Abstract
In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations
In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D x u and ∂ t u belong to the class SBV loc (Ω).
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Alberti G., Ambrosio L.: A geometrical approach to monotone functions in \({{\mathbb R}^{n}}\) . Math. Z. 230(2), 259–316 (1999)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, 2000
Ambrosio L., De Lellis C.: A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton–Jacobi equations. J. Hyperbolic Differ. Equ. 31(4), 813–826 (2004)
Bressan, A.: Viscosity Solutions of Hamilton–Jacobi Equations and Optimal Control Problems (an illustrated tutorial). Lecture Notes in Mathematics (unpublished)
Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkhäuser, Boston, 2004
Cannarsa P., Soner H.M.: On the singularities of the viscosity solutions to Hamilton–Jacobi–Bellman equations. Indiana Univ. Math. J. 36(3), 501–524 (1987)
De Giorgi E., Ambrosio L.: New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82(2), 199–210 (1988)
Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 319. AMS, 1991
Evans L.C., Souganidis P.E.: Differential games and representation formulas for solutions of Hamilton–Jacobi–Isaacs equations. Indiana Univ. Math. J. 33, 773–797 (1984)
Lions, P.L.: Generalized solutions of Hamilton–Jacobi equations. Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston, 1982
Robyr R.: SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function. J. Hyperbolic Differ. Equ. 5(2), 449–475 (2008)
Zajíček, L.: On the differentiability of convex functions in finite and infinite dimensional spaces. Czechoslovak Math. J. 29, 340–348 (1979)
Zajíček L.: On the point of multiplicity of monotone operators. Comment. Math. Univ. Carolinae 19(1)179–189 (2008)
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Communicated by S. Bianchini
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Bianchini, S., De Lellis, C. & Robyr, R. SBV Regularity for Hamilton–Jacobi Equations in \({{\mathbb R}^n}\) . Arch Rational Mech Anal 200, 1003–1021 (2011). https://doi.org/10.1007/s00205-010-0381-z
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DOI: https://doi.org/10.1007/s00205-010-0381-z