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

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March 2011, 6(1): 61-75. doi: 10.3934/nhm.2011.6.61

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

Fabio Camilli ^1, and Claudio Marchi ^2,

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

Received October 2009 Revised May 2010 Published March 2011

Abstract
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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.

Keywords: Nonlinear elliptic equations, rate of convergence., multiscale homogenization, Bellman equations.

Mathematics Subject Classification: Primary: 35B27; Secondary: 35J60, 49L2.

Citation: Fabio Camilli, Claudio Marchi. On the convergence rate in multiscale homogenization of fully nonlinear elliptic problems. Networks & Heterogeneous Media, 2011, 6 (1) : 61-75. doi: 10.3934/nhm.2011.6.61

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. doi: 10.1137/S0363012900366741. Google Scholar
[2]	O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result,, Arch. Ration. Mech. Anal., 170 (2003), 17. doi: 10.1007/s00205-003-0266-5. Google Scholar
[3]	O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation,, Mem. Amer. Math. Soc., 204 (2010). Google Scholar
[4]	O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenizations for second-order Hamilton-Jacobi equations,, J. Differential Equations, 243 (2007), 349. doi: 10.1016/j.jde.2007.05.027. Google Scholar
[5]	O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbation and homogenization of optimal control problems,, in, (2008), 1. Google Scholar
[6]	M. Arisawa and P. L. Lions, On ergodic stochastic control,, Comm. Partial Differential Equations, 23 (1998), 2187. Google Scholar
[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. doi: 10.1051/m2an:2002002. Google Scholar
[8]	A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'', Clarendon Press, (1998). Google Scholar
[9]	A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'', North-Holland, (1978). Google Scholar
[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. doi: 10.1002/cpa.20069. Google Scholar
[11]	F. Camilli and C. Marchi, Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs,, Nonlinearity, 22 (2009), 1481. doi: 10.1088/0951-7715/22/6/011. Google Scholar
[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. doi: 10.1512/iumj.2001.50.1933. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	L. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations,, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245. Google Scholar
[16]	W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'', Springer-Verlag, (1993). Google Scholar
[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. doi: 10.1512/iumj.1989.38.38015. Google Scholar
[18]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd, edition, (1983). Google Scholar
[19]	V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,'', Springer, (1994). Google Scholar
[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. doi: 10.1007/s004400050264. Google Scholar
[21]	O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,'', Academic Press, (1968). Google Scholar
[22]	P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations,, Unpublished, (1986). Google Scholar
[23]	P.L. Lions and P. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media,, Comm. Partial Differential Equations, 30 (2005), 335. doi: 10.1081/PDE-200050077. Google Scholar
[24]	P. L. Lions and P. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 667. doi: 10.1016/j.anihpc.2004.10.009. Google Scholar
[25]	C. Marchi, Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 519. doi: 10.1017/S0308210507000704. Google Scholar
[26]	M. V. Safonov, Classical solution of nonlinear elliptic equations of second-order,, Math. USSR-Izv., 33 (1989), 597. Google Scholar

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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. doi: 10.1137/S0363012900366741. Google Scholar
[2]	O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: A general convergence result,, Arch. Ration. Mech. Anal., 170 (2003), 17. doi: 10.1007/s00205-003-0266-5. Google Scholar
[3]	O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation,, Mem. Amer. Math. Soc., 204 (2010). Google Scholar
[4]	O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenizations for second-order Hamilton-Jacobi equations,, J. Differential Equations, 243 (2007), 349. doi: 10.1016/j.jde.2007.05.027. Google Scholar
[5]	O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbation and homogenization of optimal control problems,, in, (2008), 1. Google Scholar
[6]	M. Arisawa and P. L. Lions, On ergodic stochastic control,, Comm. Partial Differential Equations, 23 (1998), 2187. Google Scholar
[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. doi: 10.1051/m2an:2002002. Google Scholar
[8]	A. Braides and A. Defranceschi, "Homogenization of Multiple Integrals,'', Clarendon Press, (1998). Google Scholar
[9]	A. Bensoussan, J. L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures,'', North-Holland, (1978). Google Scholar
[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. doi: 10.1002/cpa.20069. Google Scholar
[11]	F. Camilli and C. Marchi, Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs,, Nonlinearity, 22 (2009), 1481. doi: 10.1088/0951-7715/22/6/011. Google Scholar
[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. doi: 10.1512/iumj.2001.50.1933. Google Scholar
[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. Google Scholar
[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. Google Scholar
[15]	L. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations,, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245. Google Scholar
[16]	W. H. Fleming and H. M. Soner, "Controlled Markov Processes and Viscosity Solutions,'', Springer-Verlag, (1993). Google Scholar
[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. doi: 10.1512/iumj.1989.38.38015. Google Scholar
[18]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order,'' 2nd, edition, (1983). Google Scholar
[19]	V. V. Jikov, S. M. Kozlov and O. A. Oleinik, "Homogenization of Differential Operators and Integral Functionals,'', Springer, (1994). Google Scholar
[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. doi: 10.1007/s004400050264. Google Scholar
[21]	O. A. Ladyzhenskaya and N. N. Ural'tseva, "Linear and Quasilinear Elliptic Equations,'', Academic Press, (1968). Google Scholar
[22]	P. L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations,, Unpublished, (1986). Google Scholar
[23]	P.L. Lions and P. Souganidis, Homogenization of "viscous'' Hamilton-Jacobi equations in stationary ergodic media,, Comm. Partial Differential Equations, 30 (2005), 335. doi: 10.1081/PDE-200050077. Google Scholar
[24]	P. L. Lions and P. Souganidis, Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 22 (2005), 667. doi: 10.1016/j.anihpc.2004.10.009. Google Scholar
[25]	C. Marchi, Rate of convergence for multiscale homogenization of Hamilton-Jacobi equations,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 519. doi: 10.1017/S0308210507000704. Google Scholar
[26]	M. V. Safonov, Classical solution of nonlinear elliptic equations of second-order,, Math. USSR-Izv., 33 (1989), 597. Google Scholar

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