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January  2019, 39(1): 157-183. doi: 10.3934/dcds.2019007

Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces

1. 

Department of Applied Mathematics, Faculty of Science, Fukuoka University, Fukuoka 814-0180, Japan

2. 

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-0041, Japan

Received  October 2017 Revised  July 2018 Published  October 2018

We study the convexity preserving property for a class of time-dependent Hamilton-Jacobi equations in a complete geodesic space. Assuming that the Hamiltonian is nondecreasing, we show that in a Busemann space the unique metric viscosity solution preserves the geodesic convexity of the initial value at any time. We provide two approaches and also discuss several generalizations for more general geodesic spaces including the lattice grid.

Citation: Qing Liu, Atsushi Nakayasu. Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 157-183. doi: 10.3934/dcds.2019007
References:
[1]

Y. AchdouF. CamilliA. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1.

[2]

O. AlvarezJ.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl.(9), 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7.

[3]

L. Ambrosio and J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces, J. Differential Equations, 256 (2014), 2194-2245. doi: 10.1016/j.jde.2013.12.018.

[4]

L. AmbrosioN. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490. doi: 10.1215/00127094-2681605.

[5]

S. N. Armstrong and C. K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differential Equations, 37 (2010), 381-384. doi: 10.1007/s00526-009-0267-9.

[6]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. doi: 10.1007/978-0-8176-4755-1.

[7]

M. Bardi and L. C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), 1373-1381. doi: 10.1016/0362-546X(84)90020-8.

[8]

L. CaffarelliP. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math., 60 (2007), 1769-1791. doi: 10.1002/cpa.20197.

[9]

F. CamilliR. Capitanelli and C. Marchi, Eikonal equations on the Sierpinski gasket, Math. Ann., 364 (2016), 1167-1188. doi: 10.1007/s00208-015-1251-7.

[10]

J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl., 177 (1993), 263-286. doi: 10.1006/jmaa.1993.1257.

[11]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681. doi: 10.4310/jdg/1214446559.

[12]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96. doi: 10.4310/jdg/1214439902.

[13]

W. Gangbo and A. Święch, Optimal transport and large number of particles, Discrete Contin. Dyn. Syst., 34 (2014), 1397-1441. doi: 10.3934/dcds.2014.34.1397.

[14]

W. Gangbo and A. Święch, Metric viscosity solutions of Hamilton-Jacobi equations depending on local slopes, Calc. Var. Partial Differential Equations, 54 (2015), 1183-1218. doi: 10.1007/s00526-015-0822-5.

[15]

Y. GigaS. GotoH. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. doi: 10.1512/iumj.1991.40.40023.

[16]

Y. GigaN. Hamamuki and A. Nakayasu, Eikonal equations in metric spaces, Trans. Amer. Math. Soc., 367 (2015), 49-66. doi: 10.1090/S0002-9947-2014-05893-5.

[17]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266. doi: 10.4310/jdg/1214438998.

[18]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 357-448. doi: 10.24033/asens.2323.

[19]

C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, Discrete Contin. Dyn. Syst., 37 (2017), 6405-6435. doi: 10.3934/dcds.2017278.

[20]

C. ImbertR. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002.

[21]

J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8918-6.

[22]

P. Juutinen, Concavity maximum principle for viscosity solutions of singular equations, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 601-618. doi: 10.1007/s00030-010-0071-4.

[23]

B. Kawohl, Qualitative properties of solutions to semilinear heat equations, Exposition. Math., 4 (1986), 257-270.

[24]

B. Kawohl, A remark on N. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem, Math. Methods Appl. Sci., 8 (1986), 93-101. doi: 10.1002/mma.1670080107.

[25]

A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J., 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036.

[26]

R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101.

[27]

N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042.

[28]

P.-L. Lions and M. Musiela, Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 915-921. doi: 10.1016/j.crma.2006.02.014.

[29]

Q. Liu, J. J. Manfredi and X. Zhou, Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group, Calc. Var. Partial Differential Equations, 55 (2016), Art. 80, 25 pp. doi: 10.1007/s00526-016-1024-5.

[30]

Q. LiuA. Schikorra and X. Zhou, A game-theoretic proof of convexity preserving properties for motion by curvature, Indiana Univ. Math. J., 65 (2016), 171-197. doi: 10.1512/iumj.2016.65.5740.

[31]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 215-241.

[32]

A. Nakayasu, Metric viscosity solutions for Hamilton-Jacobi equations of evolution type, Adv. Math. Sci. Appl., 24 (2014), 333-351.

[33]

A. Nakayasu and T. Namba, Stability properties and large time behavior of viscosity solutions of Hamilton-Jacobi equations on metric spaces, to appear in Nonlinearity.

[34]

A. Papadopoulos, Metric Spaces, Convexity and Non-Positive Curvature volume 6 of IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society (EMS), Zürich, second edition, 2014. doi: 10.4171/132.

[35]

Y. PeresO. SchrammS. Sheffield and D. B. Wilson, Metric Spaces, Convexity and Non-Positive Curvature, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.

[36]

S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 403-421 (1988).

[37]

D. Schieborn, Viscosity solutions of Hamilton-Jacobi equations of Eikonal type on ramified spaces, Dissertation, Eberhard-Karls-Universitat Tubingen, 2006.

[38]

D. Schieborn and F. Camilli, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equations, 46 (2013), 671-686. doi: 10.1007/s00526-012-0498-z.

[39]

H. M. Soner, Optimal control with state-space constraint. I, SIAM J. Control Optim., 24 (1986), 552-561. doi: 10.1137/0324032.

[40]

K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, In Heat Kernels And Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), volume 338 of Contemp. Math., pages 357-390. Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/conm/338/06080.

show all references

References:
[1]

Y. AchdouF. CamilliA. Cutrì and N. Tchou, Hamilton-Jacobi equations constrained on networks, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 413-445. doi: 10.1007/s00030-012-0158-1.

[2]

O. AlvarezJ.-M. Lasry and P.-L. Lions, Convex viscosity solutions and state constraints, J. Math. Pures Appl.(9), 76 (1997), 265-288. doi: 10.1016/S0021-7824(97)89952-7.

[3]

L. Ambrosio and J. Feng, On a class of first order Hamilton-Jacobi equations in metric spaces, J. Differential Equations, 256 (2014), 2194-2245. doi: 10.1016/j.jde.2013.12.018.

[4]

L. AmbrosioN. Gigli and G. Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J., 163 (2014), 1405-1490. doi: 10.1215/00127094-2681605.

[5]

S. N. Armstrong and C. K. Smart, An easy proof of Jensen's theorem on the uniqueness of infinity harmonic functions, Calc. Var. Partial Differential Equations, 37 (2010), 381-384. doi: 10.1007/s00526-009-0267-9.

[6]

M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA, 1997. With appendices by Maurizio Falcone and Pierpaolo Soravia. doi: 10.1007/978-0-8176-4755-1.

[7]

M. Bardi and L. C. Evans, On Hopf's formulas for solutions of Hamilton-Jacobi equations, Nonlinear Anal., 8 (1984), 1373-1381. doi: 10.1016/0362-546X(84)90020-8.

[8]

L. CaffarelliP. Guan and X.-N. Ma, A constant rank theorem for solutions of fully nonlinear elliptic equations, Comm. Pure Appl. Math., 60 (2007), 1769-1791. doi: 10.1002/cpa.20197.

[9]

F. CamilliR. Capitanelli and C. Marchi, Eikonal equations on the Sierpinski gasket, Math. Ann., 364 (2016), 1167-1188. doi: 10.1007/s00208-015-1251-7.

[10]

J. I. Diaz and B. Kawohl, On convexity and starshapedness of level sets for some nonlinear elliptic and parabolic problems on convex rings, J. Math. Anal. Appl., 177 (1993), 263-286. doi: 10.1006/jmaa.1993.1257.

[11]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature. I, J. Differential Geom., 33 (1991), 635-681. doi: 10.4310/jdg/1214446559.

[12]

M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96. doi: 10.4310/jdg/1214439902.

[13]

W. Gangbo and A. Święch, Optimal transport and large number of particles, Discrete Contin. Dyn. Syst., 34 (2014), 1397-1441. doi: 10.3934/dcds.2014.34.1397.

[14]

W. Gangbo and A. Święch, Metric viscosity solutions of Hamilton-Jacobi equations depending on local slopes, Calc. Var. Partial Differential Equations, 54 (2015), 1183-1218. doi: 10.1007/s00526-015-0822-5.

[15]

Y. GigaS. GotoH. Ishii and M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443-470. doi: 10.1512/iumj.1991.40.40023.

[16]

Y. GigaN. Hamamuki and A. Nakayasu, Eikonal equations in metric spaces, Trans. Amer. Math. Soc., 367 (2015), 49-66. doi: 10.1090/S0002-9947-2014-05893-5.

[17]

G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom., 20 (1984), 237-266. doi: 10.4310/jdg/1214438998.

[18]

C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), 50 (2017), 357-448. doi: 10.24033/asens.2323.

[19]

C. Imbert and R. Monneau, Quasi-convex Hamilton-Jacobi equations posed on junctions: The multi-dimensional case, Discrete Contin. Dyn. Syst., 37 (2017), 6405-6435. doi: 10.3934/dcds.2017278.

[20]

C. ImbertR. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows, ESAIM Control Optim. Calc. Var., 19 (2013), 129-166. doi: 10.1051/cocv/2012002.

[21]

J. Jost, Nonpositive Curvature: Geometric and Analytic Aspects, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8918-6.

[22]

P. Juutinen, Concavity maximum principle for viscosity solutions of singular equations, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 601-618. doi: 10.1007/s00030-010-0071-4.

[23]

B. Kawohl, Qualitative properties of solutions to semilinear heat equations, Exposition. Math., 4 (1986), 257-270.

[24]

B. Kawohl, A remark on N. Korevaar's concavity maximum principle and on the asymptotic uniqueness of solutions to the plasma problem, Math. Methods Appl. Sci., 8 (1986), 93-101. doi: 10.1002/mma.1670080107.

[25]

A. U. Kennington, Power concavity and boundary value problems, Indiana Univ. Math. J., 34 (1985), 687-704. doi: 10.1512/iumj.1985.34.34036.

[26]

R. V. Kohn and S. Serfaty, A deterministic-control-based approach to motion by curvature, Comm. Pure Appl. Math., 59 (2006), 344-407. doi: 10.1002/cpa.20101.

[27]

N. J. Korevaar, Convex solutions to nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 32 (1983), 603-614. doi: 10.1512/iumj.1983.32.32042.

[28]

P.-L. Lions and M. Musiela, Convexity of solutions of parabolic equations, C. R. Math. Acad. Sci. Paris, 342 (2006), 915-921. doi: 10.1016/j.crma.2006.02.014.

[29]

Q. Liu, J. J. Manfredi and X. Zhou, Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group, Calc. Var. Partial Differential Equations, 55 (2016), Art. 80, 25 pp. doi: 10.1007/s00526-016-1024-5.

[30]

Q. LiuA. Schikorra and X. Zhou, A game-theoretic proof of convexity preserving properties for motion by curvature, Indiana Univ. Math. J., 65 (2016), 171-197. doi: 10.1512/iumj.2016.65.5740.

[31]

J. J. ManfrediM. Parviainen and J. D. Rossi, On the definition and properties of p-harmonious functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 215-241.

[32]

A. Nakayasu, Metric viscosity solutions for Hamilton-Jacobi equations of evolution type, Adv. Math. Sci. Appl., 24 (2014), 333-351.

[33]

A. Nakayasu and T. Namba, Stability properties and large time behavior of viscosity solutions of Hamilton-Jacobi equations on metric spaces, to appear in Nonlinearity.

[34]

A. Papadopoulos, Metric Spaces, Convexity and Non-Positive Curvature volume 6 of IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society (EMS), Zürich, second edition, 2014. doi: 10.4171/132.

[35]

Y. PeresO. SchrammS. Sheffield and D. B. Wilson, Metric Spaces, Convexity and Non-Positive Curvature, J. Amer. Math. Soc., 22 (2009), 167-210. doi: 10.1090/S0894-0347-08-00606-1.

[36]

S. Sakaguchi, Concavity properties of solutions to some degenerate quasilinear elliptic Dirichlet problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14 (1987), 403-421 (1988).

[37]

D. Schieborn, Viscosity solutions of Hamilton-Jacobi equations of Eikonal type on ramified spaces, Dissertation, Eberhard-Karls-Universitat Tubingen, 2006.

[38]

D. Schieborn and F. Camilli, Viscosity solutions of Eikonal equations on topological networks, Calc. Var. Partial Differential Equations, 46 (2013), 671-686. doi: 10.1007/s00526-012-0498-z.

[39]

H. M. Soner, Optimal control with state-space constraint. I, SIAM J. Control Optim., 24 (1986), 552-561. doi: 10.1137/0324032.

[40]

K.-T. Sturm, Probability measures on metric spaces of nonpositive curvature, In Heat Kernels And Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), volume 338 of Contemp. Math., pages 357-390. Amer. Math. Soc., Providence, RI, 2003. doi: 10.1090/conm/338/06080.

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