# zbMATH — the first resource for mathematics

Some homogenization results for non-coercive Hamilton-Jacobi equations. (English) Zbl 1136.35004
The author studies the limit behaviour as $$\varepsilon \to 0$$ of the Hamilton-Jacobi equations
$U_t^{\varepsilon} + F({\varepsilon}^{-1} x, {\varepsilon}^{-1} y, {\varepsilon}^{-1}t, D_x U^{\varepsilon}, D_yU^{\varepsilon}) = 0 \quad \text{in }{\mathbb R}^{n+1} \times (0,+\infty)$ where $$x \in {\mathbb R}^{n}$$, $$y \in {\mathbb R}$$, $$t \in (0,+\infty)$$ and $$F(x,y,t,p_x,p_y)$$ is a function continuous in $${\mathbb R}^{2n+3}$$ and $${\mathbb Z}^n$$-periodic in $$x$$ and $$1$$-periodic in $$y,t$$. Usually $$F$$ is supposed to be coercive with respect to the variables $$(p_x,p_y)$$ and this is useful in solving the cell problem which provides the limit problem.
The author studies homogenization dropping coerciveness in $$(p_x,p_y)$$. He also obtains another proof for a recent result of C. Imbert and R. Monneau [Arch. Ration. Mech. Anal. 187, No. 1, 49-89 (2008; Zbl 1127.70009)], the paper that inspired the author for the present work.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35F20 Nonlinear first-order PDEs 35F25 Initial value problems for nonlinear first-order PDEs 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
##### Keywords:
homogenization; Hamilton-Jacobi equations; limit behaviour
Full Text:
##### References:
 [1] Alvarez O. (1999). Homogenization of Hamilton–Jacobi equations in perforated sets. J. Differ. Equ. 159(2): 543–577 · Zbl 0945.35010 · doi:10.1006/jdeq.1999.3665 [2] Alvarez, O., Bardi M. (2001/02). Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40(4): 1159–1188 · Zbl 1017.49028 · doi:10.1137/S0363012900366741 [3] Alvarez O. and Bardi M. (2003). Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170(1): 17–61 · Zbl 1032.35103 · doi:10.1007/s00205-003-0266-5 [4] Alvarez, O., Bardi, M.: Ergodic problems in differential games. Ann. Int. Soc. Dynam. Games (to appear) · Zbl 1153.91346 [5] Alvarez O. and Barron E.N. (2002). Homogenization in L J. Diff. Eq. 183(1): 132–164 · Zbl 1023.35010 · doi:10.1006/jdeq.2001.4118 [6] Alvarez O. and Ishii H. (2001). Hamilton–Jacobi equations with partial gradient and application to homogenization. Comm. Partial Differ. Equ. 26(5–6): 983–1002 · Zbl 1014.49021 · doi:10.1081/PDE-100002418 [7] Arisawa, M.: Ergodic problem for the Hamilton–Jacobi–Bellman equation I. Existence of the ergodic attractor. Ann. Inst. Henri Poincaré. Anal. Non Linéaire 14, 415–438 (1997) · Zbl 0892.49015 · doi:10.1016/S0294-1449(97)80134-5 [8] Arisawa, M.: Ergodic problem for the Hamilton–Jacobi–Bellman equation II. Ann. Inst. Henri Poincaré. Anal. Non Linéaire, 15, 1–24 (1998) · Zbl 0903.49018 · doi:10.1016/S0294-1449(99)80019-5 [9] Arisawa M. and Lions P.-L. (1998). On ergodic stochastic control. Comm. Partial Differ. Equ. 23(11–12): 2187–2217 · Zbl 1126.93434 · doi:10.1080/03605309808821413 [10] Artstein Z. and Gaitsgory V. (2000). The value function of singularly perturbed control systems. Appl. Math. Optim. 41(3): 425–445 · Zbl 0958.49019 · doi:10.1007/s002459911022 [11] Bardi, M.: On differential games with long-time-average cost (Preprint) · Zbl 1188.49030 [12] Barles, G.: Solutions de viscosité des équations de Hamilton–Jacobi. Collection ”Mathématiques et Applications” of SMAI, n$$\deg$$17, Springer (1994) [13] Barles, G. (1990). Uniqueness and regularity results for first-order Hamilton–Jacobi equations. Indiana Univ. Math. J. 39(2), 443–466 · Zbl 0709.35024 · doi:10.1512/iumj.1990.39.39024 [14] Barles, G., Biton, S., Ley, O. (2002). A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 162(4), 287–325 · Zbl 1052.35084 · doi:10.1007/s002050200188 [15] Barles, G., Lions, P.L.: Remarks on existence and uniqueness results for first-order Hamilton–Jacobi equations. Contributions to nonlinear partial differential equations, vol. II (Paris, 1985), pp. 4–15. Pitman Res. Notes Math. Ser., 155, Longman Sci. Tech., Harlow (1987) [16] Barles G., Soner H.M. and Souganidis P.E. (1993). Front propagation and phase field theory. SIAM J. Cont. Optim. 31(2): 439–469 · Zbl 0785.35049 · doi:10.1137/0331021 [17] Barles G. and Souganidis P.E. (2001). Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32(6): 1311–1323 · Zbl 0986.35047 · doi:10.1137/S0036141000369344 [18] Evans L.C. (1989). The perturbed test function method for viscosity solutions of nonlinear PDE. Proc. R. Soc. Edinb. Sect. A 111(3–4): 359–375 · Zbl 0679.35001 [19] Evans L.C. (1992). Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. R. Soc. Edinb. Sect. A 120(3–4): 245–265 · Zbl 0796.35011 [20] Imbert, C., Monneau, R.: Homogenization of first-order equations with u/$$\epsilon$$-periodic Hamiltonians. Part I: local equations (Preprint) · Zbl 1127.70009 [21] Giga Y. and Sato M-H. (2001). A level set approach to semicontinuous viscosity solutions for Cauchy problems. Comm. Partial Diff. Eq. 26(5–6): 813–839 · Zbl 1005.49025 · doi:10.1081/PDE-100002379 [22] Horie K. and Ishii H. (1998). Homogenization of Hamilton–Jacobi equations on domains with small scale periodic structure. Indiana Univ. Math. J. 47(3): 1011–1058 · Zbl 0924.49020 · doi:10.1512/iumj.1998.47.1385 [23] Ishii, H.: Homogenization of the Cauchy problem for Hamilton–Jacobi equations. Stochastic analysis, control, optimization and applications, pp. 305–324. Systems Control Found. Appl., Birkhäuser Boston, Boston (1999) · Zbl 0918.49024 [24] Ishii, H.: Almost periodic homogenization of Hamilton–Jacobi equations. International conference on differential equations vol. 1, 2 (Berlin, 1999), pp. 600–605. World Scientific Publishing, River Edge (2000) · Zbl 0969.35018 [25] Ishii H. (1987). Perron’s method for Hamilton–Jacobi equations. Duke Math. J. 55(2): 369–384 · Zbl 0697.35030 · doi:10.1215/S0012-7094-87-05521-9 [26] Ley O. (2001). Lower-bound gradient estimates for first-order Hamilton–Jacobi equations and applications to the regularity of propagating fronts. Adv. Differ. Equ. 6(5): 547–576 · Zbl 1015.35031 [27] Lions, P.-L., Papanicolaou, G., Varadhan, S.R.S.: Homogenization of Hamilton–Jacobi equations (unpublished work) [28] Souganidis P.E. (1999). Stochastic homogenization of Hamilton–Jacobi equations and some applications. Asymptot. Anal. 20(1): 1–11 · Zbl 0935.35008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.