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A simple proof of the propagation of singularities for solutions of Hamilton-Jacobi equations. (English) Zbl 1150.35002

The author gives a simpler proof of a result by P. Albano and P. Cannarsa [Arch. Ration. Mech. Anal. 162, No. 1, 1–23 (2002; Zbl 1043.35052)] regarding the propagation of singularities of a semiconcave viscosity solution of a Hamilton-Jacobi equation. The proof is based on an approximation of the viscosity solution by smooth functions.

MSC:

35A21 Singularity in context of PDEs
35F20 Nonlinear first-order PDEs

Citations:

Zbl 1043.35052
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References:

[1] P. Albano and P. Cannarsa, Propagation of singularities for solutions of nonlinear first order partial differential equations, Arch. Ration. Mech. Anal. 162 (2002), 1-23. Zbl1043.35052 MR1892229 · Zbl 1043.35052 · doi:10.1007/s002050100176
[2] P. Albano and P. Cannarsa, Structural properties of singularities of semiconcave functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 719-740. Zbl0957.26002 MR1760538 · Zbl 0957.26002
[3] L. Ambrosio, P. Cannarsa and H. M. Soner, On the propagation of singularities of semi-convex functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20 (1993), 597-616. Zbl0874.49041 MR1267601 · Zbl 0874.49041
[4] P. Cannarsa and C. Sinestrari, “Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control”, Progress in Nonlinear Differential Equations and their Applications, Vol. 58, Birkhäuser Boston, Inc., Boston, MA, 2004. Zbl1095.49003 MR2041617 · Zbl 1095.49003
[5] L. C. Evans, “Partial Differential Equations”, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998. Zbl0902.35002 MR1625845 · Zbl 0902.35002
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