Semiconvexity of viscosity solutions to fully nonlinear evolution equations via discrete games. (English) Zbl 1473.35103
Ferone, Vincenzo (ed.) et al., Geometric properties for parabolic and elliptic PDE’s. Contributions of the 6th Italian-Japanese workshop, Cortona, Italy, May 20–24, 2019. Cham: Springer. Springer INdAM Ser. 47, 205-231 (2021).
Summary: In this paper, by using a discrete game interpretation of fully nonlinear parabolic equations proposed by R. V. Kohn and S. Serfaty [Commun. Pure Appl. Math. 63, No. 10, 1298–1350 (2010; Zbl 1204.35070)], we show that the spatial semiconvexity of viscosity solutions is preserved for a class of fully nonlinear evolution equations with concave parabolic operators. We also reduce the game-theoretic argument to the viscous and inviscid Hamilton-Jacobi equations, categorizing the semiconvexity regularity of solutions in terms of semiconcavity of the Hamiltonian.
For the entire collection see [Zbl 1471.35003].
For the entire collection see [Zbl 1471.35003].
MSC:
| 35D40 | Viscosity solutions to PDEs |
| 35F21 | Hamilton-Jacobi equations |
| 35K55 | Nonlinear parabolic equations |
| 35K15 | Initial value problems for second-order parabolic equations |
| 91A50 | Discrete-time games |
Citations:
Zbl 1204.35070References:
| [1] | Kružkov, S.N.: Generalized solutions of nonlinear equations of the first order with several independent variables. II. Mat. Sb. (N.S.) 72(114), 108-134 (1967) · Zbl 0155.16102 |
| [2] | Alvarez, O., Lasry, J.-M., Lions, P.-L.: Convex viscosity solutions and state constraints. J. Math. Pures Appl. (9) 76(3), 265-288 (1997) · Zbl 0890.49013 |
| [3] | Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Systems & Control: Foundations & Applications. Birkhäuser, Boston (1997). With appendices by Maurizio Falcone and Pierpaolo Soravia · Zbl 0890.49011 |
| [4] | Barles, G., Biton, S., Bourgoing, M., Ley, O.: Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods. Calc. Var. Partial Differ. Equ. 18(2), 159-179 (2003) · Zbl 1036.35001 · doi:10.1007/s00526-002-0186-5 |
| [5] | Cannarsa, P., Sinestrari, C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and Their Applications, vol. 58. Birkhäuser, Boston (2004) · Zbl 1095.49003 |
| [6] | Cheridito, P., Soner, H.M., Touzi, N., Victoir, N.: Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Commun. Pure Appl. Math. 60(7), 1081-1110 (2007) · Zbl 1121.60062 · doi:10.1002/cpa.20168 |
| [7] | Crandall, M.G., Newcomb, R., Tomita, Y.: Existence and uniqueness viscosity solutions of degenerate quasilinear elliptic equations in \(R^{}\) n. Technical Summary Report, Wisconsin Univ-Madison Center for Mathematical Sciences (1988) · Zbl 0705.35044 |
| [8] | Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1-67 (1992) · Zbl 0755.35015 |
| [9] | Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence (1998) · Zbl 0902.35002 |
| [10] | Evans, L.C., Spruck, J.: Motion of level sets by mean curvature. I. J. Differ. Geom. 33(3), 635-681 (1991) · Zbl 0726.53029 |
| [11] | Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions. Stochastic Modelling and Applied Probability, vol. 25, 2nd edn. Springer, New York (2006) · Zbl 1105.60005 |
| [12] | Fujita, Y.: Hessian estimates for viscous Hamilton-Jacobi equations with the Ornstein-Uhlenbeck operator. Differ. Integr. Equ. 18(12), 1383-1396 (2005) · Zbl 1212.35207 |
| [13] | Fujita, Y., Loreti, P.: Long-time behavior of solutions to Hamilton-Jacobi equations with quadratic gradient term. Nonlinear Differ. Equ. Appl. 16(6), 771-791 (2009) · Zbl 1180.35105 · doi:10.1007/s00030-009-0034-9 |
| [14] | Giga, Y.: Surface Evolution Equations. A Level Set Approach. Monographs in Mathematics, vol. 99. Birkhäuser, Basel (2006). · Zbl 1096.53039 |
| [15] | Giga, Y., Goto, S., Ishii, H., Sato, M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443-470 (1991) · Zbl 0836.35009 · doi:10.1512/iumj.1991.40.40023 |
| [16] | Imbert, C.: Convexity of solutions and \(C^{1, 1}\) estimates for fully nonlinear elliptic equations. J. Math. Pures Appl. (9) 85(6), 791-807 (2006) · Zbl 1206.35108 |
| [17] | Ishige, K., Salani, P.: Is quasi-concavity preserved by heat flow? Arch. Math. 90, 450-460 (2008) · Zbl 1176.35012 · doi:10.1007/s00013-008-2437-y |
| [18] | Ishige, K., Salani, P.: Convexity breaking of the free boundary for porous medium equations. Interfaces Free Bound. 12, 75-84 (2010) · Zbl 1189.35387 · doi:10.4171/IFB/227 |
| [19] | Ishige, K., Liu, Q., Salani, P.: Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations. J. Math. Pures Appl. (9) 141, 342-370 (2020) · Zbl 1445.35116 |
| [20] | Ishige, K., Nakagawa, K., Salani, P.: Spatial concavity of solutions to parabolic systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20(1), 291-313 (2020) · Zbl 1447.35184 |
| [21] | Ishii, H.: Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33(5), 721-748 (1984) · Zbl 0551.49016 · doi:10.1512/iumj.1984.33.33038 |
| [22] | Juutinen, P.: Concavity maximum principle for viscosity solutions of singular equations. Nonlinear Differ. Equ. Appl. 17(5), 601-618 (2010) · Zbl 1200.35118 · doi:10.1007/s00030-010-0071-4 |
| [23] | Kawohl, B.: Rearrangements and Convexity of Level Sets in PDE. Lecture Notes in Mathematics, vol. 1150. Springer, Berlin (1985) · Zbl 0593.35002 |
| [24] | Kawohl, B.: Qualitative properties of solutions to semilinear heat equations. Expo. Math. 4(3), 257-270 (1986) · Zbl 0603.35054 |
| [25] | Kennington, A.U.: Power concavity and boundary value problems. Indiana Univ. Math. J. 34(3), 687-704 (1985) · Zbl 0549.35025 · doi:10.1512/iumj.1985.34.34036 |
| [26] | Kohn, R.V., Serfaty, S.: A deterministic-control-based approach to motion by curvature. Commun. Pure Appl. Math. 59(3), 344-407 (2006) · Zbl 1206.53072 · doi:10.1002/cpa.20101 |
| [27] | Kohn, R.V., Serfaty, S.: A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. Commun. Pure Appl. Math. 63(10), 1298-1350 (2010) · Zbl 1204.35070 · doi:10.1002/cpa.20336 |
| [28] | Korevaar, N.J.: Convex solutions to nonlinear elliptic and parabolic boundary value problems. Indiana Univ. Math. J. 32(4), 603-614 (1983) · Zbl 0481.35024 · doi:10.1512/iumj.1983.32.32042 |
| [29] | Kružkov, S.N.: Generalized solutions of Hamilton-Jacobi equations of eikonal type. I. Statement of the problems; existence, uniqueness and stability theorems; certain properties of the solutions. Mat. Sb. (N.S.) 98(140)(3(11)), 450-493, 496 (1975) · Zbl 0325.35020 |
| [30] | Lions, P.-L.: Generalized Solutions of Hamilton-Jacobi Equations. Research Notes in Mathematics, vol. 69. Pitman (Advanced Publishing Program), Boston (1982) · Zbl 0497.35001 |
| [31] | Liu, Q., Schikorra, A., Zhou, X.: A game-theoretic proof of convexity preserving properties for motion by curvature. Indiana Univ. Math. J. 65, 171-197 (2016) · Zbl 1347.49041 · doi:10.1512/iumj.2016.65.5740 |
| [32] | Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games. SIAM J. Math. Anal. 42(5), 2058-2081 (2010) · Zbl 1231.35107 · doi:10.1137/100782073 |
| [33] | Manfredi, J.J., Parviainen, M., Rossi, J.D.: An asymptotic mean value characterization for p-harmonic functions. Proc. Am. Math. Soc. 138(3), 881-889 (2010) · Zbl 1187.35115 · doi:10.1090/S0002-9939-09-10183-1 |
| [34] | Peres, Y., Sheffield, S.: Tug-of-war with noise: a game-theoretic view of the p-Laplacian. Duke Math. J. 145(1), 91-120 (2008) · Zbl 1206.35112 · doi:10.1215/00127094-2008-048 |
| [35] | Peres, Y., Schramm, O., Sheffield, S., Wilson, D.B.: Tug-of-war and the infinity Laplacian. J. Am. Math. Soc. 22(1), 167-210 (2009) · Zbl 1206.91002 · doi:10.1090/S0894-0347-08-00606-1 |
| [36] | Sakaguchi, S.: Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14(3), 403-421 (1987/1988) · Zbl 0665.35025 |
| [37] | Strömberg, T.: Semiconcavity estimates for viscous Hamilton-Jacobi equations. Arch. Math. (Basel) 94(6), 579-589 (2010) · Zbl 1200.35039 |
| [38] | Triebel, H.: Theory of Function Spaces. II. Monographs in Mathematics, vol. 84. Birkhäuser, Basel (1992) · Zbl 0763.46025 |
| [39] | Villani, C.: Optimal Transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer, Berlin (2009). Old and new · Zbl 1156.53003 |
| [40] | Ishige, K., Nakagawa, K., Salani, P.: Power concavity in weakly coupled elliptic and parabolic systems. Nonlinear Anal. 131, 81-97 (2016) · Zbl 1332.35014 · doi:10.1016/j.na.2015.08.002 |
| [41] | Bian, B., Guan, P.: A microscopic convexity principle for nonlinear partial differential equations. Invent. Math. 177(2), 307-335 (2009) · Zbl 1168.35010 · doi:10.1007/s00222-009-0179-5 |
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.
