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Spectral properties of renormalization for area-preserving maps
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 |
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. |
[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. |
[3] |
I. Babuška, Solution of interface problems by homogenization. I,, SIAM J. Math. Anal., 7 (1976), 603.
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.
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.
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.
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.
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., (1978).
|
[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. |
[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. |
[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. |
[12] |
L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, volume 43 of American Mathematical Society Colloquium Publications,, American Mathematical Society, (1995).
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[21] |
B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization,, Acta Numer., 17 (2008), 147.
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.
doi: 10.1007/BF02762047. |
[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. |
[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. |
[25] |
W. H. Fleming, The Cauchy problem for degenerate parabolic equations,, J. Math. Mech., 13 (1964), 987.
|
[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. |
[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. |
[28] |
E. Hopf, The partial differential equation $u_t + u u_x=\mu u_{x x}$,, Comm. Pure Appl. Math., 3 (1950), 201.
|
[29] |
P. Hsu, On excursions of reflecting Brownian motion,, Trans. Amer. Math. Soc., 296 (1986), 239.
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.
doi: 10.1016/0022-0396(90)90068-Z. |
[31] |
H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, In International Conference on Differential Equations, (1999), 600.
|
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[41] |
R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations,, SIAM J. Math. Anal., 42 (2010), 2652.
doi: 10.1137/080737897. |
[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. |
[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. |
[46] |
Hiroshi Tanaka, Homogenization of diffusion processes with boundary conditions,, In Stochastic analysis and applications, (1984), 411.
|
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. |
[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. |
[3] |
I. Babuška, Solution of interface problems by homogenization. I,, SIAM J. Math. Anal., 7 (1976), 603.
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.
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.
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.
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.
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., (1978).
|
[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. |
[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. |
[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. |
[12] |
L. A. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, volume 43 of American Mathematical Society Colloquium Publications,, American Mathematical Society, (1995).
|
[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. |
[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. |
[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. |
[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.
|
[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. |
[21] |
B. Engquist and P. E. Souganidis, Asymptotic and numerical homogenization,, Acta Numer., 17 (2008), 147.
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.
doi: 10.1007/BF02762047. |
[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. |
[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. |
[25] |
W. H. Fleming, The Cauchy problem for degenerate parabolic equations,, J. Math. Mech., 13 (1964), 987.
|
[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. |
[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. |
[28] |
E. Hopf, The partial differential equation $u_t + u u_x=\mu u_{x x}$,, Comm. Pure Appl. Math., 3 (1950), 201.
|
[29] |
P. Hsu, On excursions of reflecting Brownian motion,, Trans. Amer. Math. Soc., 296 (1986), 239.
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.
doi: 10.1016/0022-0396(90)90068-Z. |
[31] |
H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations,, In International Conference on Differential Equations, (1999), 600.
|
[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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[41] |
R. W. Schwab, Periodic homogenization for nonlinear integro-differential equations,, SIAM J. Math. Anal., 42 (2010), 2652.
doi: 10.1137/080737897. |
[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. |
[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. |
[46] |
Hiroshi Tanaka, Homogenization of diffusion processes with boundary conditions,, In Stochastic analysis and applications, (1984), 411.
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