# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 199-257. doi: 10.1007/BF00375127. [2] M. Arisawa, Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 293-332. doi: 10.1016/S0294-1449(02)00025-2. [3] I. Babuška, Solution of interface problems by homogenization. I, SIAM J. Math. Anal., 7 (1976), 603-634. doi: 10.1137/0507048. [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), 2355-2375. doi: 10.1512/iumj.2008.57.3363. [5] G. Barles, Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differential Equations, 154 (1999), 191-224. doi: 10.1006/jdeq.1998.3568. [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), 202-241. doi: 10.1016/j.jde.2005.09.004. [7] G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Rational Mech. Anal., 141 (1998), 237-296. doi: 10.1007/s002050050077. [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., Amsterdam, 1978. [9] S. Biton, Nonlinear monotone semigroups and viscosity solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 383-402. doi: 10.1016/S0294-1449(00)00057-3. [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), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. [11] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274. [12] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, volume 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995. [13] H. Chang Lara, Regularity for fully non linear equations with non local drift, arXiv:1210.4242, 2012. [14] H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal, nonsymmetric equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 833-859. doi: 10.1016/j.anihpc.2012.04.006. [15] S. Choi and I. Kim, Homogenization for nonlinear pdes in general domains with oscillatory neumann boundary data, J. Math. Pures Appl., 102 (2014), 419-448, arXiv:1302.5386 [math.AP], 2013. doi: 10.1016/j.matpur.2013.11.015. [16] S. Choi, I. Kim and K.-A. Lee, Homogenization of Neumann boundary data with fully nonlinear operator, Anal. PDE, 6 (2013), 951-972. doi: 10.2140/apde.2013.6.951. [17] F. H. Clarke, Optimization and Nonsmooth Analysis, volume 5. SIAM, 1990. doi: 10.1137/1.9781611971309. [18] E. D. Conway and E. Hopf, Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation, J. Math. Mech., 13 (1964), 939-986. [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), 1-38. [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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [21] B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numer., 17 (2008), 147-190. doi: 10.1017/S0962492906360011. [22] L. C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods, Israel J. Math., 36 (1980), 225-247. doi: 10.1007/BF02762047. [23] L. C. Evans, Some min-max methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J., 33 (1984), 31-50. doi: 10.1512/iumj.1984.33.33002. [24] L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121. [25] W. H. Fleming, The Cauchy problem for degenerate parabolic equations, J. Math. Mech., 13 (1964), 987-1008. [26] W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations, 5 (1969), 515-530. doi: 10.1016/0022-0396(69)90091-6. [27] N. Guillen and R. W. Schwab, Aleksandrov-bakelman-pucci type estimates for integro-differential equations, Archive for Rational Mechanics and Analysis, 206 (2012), 111-157. doi: 10.1007/s00205-012-0529-0. [28] E. Hopf, The partial differential equation $u_t + u u_x=\mu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230. [29] P. Hsu, On excursions of reflecting Brownian motion, Trans. Amer. Math. Soc., 296 (1986), 239-264. doi: 10.1090/S0002-9947-1986-0837810-X. [30] H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z. [31] H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 600-605. World Sci. Publ., River Edge, NJ, 2000. [32] V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif’yan]. doi: 10.1007/978-3-642-84659-5. [33] M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators,, J. Eur. Math. Soc. (JEMS), (). [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), 1467-1498. doi: 10.1512/iumj.2014.63.5394. [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), 147-158. doi: 10.1016/0362-546X(94)00170-M. [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), 18-20. [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), 161-175, 239. [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), 539-563. doi: 10.1002/cpa.3160390405. [39] P.-L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic hamilton-jacobi-bellman equation, Mathematische Zeitschrift, 191 (1986), 1-15. doi: 10.1007/BF01163605. [40] E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm. Partial Differential Equations, 31 (2006), 1227-1252. doi: 10.1080/03605300600634999. [41] R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680. doi: 10.1137/080737897. [42] M. A. Shubin, Almost periodic functions and partial differential operators, Russian Mathematical Surveys, 33 (1978), 1-52. [43] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706. [44] P. E. Souganidis, Personal, communication., (). [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), 217-257. doi: 10.1016/0362-546X(85)90062-8. [46] Hiroshi Tanaka, Homogenization of diffusion processes with boundary conditions, In Stochastic analysis and applications, volume 7 of Adv. Probab. Related Topics, pages 411-437. Dekker, New York, 1984.

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), 199-257. doi: 10.1007/BF00375127. [2] M. Arisawa, Long time averaged reflection force and homogenization of oscillating Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20 (2003), 293-332. doi: 10.1016/S0294-1449(02)00025-2. [3] I. Babuška, Solution of interface problems by homogenization. I, SIAM J. Math. Anal., 7 (1976), 603-634. doi: 10.1137/0507048. [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), 2355-2375. doi: 10.1512/iumj.2008.57.3363. [5] G. Barles, Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications, J. Differential Equations, 154 (1999), 191-224. doi: 10.1006/jdeq.1998.3568. [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), 202-241. doi: 10.1016/j.jde.2005.09.004. [7] G. Barles and P. E. Souganidis, A new approach to front propagation problems: Theory and applications, Arch. Rational Mech. Anal., 141 (1998), 237-296. doi: 10.1007/s002050050077. [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., Amsterdam, 1978. [9] S. Biton, Nonlinear monotone semigroups and viscosity solutions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 383-402. doi: 10.1016/S0294-1449(00)00057-3. [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), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A. [11] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638. doi: 10.1002/cpa.20274. [12] L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, volume 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995. [13] H. Chang Lara, Regularity for fully non linear equations with non local drift, arXiv:1210.4242, 2012. [14] H. Chang Lara and G. Dávila, Regularity for solutions of nonlocal, nonsymmetric equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 833-859. doi: 10.1016/j.anihpc.2012.04.006. [15] S. Choi and I. Kim, Homogenization for nonlinear pdes in general domains with oscillatory neumann boundary data, J. Math. Pures Appl., 102 (2014), 419-448, arXiv:1302.5386 [math.AP], 2013. doi: 10.1016/j.matpur.2013.11.015. [16] S. Choi, I. Kim and K.-A. Lee, Homogenization of Neumann boundary data with fully nonlinear operator, Anal. PDE, 6 (2013), 951-972. doi: 10.2140/apde.2013.6.951. [17] F. H. Clarke, Optimization and Nonsmooth Analysis, volume 5. SIAM, 1990. doi: 10.1137/1.9781611971309. [18] E. D. Conway and E. Hopf, Hamilton's theory and generalized solutions of the Hamilton-Jacobi equation, J. Math. Mech., 13 (1964), 939-986. [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), 1-38. [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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5. [21] B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization, Acta Numer., 17 (2008), 147-190. doi: 10.1017/S0962492906360011. [22] L. C. Evans, On solving certain nonlinear partial differential equations by accretive operator methods, Israel J. Math., 36 (1980), 225-247. doi: 10.1007/BF02762047. [23] L. C. Evans, Some min-max methods for the Hamilton-Jacobi equation, Indiana Univ. Math. J., 33 (1984), 31-50. doi: 10.1512/iumj.1984.33.33002. [24] L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265. doi: 10.1017/S0308210500032121. [25] W. H. Fleming, The Cauchy problem for degenerate parabolic equations, J. Math. Mech., 13 (1964), 987-1008. [26] W. H. Fleming, The Cauchy problem for a nonlinear first order partial differential equation, J. Differential Equations, 5 (1969), 515-530. doi: 10.1016/0022-0396(69)90091-6. [27] N. Guillen and R. W. Schwab, Aleksandrov-bakelman-pucci type estimates for integro-differential equations, Archive for Rational Mechanics and Analysis, 206 (2012), 111-157. doi: 10.1007/s00205-012-0529-0. [28] E. Hopf, The partial differential equation $u_t + u u_x=\mu u_{x x}$, Comm. Pure Appl. Math., 3 (1950), 201-230. [29] P. Hsu, On excursions of reflecting Brownian motion, Trans. Amer. Math. Soc., 296 (1986), 239-264. doi: 10.1090/S0002-9947-1986-0837810-X. [30] H. Ishii and P.-L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z. [31] H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 600-605. World Sci. Publ., River Edge, NJ, 2000. [32] V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif’yan]. doi: 10.1007/978-3-642-84659-5. [33] M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators,, J. Eur. Math. Soc. (JEMS), (). [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), 1467-1498. doi: 10.1512/iumj.2014.63.5394. [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), 147-158. doi: 10.1016/0362-546X(94)00170-M. [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), 18-20. [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), 161-175, 239. [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), 539-563. doi: 10.1002/cpa.3160390405. [39] P.-L. Lions and N. S. Trudinger, Linear oblique derivative problems for the uniformly elliptic hamilton-jacobi-bellman equation, Mathematische Zeitschrift, 191 (1986), 1-15. doi: 10.1007/BF01163605. [40] E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Comm. Partial Differential Equations, 31 (2006), 1227-1252. doi: 10.1080/03605300600634999. [41] R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations, SIAM J. Math. Anal., 42 (2010), 2652-2680. doi: 10.1137/080737897. [42] M. A. Shubin, Almost periodic functions and partial differential operators, Russian Mathematical Surveys, 33 (1978), 1-52. [43] L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional Laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174. doi: 10.1512/iumj.2006.55.2706. [44] P. E. Souganidis, Personal, communication., (). [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), 217-257. doi: 10.1016/0362-546X(85)90062-8. [46] Hiroshi Tanaka, Homogenization of diffusion processes with boundary conditions, In Stochastic analysis and applications, volume 7 of Adv. Probab. Related Topics, pages 411-437. Dekker, New York, 1984.
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