no. 2

A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 367-376.

We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the hamiltonian. The proof relies on a reverse Hölder inequality.

DOI : https://doi.org/10.1051/cocv:2008028
Classification : 35F20,  49L25
Mots clés : Hamilton-Jacobi equation, viscosity solutions, optimal control, regularity, reverse Hölder inequality 
@article{COCV_2009__15_2_367_0,
     author = {Cardaliaguet, Pierre},
     title = {A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {367--376},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {2},
     year = {2009},
     doi = {10.1051/cocv:2008028},
     zbl = {1175.35030},
     mrnumber = {2513090},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2009__15_2_367_0/}
}
Cardaliaguet, Pierre. A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 2, pp. 367-376. doi : 10.1051/cocv:2008028. http://www.numdam.org/item/COCV_2009__15_2_367_0/

[1] M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser (1996). | MR 1484411 | Zbl 0890.49011

[2] G. Barles, Regularity results for first order Hamilton-Jacobi equations. Differ. Integral Equ. 3 (1990) 103-125. | MR 1014729 | Zbl 0739.35012

[3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Berlin (1994). | MR 1613876 | Zbl 0819.35002

[4] A. Bensoussan and J. Frehse, Regularity results for nonlinear elliptic systems and applications, Applied Mathematical Sciences 151. Springer-Verlag, Berlin (2002). | MR 1917320 | Zbl 1055.35002

[5] F.W. Gehring, The L p -integrability of the partial derivatives of a quasiconformal mapping. Acta Math. 130 (1973) 265-277. | MR 402038 | Zbl 0258.30021

[6] P.-L. Lions, Regularizing effects for first-order Hamilton-Jacobi equations. Applicable Anal. 20 (1985) 283-307. | MR 814955 | Zbl 0551.35014

[7] F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence. Indiana Univ. Math. J. 49 (2000) 1043-1077. | MR 1803221 | Zbl 0987.35024