×

SBV regularity for Hamilton-Jacobi equations in \(\mathbb R^n\). (English) Zbl 1223.49039

Summary: We study the regularity of viscosity solutions to the following Hamilton-Jacobi equations \[ \partial_tu+H(D_xu)=0\quad\text{in }\Omega\subset \mathbb R\times\mathbb R^n. \]
In particular, under the assumption that the Hamiltonian \(H\in C^2(\mathbb R^n)\) is uniformly convex, we prove that \(D_xu\) and \(\partial_tu\) belong to the class \(SBV_{\text{loc}}(\Omega)\).

MSC:

49N60 Regularity of solutions in optimal control
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
70H20 Hamilton-Jacobi equations in mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alberti G., Ambrosio L.: A geometrical approach to monotone functions in $${{\(\backslash\)mathbb R}\^{n}}$$ . Math. Z. 230(2), 259–316 (1999) · Zbl 0934.49025 · doi:10.1007/PL00004691
[2] Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs, 2000 · Zbl 0957.49001
[3] 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) · Zbl 1071.35032 · doi:10.1142/S0219891604000263
[4] Bressan, A.: Viscosity Solutions of Hamilton–Jacobi Equations and Optimal Control Problems (an illustrated tutorial). Lecture Notes in Mathematics (unpublished)
[5] Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton–Jacobi Equations, and Optimal Control. Birkhäuser, Boston, 2004 · Zbl 1095.49003
[6] 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) · Zbl 0612.70016 · doi:10.1512/iumj.1987.36.36028
[7] 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) · Zbl 0715.49014
[8] Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 319. AMS, 1991 · Zbl 0767.65015
[9] 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) · Zbl 1169.91317 · doi:10.1512/iumj.1984.33.33040
[10] Lions, P.L.: Generalized solutions of Hamilton–Jacobi equations. Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston, 1982 · Zbl 0497.35001
[11] 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) · Zbl 1152.35074 · doi:10.1142/S0219891608001544
[12] Zajíček, L.: On the differentiability of convex functions in finite and infinite dimensional spaces. Czechoslovak Math. J. 29, 340–348 (1979)
[13] Zajíček L.: On the point of multiplicity of monotone operators. Comment. Math. Univ. Carolinae 19(1)179–189 (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.