Hostname: page-component-cb9f654ff-mx8w7 Total loading time: 0 Render date: 2025-08-19T08:42:00.908Z Has data issue: false hasContentIssue false
  • English
  • Français

Cauchy problem and periodic homogenization for nonlocal Hamilton–Jacobi equations with coercive gradient terms

Part of: Miscellaneous topics - Partial differential equations Representations of solutions Partial differential equations General first-order equations and systems

Published online by Cambridge University Press:  17 September 2019

Martino Bardi ,
Annalisa Cesaroni  and
Erwin Topp
Martino Bardi
Affiliation:
Dipartimento di Matematica Tullio Levi Civita, Università di Padova, Via Trieste 63, Padova, Italy (bardi@math.unipd.it)
Annalisa Cesaroni
Affiliation:
Dipartimento di Scienze Statistiche, Università di Padova, Via Cesare Battisti 141, Padova, Italy (annalisa.cesaroni@unipd.it)
Erwin Topp
Affiliation:
Departamento de Matemática y C.C., Universidad de Santiago de Chile, Casilla 307, Santiago, Chile (erwin.topp@usach.cl)
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

Keywords

HomogenizationHamilton–Jacobi equationsintegro-differential equationsfractional Laplacianscomparison principleviscosity solutions

MSC classification

Primary: 35R09: Integro-partial differential equations
Secondary: 35B27: Homogenization; equations in media with periodic structure 35F25: Initial value problems for nonlinear first-order equations 35D40: Viscosity solutions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3028 - 3059
Check for updates
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Alvarez, O. and Bardi, M.. Viscosity solutions method for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2001), 1159–1188.CrossRefGoogle Scholar
2Alvarez, O. and Bardi, M.. Ergodicity, stabilization, and singular perturbations for Bellman–Isaacs equations. Mem. Am. Math. Soc. 204 (2010), 960.Google Scholar
3Alvarez, O., Bardi, M. and Marchi, C.. Multiscale problems and homogenization for second-order Hamilton–Jacobi equations. J. Differ. Equ. 243 (2007), 349–387.Google Scholar
4Arisawa, M.. Homogenization of a class of integro-differential equations with Lévy operators. Commun. Partial Differ. Equ. 34 (2009), 617–624.Google Scholar
5Arisawa, M.. Homogenization of integro-differential equations with Lévy operators with asymmetric and degenerate kernels. Proc. Royal Soc. Edinburgh 142A (2012), 917–943.CrossRefGoogle Scholar
6Arisawa, M.. Quasi periodic and almost periodic homogenizations of integro-differential equations with Lévy operators. Preprint 2011, https://arxiv.org/abs/1111.1042.Google Scholar
7Bardi, M. and Capuzzo-Dolcetta, I.. Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Boston: Birkhäuser, 1997.Google Scholar
8Barles, G.. Solutions de Viscosite des Equations de Hamilton–Jacobi Collection ‘Mathematiques et Applications’ de la SMAI, n. 17, Springer-Verlag (1994).Google Scholar
9Barles, G., Chasseigne, E., Ciomaga, A. and Imbert, C.. Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equ. 252 (2012), 6012–6060.Google Scholar
10Barles, G. and Imbert, C.. Second-order elliptic integro-differential equations: viscosity solutions' theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéare 25 (2008), 567–585.Google Scholar
11Barles, G., Koike, S., Ley, O. and Topp, E.. Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians. Calc. Var. Partial Differ. Equ. 54 (2015), 539–572.Google Scholar
12Barles, G., Ley, O. and Topp, E.. Lipschitz regularity for integro-differential equations with coercive Hamiltonians and application to large time behavior. Nonlinearity 30 (2017), 703–734.Google Scholar
13Barles, G. and Perthame, B.. Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988), 1133–1148.CrossRefGoogle Scholar
14Caffarelli, L. and Silvestre, L.. Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62 (2009), 597–638.Google Scholar
15Capuzzo Dolcetta, I., Leoni, F. and Porretta, A.. Hölder estimates for degenerate elliptic equations with coercive Hamiltonians.Trans. Am. Math. Soc. 362 (2010), 4511–4536.CrossRefGoogle Scholar
16Chang-Lara, H. and Dávila, G.. Regularity for solutions of nonlocal, nonsymmetric equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), 833–859.CrossRefGoogle Scholar
17Ciomaga, A.. On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17 (2012), 635–671.Google Scholar
18Ciomaga, A., Ghilli, D. and Topp, E.. Periodic homogenization for nonlocal Hamilton–Jacobi equations with critical fractional diffusion, preprint 2019.Google Scholar
19Dávila, G., Quaas, A. and Topp, E.. Existence, nonexistence and multiplicity results for fully nonlinear nonlocal Dirichlet problems. J. Differ. Equ. (2018) https://doi.org/10.1016/j.jde.2018.10.046.Google Scholar
20Di Nezza, E., Palatucci, G. and Valdinoci, E.. Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521–573.Google Scholar
21Evans, L. C.. The perturbed test function method for viscosity solutions of nonlinear PDEs. Proc. R. Soc. Edinb. A 111 (1989), 359–375.Google Scholar
22Evans, L. C.. Periodic Homogenization of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinb. A 120 (1992), 245–265.Google Scholar
23Fernández Bonder, J., Ritorto, A. and Salort, A.. H-convergence results for nonlocal elliptic-type problems via Tartar's method. SIAM J. Math. Anal. 49 (2017), 2387–2408.CrossRefGoogle Scholar
24Lions, P.-L., Papanicolaou, G. and Varadhan, S. R. S.. Homogenization of Hamilton–Jacobi Equations. Manuscript, 1986.Google Scholar
25Piatnitski, A. and Zhizhina, E.. Periodic homogenization of nonlocal operators with a convolution-type kernel. SIAM J. Math. Anal. 49 (2017), 64–81.Google Scholar
26Schwab, R. W.. Periodic homogenization for nonlinear integro-differential equations. SIAM J. Math. Anal. 42 (2010), 2652–2680.CrossRefGoogle Scholar
27Schwab, R. W.. Stochastic homogenization for some nonlinear integro-differential equations. Commun. Partial Differ. Equ. 38 (2013), 171–198.CrossRefGoogle Scholar
28Silvestre, L.. Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60 (2007), 67–112.CrossRefGoogle Scholar
29Silvestre, L.. On the differentiability of the solution to the Hamilton–Jacobi equation with critical fractional diffusion. Adv. Math. 226 (2011), 2020–2039.CrossRefGoogle Scholar