On the convergence rate in  multiscale homogenization of  fully nonlinear elliptic problems

Fabio Camilli; Claudio Marchi; Fabio Camilli; Claudio Marchi

doi:10.3934/nhm.2011.6.61

Networks and Heterogeneous Media

2011, Volume 6, Issue 1: 61-75. doi: 10.3934/nhm.2011.6.61

Previous Article Next Article

On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems

Fabio Camilli ^{1
,},
Claudio Marchi ^{2
,}

1.
Dip. di Matematica Pura e Applicata, Univ. dell'Aquila, loc. Monteluco di Roio, 67040 l'Aquila
2.
Dipartimento di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova

Received: 01 October 2009 Revised: 01 May 2010
Primary: 35B27; Secondary: 35J60, 49L25.

This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form $u^\epsilon+H^\epsilon (x,\frac{x}{\epsilon},\ldots,\frac{x}{epsilon^k},Du^\epsilon,D^2u^\epsilon)=0$. The operators $H^\epsilon$ are a regular perturbations of some uniformly elliptic, convex operator $H$. As $\epsilon\to 0^+$, the solutions $u^\epsilon$ converge locally uniformly to the solution $u$ of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed.
- Nonlinear elliptic equations,
- Bellman equations,
- multiscale homogenization,
- rate of convergence
Citation: Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems[J]. Networks and Heterogeneous Media, 2011, 6(1): 61-75. doi: 10.3934/nhm.2011.6.61

Related Papers:

Abstract

This paper concerns periodic multiscale homogenization for fully nonlinear equations of the form $u^\epsilon+H^\epsilon (x,\frac{x}{\epsilon},\ldots,\frac{x}{epsilon^k},Du^\epsilon,D^2u^\epsilon)=0$. The operators $H^\epsilon$ are a regular perturbations of some uniformly elliptic, convex operator $H$. As $\epsilon\to 0^+$, the solutions $u^\epsilon$ converge locally uniformly to the solution $u$ of a suitably defined effective problem. The purpose of this paper is to obtain an estimate of the corresponding rate of convergence. Finally, some examples are discussed.

References

[1]	O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deterministic and stochastic control, SIAM J. Control Optim., 40 (2001), 1159-1188. doi: 10.1137/S0363012900366741
[2]	O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61. doi: 10.1007/s00205-003-0266-5
[3]	O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation, Mem. Amer. Math. Soc., 204 (2010).
[4]	O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenizations for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387. doi: 10.1016/j.jde.2007.05.027
[5]	O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbation and homogenization of optimal control problems, in "Geometric Control and Nonsmooth Analysis" (F. Ancona, A. Bressan, P. Cannarsa, F. Clarke, P.R. Wolenski; Eds.), World Scientific, Singapore, 2008, 1-27.
[6]	M. Arisawa and P. L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217.
[7]	G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations, M2AN Math. Model. Numer. Anal., 36 (2002), 33-54. doi: 10.1051/m2an:2002002
[8]	A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'' Clarendon Press, Oxford, 1998.
[9]	A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'' North-Holland, Amsterdam, 1978.
[10]	L. A. Caffarelli, P. Souganidis and L. Wang, Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media, Comm. Pure Appl. Math., 58 (2005), 319-361. doi: 10.1002/cpa.20069
[11]	F. Camilli and C. Marchi, Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs, Nonlinearity, 22 (2009), 1481-1498. doi: 10.1088/0951-7715/22/6/011
[12]	I. Capuzzo Dolcetta and H. Ishii, On the rate of convergence in Homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), 1113-1129. doi: 10.1512/iumj.2001.50.1933
[13]	M. G. Crandall, H. Ishii and P.-L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67.
[14]	L. Evans, The perturbed test function method for viscosity solutions of nonlinear P.D.E., Proc. Roy. Soc. Edinburgh Sect. A, 111 (1989), 359-375.
[15]	L. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265.
[16]	W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'' Springer-Verlag, Berlin, 1993.
[17]	W. H. Fleming and P. E. Souganidis, On the existence of value functions of two-players, zero-sum stochastic differential games, Indiana Univ. Math. J., 38 (1989), 293-314. doi: 10.1512/iumj.1989.38.38015
[18]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd edition, Springer, Berlin, 1983.
[19]	V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,'' Springer, Berlin, 1994.
[20]	N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman's equations with variable coefficients, Probab. Theory Related Fields, 117 (2000), 1-16. doi: 10.1007/s004400050264
[21]	O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,'' Academic Press, New York, 1968.
[22]	P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986.
[23]	P.L. Lions and P. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media, Comm. Partial Differential Equations, 30 (2005), 335-375. doi: 10.1081/PDE-200050077
[24]	P. L. Lions and P. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications, Ann. Inst. H. Poincarè Anal. Non Linèaire, 22 (2005), 667-677. doi: 10.1016/j.anihpc.2004.10.009
[25]	C. Marchi, Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 519-539. doi: 10.1017/S0308210507000704
[26]	M. V. Safonov, Classical solution of nonlinear elliptic equations of second-order, Math. USSR-Izv., 33 (1989), 597-612. (Engl. transl. of Izv. Akad. Nauk SSSR Ser. Mat., 52 (1988)).

Reader Comments

Your name:*

Email:*
© 2011 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)