July  2016, 36(7): 3677-3703. doi: 10.3934/dcds.2016.36.3677

Neumann homogenization via integro-differential operators

1. 

Department of Mathematics, University of Massachusetts, Amherst, Amherst, MA 90095, United States

2. 

Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824, United States

Received  July 2015 Revised  January 2016 Published  March 2016

In this note we describe how the Neumann homogenization of fully nonlinear elliptic equations can be recast as the study of nonlocal (integro-differential) equations involving elliptic integro-differential operators on the boundary. This is motivated by a new integro-differential representation for nonlinear operators with a comparison principle which we also introduce. In the simple case that the original domain is an infinite strip with almost periodic Neumann data, this leads to an almost periodic homogenization problem involving a fully nonlinear integro-differential operator on the Neumann boundary. This method gives a new proof-- which was left as an open question in the earlier work of Barles- Da Lio- Lions- Souganidis (2008)-- of the result obtained recently by Choi-Kim-Lee (2013), and we anticipate that it will generalize to other contexts.
Citation: Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677
References:
[1]

L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, Arch. Rational Mech. Anal., 123 (1993), 199.  doi: 10.1007/BF00375127.  Google Scholar

[2]

M. Arisawa, Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 293.  doi: 10.1016/S0294-1449(02)00025-2.  Google Scholar

[3]

I. Babuška, Solution of interface problems by homogenization. I,, SIAM J. Math. Anal., 7 (1976), 603.  doi: 10.1137/0507048.  Google Scholar

[4]

G. Barles, F. Da Lio, P.-L. Lions and P. E. Souganidis, Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions,, Indiana Univ. Math. J., 57 (2008), 2355.  doi: 10.1512/iumj.2008.57.3363.  Google Scholar

[5]

G. Barles, Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications,, J. Differential Equations, 154 (1999), 191.  doi: 10.1006/jdeq.1998.3568.  Google Scholar

[6]

G. Barles and F. Da Lio, Local $C^{0,\alpha}$ estimates for viscosity solutions of Neumann-type boundary value problems,, J. Differential Equations, 225 (2006), 202.  doi: 10.1016/j.jde.2005.09.004.  Google Scholar

[7]

G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications,, Arch. Rational Mech. Anal., 141 (1998), 237.  doi: 10.1007/s002050050077.  Google Scholar

[8]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications,, North-Holland Publishing Co., (1978).   Google Scholar

[9]

S. Biton, Nonlinear monotone semigroups and viscosity solutions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 383.  doi: 10.1016/S0294-1449(00)00057-3.  Google Scholar

[10]

L. Caffarelli, M. G. Crandall, M. Kocan and A. Swięch, On viscosity solutions of fully nonlinear equations with measurable ingredients,, Comm. Pure Appl. Math., 49 (1996), 365.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[11]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar

[12]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, volume 43 of American Mathematical Society Colloquium Publications,, American Mathematical Society, (1995).   Google Scholar

[13]

H. Chang Lara, Regularity for fully non linear equations with non local drift,, , (2012).   Google Scholar

[14]

H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal, nonsymmetric equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 833.  doi: 10.1016/j.anihpc.2012.04.006.  Google Scholar

[15]

S. Choi and I. Kim, Homogenization for nonlinear pdes in general domains with oscillatory neumann boundary data,, J. Math. Pures Appl., 102 (2014), 419.  doi: 10.1016/j.matpur.2013.11.015.  Google Scholar

[16]

S. Choi, I. Kim and K.-A. Lee, Homogenization of Neumann boundary data with fully nonlinear operator,, Anal. PDE, 6 (2013), 951.  doi: 10.2140/apde.2013.6.951.  Google Scholar

[17]

F. H. Clarke, Optimization and Nonsmooth Analysis, volume 5., SIAM, (1990).  doi: 10.1137/1.9781611971309.  Google Scholar

[18]

E. D. Conway and E. Hopf, Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation,, J. Math. Mech., 13 (1964), 939.   Google Scholar

[19]

P. Courrege, Sur la forme intégro-différentielle des opérateurs de $ c^{\infty}_k$ dans $c $ satisfaisant au principe du maximum,, Séminaire Brelot-Choquet-Deny. Théorie du Potentiel, 10 (1965), 1.   Google Scholar

[20]

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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[21]

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization,, Acta Numer., 17 (2008), 147.  doi: 10.1017/S0962492906360011.  Google Scholar

[22]

L. C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods,, Israel J. Math., 36 (1980), 225.  doi: 10.1007/BF02762047.  Google Scholar

[23]

L. C. Evans, Some min-max methods for the Hamilton-Jacobi equation,, Indiana Univ. Math. J., 33 (1984), 31.  doi: 10.1512/iumj.1984.33.33002.  Google Scholar

[24]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations,, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245.  doi: 10.1017/S0308210500032121.  Google Scholar

[25]

W. H. Fleming, The Cauchy problem for degenerate parabolic equations,, J. Math. Mech., 13 (1964), 987.   Google Scholar

[26]

W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation,, J. Differential Equations, 5 (1969), 515.  doi: 10.1016/0022-0396(69)90091-6.  Google Scholar

[27]

N. Guillen and R. W. Schwab, Aleksandrov-bakelman-pucci type estimates for integro-differential equations,, Archive for Rational Mechanics and Analysis, 206 (2012), 111.  doi: 10.1007/s00205-012-0529-0.  Google Scholar

[28]

E. Hopf, The partial differential equation $u_t + u u_x=\mu u_{x x}$,, Comm. Pure Appl. Math., 3 (1950), 201.   Google Scholar

[29]

P. Hsu, On excursions of reflecting Brownian motion,, Trans. Amer. Math. Soc., 296 (1986), 239.  doi: 10.1090/S0002-9947-1986-0837810-X.  Google Scholar

[30]

H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Differential Equations, 83 (1990), 26.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[31]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, In International Conference on Differential Equations, (1999), 600.   Google Scholar

[32]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994).  doi: 10.1007/978-3-642-84659-5.  Google Scholar

[33]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators,, J. Eur. Math. Soc. (JEMS), ().   Google Scholar

[34]

M. Kassmann, M. Rang and R. W. Schwab, Hölder regularity for integro-differential equations with nonlinear directional dependence,, Indiana Univ. Math. J., 63 (2014), 1467.  doi: 10.1512/iumj.2014.63.5394.  Google Scholar

[35]

M. A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations,, Nonlinear Anal., 24 (1995), 147.  doi: 10.1016/0362-546X(94)00170-M.  Google Scholar

[36]

N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure,, Dokl. Akad. Nauk SSSR, 245 (1979), 18.   Google Scholar

[37]

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients,, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161.   Google Scholar

[38]

P.-L. Lions, N. S. Trudinger and J. IE Urbas, The neumann problem for equations of monge-ampère type,, Communications on pure and applied mathematics, 39 (1986), 539.  doi: 10.1002/cpa.3160390405.  Google Scholar

[39]

P.-L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic hamilton-jacobi-bellman equation,, Mathematische Zeitschrift, 191 (1986), 1.  doi: 10.1007/BF01163605.  Google Scholar

[40]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data,, Comm. Partial Differential Equations, 31 (2006), 1227.  doi: 10.1080/03605300600634999.  Google Scholar

[41]

R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations,, SIAM J. Math. Anal., 42 (2010), 2652.  doi: 10.1137/080737897.  Google Scholar

[42]

M. A. Shubin, Almost periodic functions and partial differential operators,, Russian Mathematical Surveys, 33 (1978), 1.   Google Scholar

[43]

L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace,, Indiana Univ. Math. J., 55 (2006), 1155.  doi: 10.1512/iumj.2006.55.2706.  Google Scholar

[44]

P. E. Souganidis, Personal, communication., ().   Google Scholar

[45]

P. E. Souganidis, Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games,, Nonlinear Anal., 9 (1985), 217.  doi: 10.1016/0362-546X(85)90062-8.  Google Scholar

[46]

Hiroshi Tanaka, Homogenization of diffusion processes with boundary conditions,, In Stochastic analysis and applications, (1984), 411.   Google Scholar

show all references

References:
[1]

L. Alvarez, F. Guichard, P.-L. Lions and J.-M. Morel, Axioms and fundamental equations of image processing,, Arch. Rational Mech. Anal., 123 (1993), 199.  doi: 10.1007/BF00375127.  Google Scholar

[2]

M. Arisawa, Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 293.  doi: 10.1016/S0294-1449(02)00025-2.  Google Scholar

[3]

I. Babuška, Solution of interface problems by homogenization. I,, SIAM J. Math. Anal., 7 (1976), 603.  doi: 10.1137/0507048.  Google Scholar

[4]

G. Barles, F. Da Lio, P.-L. Lions and P. E. Souganidis, Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions,, Indiana Univ. Math. J., 57 (2008), 2355.  doi: 10.1512/iumj.2008.57.3363.  Google Scholar

[5]

G. Barles, Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications,, J. Differential Equations, 154 (1999), 191.  doi: 10.1006/jdeq.1998.3568.  Google Scholar

[6]

G. Barles and F. Da Lio, Local $C^{0,\alpha}$ estimates for viscosity solutions of Neumann-type boundary value problems,, J. Differential Equations, 225 (2006), 202.  doi: 10.1016/j.jde.2005.09.004.  Google Scholar

[7]

G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications,, Arch. Rational Mech. Anal., 141 (1998), 237.  doi: 10.1007/s002050050077.  Google Scholar

[8]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, volume 5 of Studies in Mathematics and its Applications,, North-Holland Publishing Co., (1978).   Google Scholar

[9]

S. Biton, Nonlinear monotone semigroups and viscosity solutions,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 383.  doi: 10.1016/S0294-1449(00)00057-3.  Google Scholar

[10]

L. Caffarelli, M. G. Crandall, M. Kocan and A. Swięch, On viscosity solutions of fully nonlinear equations with measurable ingredients,, Comm. Pure Appl. Math., 49 (1996), 365.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[11]

L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar

[12]

L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, volume 43 of American Mathematical Society Colloquium Publications,, American Mathematical Society, (1995).   Google Scholar

[13]

H. Chang Lara, Regularity for fully non linear equations with non local drift,, , (2012).   Google Scholar

[14]

H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal, nonsymmetric equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 833.  doi: 10.1016/j.anihpc.2012.04.006.  Google Scholar

[15]

S. Choi and I. Kim, Homogenization for nonlinear pdes in general domains with oscillatory neumann boundary data,, J. Math. Pures Appl., 102 (2014), 419.  doi: 10.1016/j.matpur.2013.11.015.  Google Scholar

[16]

S. Choi, I. Kim and K.-A. Lee, Homogenization of Neumann boundary data with fully nonlinear operator,, Anal. PDE, 6 (2013), 951.  doi: 10.2140/apde.2013.6.951.  Google Scholar

[17]

F. H. Clarke, Optimization and Nonsmooth Analysis, volume 5., SIAM, (1990).  doi: 10.1137/1.9781611971309.  Google Scholar

[18]

E. D. Conway and E. Hopf, Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation,, J. Math. Mech., 13 (1964), 939.   Google Scholar

[19]

P. Courrege, Sur la forme intégro-différentielle des opérateurs de $ c^{\infty}_k$ dans $c $ satisfaisant au principe du maximum,, Séminaire Brelot-Choquet-Deny. Théorie du Potentiel, 10 (1965), 1.   Google Scholar

[20]

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.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[21]

B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization,, Acta Numer., 17 (2008), 147.  doi: 10.1017/S0962492906360011.  Google Scholar

[22]

L. C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods,, Israel J. Math., 36 (1980), 225.  doi: 10.1007/BF02762047.  Google Scholar

[23]

L. C. Evans, Some min-max methods for the Hamilton-Jacobi equation,, Indiana Univ. Math. J., 33 (1984), 31.  doi: 10.1512/iumj.1984.33.33002.  Google Scholar

[24]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations,, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245.  doi: 10.1017/S0308210500032121.  Google Scholar

[25]

W. H. Fleming, The Cauchy problem for degenerate parabolic equations,, J. Math. Mech., 13 (1964), 987.   Google Scholar

[26]

W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation,, J. Differential Equations, 5 (1969), 515.  doi: 10.1016/0022-0396(69)90091-6.  Google Scholar

[27]

N. Guillen and R. W. Schwab, Aleksandrov-bakelman-pucci type estimates for integro-differential equations,, Archive for Rational Mechanics and Analysis, 206 (2012), 111.  doi: 10.1007/s00205-012-0529-0.  Google Scholar

[28]

E. Hopf, The partial differential equation $u_t + u u_x=\mu u_{x x}$,, Comm. Pure Appl. Math., 3 (1950), 201.   Google Scholar

[29]

P. Hsu, On excursions of reflecting Brownian motion,, Trans. Amer. Math. Soc., 296 (1986), 239.  doi: 10.1090/S0002-9947-1986-0837810-X.  Google Scholar

[30]

H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Differential Equations, 83 (1990), 26.  doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[31]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, In International Conference on Differential Equations, (1999), 600.   Google Scholar

[32]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994).  doi: 10.1007/978-3-642-84659-5.  Google Scholar

[33]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators,, J. Eur. Math. Soc. (JEMS), ().   Google Scholar

[34]

M. Kassmann, M. Rang and R. W. Schwab, Hölder regularity for integro-differential equations with nonlinear directional dependence,, Indiana Univ. Math. J., 63 (2014), 1467.  doi: 10.1512/iumj.2014.63.5394.  Google Scholar

[35]

M. A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of second-order fully nonlinear degenerate parabolic equations,, Nonlinear Anal., 24 (1995), 147.  doi: 10.1016/0362-546X(94)00170-M.  Google Scholar

[36]

N. V. Krylov and M. V. Safonov, An estimate for the probability of a diffusion process hitting a set of positive measure,, Dokl. Akad. Nauk SSSR, 245 (1979), 18.   Google Scholar

[37]

N. V. Krylov and M. V. Safonov, A property of the solutions of parabolic equations with measurable coefficients,, Izv. Akad. Nauk SSSR Ser. Mat., 44 (1980), 161.   Google Scholar

[38]

P.-L. Lions, N. S. Trudinger and J. IE Urbas, The neumann problem for equations of monge-ampère type,, Communications on pure and applied mathematics, 39 (1986), 539.  doi: 10.1002/cpa.3160390405.  Google Scholar

[39]

P.-L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic hamilton-jacobi-bellman equation,, Mathematische Zeitschrift, 191 (1986), 1.  doi: 10.1007/BF01163605.  Google Scholar

[40]

E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data,, Comm. Partial Differential Equations, 31 (2006), 1227.  doi: 10.1080/03605300600634999.  Google Scholar

[41]

R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations,, SIAM J. Math. Anal., 42 (2010), 2652.  doi: 10.1137/080737897.  Google Scholar

[42]

M. A. Shubin, Almost periodic functions and partial differential operators,, Russian Mathematical Surveys, 33 (1978), 1.   Google Scholar

[43]

L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace,, Indiana Univ. Math. J., 55 (2006), 1155.  doi: 10.1512/iumj.2006.55.2706.  Google Scholar

[44]

P. E. Souganidis, Personal, communication., ().   Google Scholar

[45]

P. E. Souganidis, Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games,, Nonlinear Anal., 9 (1985), 217.  doi: 10.1016/0362-546X(85)90062-8.  Google Scholar

[46]

Hiroshi Tanaka, Homogenization of diffusion processes with boundary conditions,, In Stochastic analysis and applications, (1984), 411.   Google Scholar

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