Envelopes and nonconvex Hamilton-Jacobi equations. (English) Zbl 1302.49038
The author derives a new representation formula for viscosity solutions of nonconvex Hamilton-Jacobi PDEs given by
\[
\begin{cases} u_{t}(x,t)+H(Du(x,t)) =0, &(x,t)\in \mathbb{R}^{n}\times (0,+\infty)\\ u(x,0) =g(x), &x\in \mathbb{R}^{n}, \end{cases}
\]
where \( H:\mathbb{R}^{n}\rightarrow \mathbb{R} \) is a smooth nonconvex Hamiltonian and \( g:\mathbb{R}^{n}\rightarrow \mathbb{R} \) is a smooth nonconvex initial data. As an application, the author studies “envelope and singular characteristic constructions of equivocal surfaces and [...] differential game theoretic interpretations.”
Reviewer: Ovidiu Cârjă (Iaşi)
MSC:
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 49N70 | Differential games and control |
| 35L65 | Hyperbolic conservation laws |
References:
| [1] | Ballou, D.P.: Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans. AMS 152, 441-460 (1970) · Zbl 0207.40401 · doi:10.1090/S0002-9947-1970-0435615-3 |
| [2] | Ballou, D.P.: Weak solutions with a dense set of discontinuities. J Differ. Equ. 10, 270-280 (1971) · Zbl 0206.39301 · doi:10.1016/0022-0396(71)90051-9 |
| [3] | Basar, T., Olsder, G.J.: Dynamic Noncooperative Game Theory. 2nd edn., SIAM, Academic Press, New York (1999) · Zbl 0946.91001 |
| [4] | Bardi, M., Evans, L.C.: On Hopf’s formulas for solutions of Hamilton-Jacobi equations. Nonlinear Anal. 8, 1373-1381 (1984) · Zbl 0569.35011 · doi:10.1016/0362-546X(84)90020-8 |
| [5] | Bardi, M., Faggian, S.: Hopf-type estimates and formulas for nonconvex nonconcave Hamilton-Jacobi equations. SIAM J. Math. Anal. 29, 1067-1086 (1998) · Zbl 0914.35026 · doi:10.1137/S0036141096309629 |
| [6] | Bernhard, P.: Singular surfaces in differential games: an introduction. In: Differential Games and Applications. Proceedings of workshop, Enschede: lecture notes in control and information Sciences. Springer, Berlin, 3, 1-33 (1977) · Zbl 0914.35026 |
| [7] | Cagnetti, F., Gomes, D., Tran, H.V.: Aubry-Mather measures in the nonconvex setting. SIAM J. Math. Anal. 43, 2601-2629 (2011) · Zbl 1258.35059 · doi:10.1137/100817656 |
| [8] | Cardaliaguet, P.: Introduction to Differential Games. www.ceremade.dauphine.fr/ cardalia/CoursJeuxDiff.pdf · Zbl 1293.91020 |
| [9] | Dafermos, C.: Regularity and large time behavior of solutions of a conservation law without convexity. Proc. R. Soc. Edinburgh Sect. A 99, 201-239 (1985) · Zbl 0616.35054 · doi:10.1017/S0308210500014256 |
| [10] | Evans, L.C.: Adjoint and compensated compactness methods for Hamilton-Jacobi PDE. Arch. Ration Mech. Anal. 197, 1053-1088 (2010) · Zbl 1273.70030 · doi:10.1007/s00205-010-0307-9 |
| [11] | 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 |
| [12] | Friedman, A.: Differential Games. Wiley, New York (1971) · Zbl 0229.90060 |
| [13] | Hopf, E.: Generalized solutions of nonlinear equations of first order. J. Math. Mech. 14, 951-973 (1965) · Zbl 0168.35101 |
| [14] | Isaacs, R.: Differential Games. A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Wiley, New York (1965) · Zbl 0125.38001 |
| [15] | Lewin, J.: Differential Games. Theory and Methods for Solving Game Problems with Singular Surfaces. Springer, Berlin (1994) · Zbl 0907.90296 |
| [16] | Lions, P.-L., Rochet, J.-C.: Hopf’s formula and multitime Hamilton-Jacobi equations. Proc. Am. Math. Soc. 96, 79-84 (1986) · Zbl 0597.35079 · doi:10.1090/S0002-9939-1986-0813815-5 |
| [17] | Melikyan, A.: Generalized Characteristics of First Order PDEs. Applications in Optimal Control and Differential Games. Birkhauser, Basel (1998) · Zbl 0936.35004 |
| [18] | Osher, S.: The Riemann problem for nonconvex scalar conservation laws and Hamilton-Jacobi equations. Proc. Am. Math. Soc. 89, 641-646 (1983) · Zbl 0555.35086 · doi:10.1090/S0002-9939-1983-0718989-X |
| [19] | Rapaport, A.E.: Characterization of barriers of differential games. J. Optim. Theory 97, 151-179 (1998) · Zbl 0907.90296 · doi:10.1023/A:1022631318424 |
| [20] | Tran, H.V.: Adjoint methods for static Hamilton-Jacobi equations. Calc. Var. PDE 41, 301-319 (2011) · Zbl 1231.35043 · doi:10.1007/s00526-010-0363-x |
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.
