Abstract
We use the adjoint methods to study the static Hamilton–Jacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of regularized equations of vanishing viscosity type, and from the solutions σ ε of those we can get the properties of the solutions u of the Hamilton–Jacobi equations. We classify the static equations into two types and present two new ways to deal with each type. The methods can be applied to various static problems and point out the new ways to look at those PDE.
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Acknowledgments
The author would like to express his appreciation to his advisor, Lawrence C. Evans for giving him the problems and plenty of fruitful discussions. The author thank Scott Armstrong, Filippo Cagnetti, Charlie Smart for their helpful discussions and suggestions. Finally, the author would like to thank the anonymous referee for his kind comments and suggestions. The author is supported in part by VEF fellowship.
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Communicated by L. Ambrosio.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.