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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[32]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[37]

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

[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.  Google Scholar

[39]

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

[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.  Google Scholar

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