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 and 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), 1159-1188. 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)).

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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), 1159-1188. 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)).

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