN Guiqiang; SU Bo |
Foundation： |
Project supported by the National Science Foundation (DMS-9971793, DMS-9708261). |
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| Abstract： |
An approach is introduced to construct global discontinuous solutions
in $L^{\infty}$ for Hamilton-Jacobi equations. This approach allows the
initial data only in $L^{\infty}$ and applies to the equations with
nonconvex Hamiltonians. The profit functions are introduced to formulate
the notion of discontinuous solutions in $L^\infty$.
The existence of global discontinuous solutions in $L^{\infty}$ is
established.
These solutions in $L^{\infty}$ coincide with the viscosity solutions
and the minimax solutions, provided that the initial data are continuous.
A prototypical equation is analyzed to examine the $L^{\infty}$ stability
of our $L^{\infty}$ solutions. The analysis also shows that global
discontinuous solutions are determined by the topology in which the
initial data are approximated. |
Keywords： |
Hamilton-Jacobi equations, Discontinuous solutions, Profit functions,
Viscosity solutions, Minimax solutions, Stability |
Classification： |
35F25, 35D05, 35B35, 49L99, 90D25 |
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