doi: 10.3934/dcds.2022061
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Comparison principles for nonlocal Hamilton-Jacobi equations

Universidad Técnica Federico Santa María, Departamento de Matemática, Chile

*Corresponding author: Gonzalo Dávila

Received  March 2022 Revised  April 2022 Early access April 2022

Fund Project: The first author is supported by Fondecyt grant 1190209

We prove the comparison principle for viscosity sub and super solutions of degenerate nonlocal operators with general nonlocal gradient nonlinearities. The proofs apply to purely Hamilton-Jacobi equations of order $ 0<s<1 $.

Citation: Gonzalo Dávila. Comparison principles for nonlocal Hamilton-Jacobi equations. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022061
References:
[1]

B. Abdellaoui and A. J. Fernández, Nonlinear fractional Laplacian problems with nonlocal "gradient terms", Proc. Royal Soc. Edinburgh Sect. A, 150 (2020), 2682-2718.  doi: 10.1017/prm.2019.60.

[2]

M. Arisawa, A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators, J. Math. Pures Appl., 89 (2008), 567-574.  doi: 10.1016/j.matpur.2008.02.005.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2$^{nd}$ edition. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[4]

D. Bakry and M. Émery, Diffusions hypercontractives, In Séminaire de Probabilit és, XIX, 1983/84, Lecture Notes in Math, Springer, Berlin, 1123 (1985), 177–206. doi: 10.1007/BFb0075847.

[5]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348, Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[6]

A. Banerjee, G. Dávila and Y. Sire, Regularity for parabolic systems with critical growth in the gradient and applications, To appear, Journal d'Analyse Mathématique.

[7]

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. doi: 10.1007/978-0-8176-4755-1.

[8]

G. Barles, Existence results for first order Hamilton Jacobi equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 325-340.  doi: 10.1016/s0294-1449(16)30415-2.

[9]

G. Barles, Uniqueness for first-order Hamilton-Jacobi equations and Hopf formula, J. Differential Equations, 69 (1987), 346-367.  doi: 10.1016/0022-0396(87)90124-0.

[10]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin), 17. Springer-Verlag, Paris, 1994.

[11]

G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math., 2074 (2013), 49–109 doi: 10.1007/978-3-642-36433-4_2.

[12]

G. BarlesE. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246.  doi: 10.1512/iumj.2008.57.3315.

[13]

G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.  doi: 10.1016/j.anihpc.2007.02.007.

[14]

G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939.  doi: 10.1137/S0036141099350869.

[15]

A. Barrasso and F. Russo, Martingale driven BSDEs, PDEs and other related deterministic problems, Stochastic Process. Appl., 133 (2021), 193-228.  doi: 10.1016/j.spa.2020.11.007.

[16]

A. Barrasso and F. Russo, Decoupled mild solutions of path-dependent PDEs and integro PDEs represented by BSDEs driven by cadlag martingales, Potential Anal., 53 (2020), 449-481.  doi: 10.1007/s11118-019-09775-x.

[17]

B. Barrios and M. Medina, Equivalence of weak and viscosity solutions in fractional non-homogeneous problems, Math. Ann., 381 (2021), 1979-2012.  doi: 10.1007/s00208-020-02119-w.

[18]

L. A. Caffarelli and G. Dávila, Interior regularity for fractional systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 36 (2019), 165-180.  doi: 10.1016/j.anihpc.2018.04.004.

[19]

E. Chasseigne and E. R. Jakobsen, On nonlocal quasilinear equations and their local limits, J. Differential Equations, 262 (2017), 3759-3804.  doi: 10.1016/j.jde.2016.12.001.

[20]

M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential Integral Equations, 3 (1990), 1001-1014. 

[21]

F. Da Lio and A. Pigati, Free boundary minimal surfaces: A nonlocal approach, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (2020), 437-489.  doi: 10.2422/2036-2145.201801_008.

[22]

F. Da Lio and T. Rivière, Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190.  doi: 10.2140/apde.2011.4.149.

[23]

F. Da Lio and T. Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math., 227 (2011), 1300-1348.  doi: 10.1016/j.aim.2011.03.011.

[24]

G. Dávila, A. Quaas and E. Topp, Harnack Inequality and self-similar solutions for fully nonlinear fractional parabolic equations, preprint, 2019, arXiv: 1909.02624

[25]

G. Dávila and E. Topp, The Nonlocal Inverse Problem of Donsker and Varadhan, preprint, 2019, arXiv: 2011.13295

[26]

B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555.  doi: 10.2478/s13540-012-0038-8.

[27]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[28]

H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.  doi: 10.1215/S0012-7094-87-05521-9.

[29]

H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38 (1995), 101-120. 

[30]

E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 137-165.  doi: 10.1007/s00030-005-0031-6.

[31]

Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Universitext. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-77739-4.

[32]

V. Millot and Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., 215 (2015), 125-210.  doi: 10.1007/s00205-014-0776-3.

[33]

G. Namah and J. M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.  doi: 10.1080/03605309908821451.

[34]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[35]

X. Ros-Oton and J. Serra, The pohozaev identity for the fractional laplacian, Arch Rational Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.

[36]

A. SpenerF. Weber and R. Zacher, The fractional Laplacian has infinite dimension, Comm. Partial Differential Equations, 45 (2020), 57-75.  doi: 10.1080/03605302.2019.1663434.

show all references

References:
[1]

B. Abdellaoui and A. J. Fernández, Nonlinear fractional Laplacian problems with nonlocal "gradient terms", Proc. Royal Soc. Edinburgh Sect. A, 150 (2020), 2682-2718.  doi: 10.1017/prm.2019.60.

[2]

M. Arisawa, A remark on the definitions of viscosity solutions for the integro-differential equations with Lévy operators, J. Math. Pures Appl., 89 (2008), 567-574.  doi: 10.1016/j.matpur.2008.02.005.

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics, 2$^{nd}$ edition. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[4]

D. Bakry and M. Émery, Diffusions hypercontractives, In Séminaire de Probabilit és, XIX, 1983/84, Lecture Notes in Math, Springer, Berlin, 1123 (1985), 177–206. doi: 10.1007/BFb0075847.

[5]

D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 348, Springer, Cham, 2014. doi: 10.1007/978-3-319-00227-9.

[6]

A. Banerjee, G. Dávila and Y. Sire, Regularity for parabolic systems with critical growth in the gradient and applications, To appear, Journal d'Analyse Mathématique.

[7]

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. doi: 10.1007/978-0-8176-4755-1.

[8]

G. Barles, Existence results for first order Hamilton Jacobi equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 325-340.  doi: 10.1016/s0294-1449(16)30415-2.

[9]

G. Barles, Uniqueness for first-order Hamilton-Jacobi equations and Hopf formula, J. Differential Equations, 69 (1987), 346-367.  doi: 10.1016/0022-0396(87)90124-0.

[10]

G. Barles, Solutions de Viscosité des Équations de Hamilton-Jacobi, Mathématiques & Applications (Berlin), 17. Springer-Verlag, Paris, 1994.

[11]

G. Barles, An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications, Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications, Lecture Notes in Math., 2074 (2013), 49–109 doi: 10.1007/978-3-642-36433-4_2.

[12]

G. BarlesE. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246.  doi: 10.1512/iumj.2008.57.3315.

[13]

G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 567-585.  doi: 10.1016/j.anihpc.2007.02.007.

[14]

G. Barles and P. E. Souganidis, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), 925-939.  doi: 10.1137/S0036141099350869.

[15]

A. Barrasso and F. Russo, Martingale driven BSDEs, PDEs and other related deterministic problems, Stochastic Process. Appl., 133 (2021), 193-228.  doi: 10.1016/j.spa.2020.11.007.

[16]

A. Barrasso and F. Russo, Decoupled mild solutions of path-dependent PDEs and integro PDEs represented by BSDEs driven by cadlag martingales, Potential Anal., 53 (2020), 449-481.  doi: 10.1007/s11118-019-09775-x.

[17]

B. Barrios and M. Medina, Equivalence of weak and viscosity solutions in fractional non-homogeneous problems, Math. Ann., 381 (2021), 1979-2012.  doi: 10.1007/s00208-020-02119-w.

[18]

L. A. Caffarelli and G. Dávila, Interior regularity for fractional systems, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 36 (2019), 165-180.  doi: 10.1016/j.anihpc.2018.04.004.

[19]

E. Chasseigne and E. R. Jakobsen, On nonlocal quasilinear equations and their local limits, J. Differential Equations, 262 (2017), 3759-3804.  doi: 10.1016/j.jde.2016.12.001.

[20]

M. G. Crandall and H. Ishii, The maximum principle for semicontinuous functions, Differential Integral Equations, 3 (1990), 1001-1014. 

[21]

F. Da Lio and A. Pigati, Free boundary minimal surfaces: A nonlocal approach, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (2020), 437-489.  doi: 10.2422/2036-2145.201801_008.

[22]

F. Da Lio and T. Rivière, Three-term commutator estimates and the regularity of 1/2-harmonic maps into spheres, Anal. PDE, 4 (2011), 149-190.  doi: 10.2140/apde.2011.4.149.

[23]

F. Da Lio and T. Rivière, Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps, Adv. Math., 227 (2011), 1300-1348.  doi: 10.1016/j.aim.2011.03.011.

[24]

G. Dávila, A. Quaas and E. Topp, Harnack Inequality and self-similar solutions for fully nonlinear fractional parabolic equations, preprint, 2019, arXiv: 1909.02624

[25]

G. Dávila and E. Topp, The Nonlocal Inverse Problem of Donsker and Varadhan, preprint, 2019, arXiv: 2011.13295

[26]

B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555.  doi: 10.2478/s13540-012-0038-8.

[27]

N. El KarouiS. Peng and M. C. Quenez, Backward stochastic differential equations in finance, Math. Finance, 7 (1997), 1-71.  doi: 10.1111/1467-9965.00022.

[28]

H. Ishii, Perron's method for Hamilton-Jacobi equations, Duke Math. J., 55 (1987), 369-384.  doi: 10.1215/S0012-7094-87-05521-9.

[29]

H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcial. Ekvac., 38 (1995), 101-120. 

[30]

E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl., 13 (2006), 137-165.  doi: 10.1007/s00030-005-0031-6.

[31]

Z. M. Ma and M. Röckner, Introduction to the Theory of (Nonsymmetric) Dirichlet Forms, Universitext. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-77739-4.

[32]

V. Millot and Y. Sire, On a fractional Ginzburg-Landau equation and 1/2-harmonic maps into spheres, Arch. Ration. Mech. Anal., 215 (2015), 125-210.  doi: 10.1007/s00205-014-0776-3.

[33]

G. Namah and J. M. Roquejoffre, Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations, Comm. Partial Differential Equations, 24 (1999), 883-893.  doi: 10.1080/03605309908821451.

[34]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.  doi: 10.1016/j.matpur.2013.06.003.

[35]

X. Ros-Oton and J. Serra, The pohozaev identity for the fractional laplacian, Arch Rational Mech. Anal., 213 (2014), 587-628.  doi: 10.1007/s00205-014-0740-2.

[36]

A. SpenerF. Weber and R. Zacher, The fractional Laplacian has infinite dimension, Comm. Partial Differential Equations, 45 (2020), 57-75.  doi: 10.1080/03605302.2019.1663434.

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