• Previous Article
    Existence and continuity of global attractors for ternary mixtures of solids
  • DCDS-B Home
  • This Issue
  • Next Article
    Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations
doi: 10.3934/dcdsb.2021119
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Bloch wave approach to almost periodic homogenization and approximations of effective coefficients

Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India

* Corresponding author: Vivek Tewary, Present Address: TIFR Centre for Applicable Mathematics, Bengaluru, India

Received  August 2020 Revised  February 2021 Early access April 2021

Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the almost periodic homogenization result by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.

Citation: Sista Sivaji Ganesh, Vivek Tewary. Bloch wave approach to almost periodic homogenization and approximations of effective coefficients. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021119
References:
[1]

A. AbdulleD. Arjmand and E. Paganoni, Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems, C. R. Math. Acad. Sci. Paris, 357 (2019), 545-551.  doi: 10.1016/j.crma.2019.05.011.  Google Scholar

[2]

G. AllaireY. CapdeboscqA. PiatnitskiV. Siess and M. Vanninathan, Homogenization of periodic systems with large potentials, Arch. Ration. Mech. Anal., 174 (2004), 179-220.  doi: 10.1007/s00205-004-0332-7.  Google Scholar

[3]

M. Allais, Sur la distribution normale des valeurs à des instants régulièrement espacés d'une somme de sinusoïdes, C. R. Acad. Sci. Paris Sér. I Math., 296 (1983), 829-832.   Google Scholar

[4]

M. S. AlnæsJ. BlechtaJ. HakeA. JohanssonB. KehletA. LoggC. RichardsonJ. RingM. E. Rognes and G. N. Wells, The fenics project version 1.5, Archive of Numerical Software, 3 (2015), 9-23.   Google Scholar

[5]

S. N. ArmstrongP. Cardaliaguet and P. E. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations, J. Amer. Math. Soc., 27 (2014), 479-540.  doi: 10.1090/S0894-0347-2014-00783-9.  Google Scholar

[6]

S. N. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923.  doi: 10.1002/cpa.21616.  Google Scholar

[7]

M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607.  Google Scholar

[8]

A. Benoit and A. Gloria, Long-time homogenization and asymptotic ballistic transport of classical waves, Ann. Sci. Éc. Norm. Supér. (4), 52 (2019), 703–759. doi: 10.24033/asens.2395.  Google Scholar

[9]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.  Google Scholar

[10]

A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.  Google Scholar

[11]

X. BlancC. Le Bris and P.-L. Lions, Local profiles for elliptic problems at different scales: Defects in, and interfaces between periodic structures, Comm. Partial Differential Equations, 40 (2015), 2173-2236.  doi: 10.1080/03605302.2015.1043464.  Google Scholar

[12]

H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.  Google Scholar

[13]

A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. H. Poincaré Probab. Statist., 40 (2004), 153-165.  doi: 10.1016/S0246-0203(03)00065-7.  Google Scholar

[14]

L. BălilescuC. ConcaT. GhoshJ. San Martín and M. Vanninathan, The dispersion tensor and its unique minimizer in Hashin–Shtrikman micro-structures, Arch. Ration. Mech. Anal., 230 (2018), 665-700.  doi: 10.1007/s00205-018-1255-z.  Google Scholar

[15]

T. O. Carvalho and C. R. de Oliveira, Spectra and transport in almost periodic dimers, J. Statist. Phys., 107 (2002), 1015-1030.  doi: 10.1023/A:1015153523475.  Google Scholar

[16]

J. Casado-Díaz and I. Gayte, A derivation theory for generalized Besicovitch spaces and its application for partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 283-315.  doi: 10.1017/S0308210500001633.  Google Scholar

[17]

Andrej Cherkaev and Robert Kohn (eds.), Topics in the Mathematical Modelling of Composite Materials, vol. 31, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-2032-9.  Google Scholar

[18]

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.  Google Scholar

[19]

D. Damanik, J. Fillman and A. Gorodetski, Multidimensional almost-periodic Schrödinger operators with Cantor spectrum, Annales Henri Poincaré, 20 (2019), 1393–1402. doi: 10.1007/s00023-019-00768-5.  Google Scholar

[20]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[21]

A. Gloria, Reduction of the resonance error-Part 1: Approximation of homogenized coefficients, Math. Models Methods Appl. Sci., 21 (2011), 1601-1630.  doi: 10.1142/S0218202511005507.  Google Scholar

[22]

A. Gloria and Z. Habibi, Reduction in the resonance error in numerical homogenization Ⅱ: Correctors and extrapolation, Found. Comput. Math., 16 (2016), 217-296.  doi: 10.1007/s10208-015-9246-z.  Google Scholar

[23]

A. Gloria and F. Otto, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc., 19 (2017), 3489-3548.  doi: 10.4171/JEMS/745.  Google Scholar

[24]

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

[25]

S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), 107 (1978), 199–217,317.  Google Scholar

[26]

S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109 (1979), 188–202,327.  Google Scholar

[27] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982.   Google Scholar
[28]

O. A. Oleinik and V. V. Zhikov, On the homogenization of elliptic operators with almost-periodic coefficients, Rend. Sem. Mat. Fis. Milano, 52 (1982), 149-166.  doi: 10.1007/BF02925004.  Google Scholar

[29]

A. Pankov, Periodic approximations of homogenization problems, Math. Methods Appl. Sci., 36 (2013), 2018-2022.  doi: 10.1002/mma.1528.  Google Scholar

[30]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292.  doi: 10.1007/BF00252910.  Google Scholar

[31]

M. Reed and B. Simon, Methods of modern mathematical physics. Ⅳ. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[32]

D. ShechtmanI. BlechD. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., 53 (1984), 1951-1953.  doi: 10.1103/PhysRevLett.53.1951.  Google Scholar

[33]

Z. Shen, Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems, Anal. PDE, 8 (2015), 1565-1601.  doi: 10.2140/apde.2015.8.1565.  Google Scholar

[34]

Z. Shen and J. Zhuge, Approximate correctors and convergence rates in almost-periodic homogenization, J. Math. Pures Appl., 110 (2018), 187-238.  doi: 10.1016/j.matpur.2017.09.014.  Google Scholar

[35]

M. A. Shubin, Almost periodic functions and partial differential operators, Russian Mathematical Surveys, 33 (1978), 1-52.   Google Scholar

[36]

B. Simon, Almost periodic Schrödinger operators: A review, Adv. in Appl. Math., 3 (1982), 463-490.  doi: 10.1016/S0196-8858(82)80018-3.  Google Scholar

[37]

S. Sivaji Ganesh and V. Tewary, Bloch wave homogenisation of quasiperiodic media, European Journal of Applied Mathematics, (2020), 1–21. doi: 10.1017/S0956792520000352.  Google Scholar

[38]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptot. Anal., 39 (2004), 15-44.   Google Scholar

[39]

S. Whitaker, The Method of Volume Averaging, vol. 13, Springer Science & Business Media, 2013. Google Scholar

[40]

V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh., 27 (1986), 167–180,215.  Google Scholar

show all references

References:
[1]

A. AbdulleD. Arjmand and E. Paganoni, Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems, C. R. Math. Acad. Sci. Paris, 357 (2019), 545-551.  doi: 10.1016/j.crma.2019.05.011.  Google Scholar

[2]

G. AllaireY. CapdeboscqA. PiatnitskiV. Siess and M. Vanninathan, Homogenization of periodic systems with large potentials, Arch. Ration. Mech. Anal., 174 (2004), 179-220.  doi: 10.1007/s00205-004-0332-7.  Google Scholar

[3]

M. Allais, Sur la distribution normale des valeurs à des instants régulièrement espacés d'une somme de sinusoïdes, C. R. Acad. Sci. Paris Sér. I Math., 296 (1983), 829-832.   Google Scholar

[4]

M. S. AlnæsJ. BlechtaJ. HakeA. JohanssonB. KehletA. LoggC. RichardsonJ. RingM. E. Rognes and G. N. Wells, The fenics project version 1.5, Archive of Numerical Software, 3 (2015), 9-23.   Google Scholar

[5]

S. N. ArmstrongP. Cardaliaguet and P. E. Souganidis, Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations, J. Amer. Math. Soc., 27 (2014), 479-540.  doi: 10.1090/S0894-0347-2014-00783-9.  Google Scholar

[6]

S. N. Armstrong and Z. Shen, Lipschitz estimates in almost-periodic homogenization, Comm. Pure Appl. Math., 69 (2016), 1882-1923.  doi: 10.1002/cpa.21616.  Google Scholar

[7]

M. Avellaneda and F.-H. Lin, Compactness methods in the theory of homogenization, Comm. Pure Appl. Math., 40 (1987), 803-847.  doi: 10.1002/cpa.3160400607.  Google Scholar

[8]

A. Benoit and A. Gloria, Long-time homogenization and asymptotic ballistic transport of classical waves, Ann. Sci. Éc. Norm. Supér. (4), 52 (2019), 703–759. doi: 10.24033/asens.2395.  Google Scholar

[9]

A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI, 2011. doi: 10.1090/chel/374.  Google Scholar

[10]

A. S. Besicovitch, Almost Periodic Functions, Dover Publications, Inc., New York, 1955.  Google Scholar

[11]

X. BlancC. Le Bris and P.-L. Lions, Local profiles for elliptic problems at different scales: Defects in, and interfaces between periodic structures, Comm. Partial Differential Equations, 40 (2015), 2173-2236.  doi: 10.1080/03605302.2015.1043464.  Google Scholar

[12]

H. Bohr, Almost Periodic Functions, Chelsea Publishing Company, New York, N.Y., 1947.  Google Scholar

[13]

A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. Inst. H. Poincaré Probab. Statist., 40 (2004), 153-165.  doi: 10.1016/S0246-0203(03)00065-7.  Google Scholar

[14]

L. BălilescuC. ConcaT. GhoshJ. San Martín and M. Vanninathan, The dispersion tensor and its unique minimizer in Hashin–Shtrikman micro-structures, Arch. Ration. Mech. Anal., 230 (2018), 665-700.  doi: 10.1007/s00205-018-1255-z.  Google Scholar

[15]

T. O. Carvalho and C. R. de Oliveira, Spectra and transport in almost periodic dimers, J. Statist. Phys., 107 (2002), 1015-1030.  doi: 10.1023/A:1015153523475.  Google Scholar

[16]

J. Casado-Díaz and I. Gayte, A derivation theory for generalized Besicovitch spaces and its application for partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 283-315.  doi: 10.1017/S0308210500001633.  Google Scholar

[17]

Andrej Cherkaev and Robert Kohn (eds.), Topics in the Mathematical Modelling of Composite Materials, vol. 31, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-1-4612-2032-9.  Google Scholar

[18]

C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math., 57 (1997), 1639-1659.  doi: 10.1137/S0036139995294743.  Google Scholar

[19]

D. Damanik, J. Fillman and A. Gorodetski, Multidimensional almost-periodic Schrödinger operators with Cantor spectrum, Annales Henri Poincaré, 20 (2019), 1393–1402. doi: 10.1007/s00023-019-00768-5.  Google Scholar

[20]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer-Verlag, Berlin-New York, 1974.  Google Scholar

[21]

A. Gloria, Reduction of the resonance error-Part 1: Approximation of homogenized coefficients, Math. Models Methods Appl. Sci., 21 (2011), 1601-1630.  doi: 10.1142/S0218202511005507.  Google Scholar

[22]

A. Gloria and Z. Habibi, Reduction in the resonance error in numerical homogenization Ⅱ: Correctors and extrapolation, Found. Comput. Math., 16 (2016), 217-296.  doi: 10.1007/s10208-015-9246-z.  Google Scholar

[23]

A. Gloria and F. Otto, Quantitative results on the corrector equation in stochastic homogenization, J. Eur. Math. Soc., 19 (2017), 3489-3548.  doi: 10.4171/JEMS/745.  Google Scholar

[24]

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

[25]

S. M. Kozlov, Averaging of differential operators with almost periodic rapidly oscillating coefficients, Mat. Sb. (N.S.), 107 (1978), 199–217,317.  Google Scholar

[26]

S. M. Kozlov, The averaging of random operators, Mat. Sb. (N.S.), 109 (1979), 188–202,327.  Google Scholar

[27] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge-New York, 1982.   Google Scholar
[28]

O. A. Oleinik and V. V. Zhikov, On the homogenization of elliptic operators with almost-periodic coefficients, Rend. Sem. Mat. Fis. Milano, 52 (1982), 149-166.  doi: 10.1007/BF02925004.  Google Scholar

[29]

A. Pankov, Periodic approximations of homogenization problems, Math. Methods Appl. Sci., 36 (2013), 2018-2022.  doi: 10.1002/mma.1528.  Google Scholar

[30]

L. E. Payne and H. F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rational Mech. Anal., 5 (1960), 286-292.  doi: 10.1007/BF00252910.  Google Scholar

[31]

M. Reed and B. Simon, Methods of modern mathematical physics. Ⅳ. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar

[32]

D. ShechtmanI. BlechD. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., 53 (1984), 1951-1953.  doi: 10.1103/PhysRevLett.53.1951.  Google Scholar

[33]

Z. Shen, Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems, Anal. PDE, 8 (2015), 1565-1601.  doi: 10.2140/apde.2015.8.1565.  Google Scholar

[34]

Z. Shen and J. Zhuge, Approximate correctors and convergence rates in almost-periodic homogenization, J. Math. Pures Appl., 110 (2018), 187-238.  doi: 10.1016/j.matpur.2017.09.014.  Google Scholar

[35]

M. A. Shubin, Almost periodic functions and partial differential operators, Russian Mathematical Surveys, 33 (1978), 1-52.   Google Scholar

[36]

B. Simon, Almost periodic Schrödinger operators: A review, Adv. in Appl. Math., 3 (1982), 463-490.  doi: 10.1016/S0196-8858(82)80018-3.  Google Scholar

[37]

S. Sivaji Ganesh and V. Tewary, Bloch wave homogenisation of quasiperiodic media, European Journal of Applied Mathematics, (2020), 1–21. doi: 10.1017/S0956792520000352.  Google Scholar

[38]

S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators, Asymptot. Anal., 39 (2004), 15-44.   Google Scholar

[39]

S. Whitaker, The Method of Volume Averaging, vol. 13, Springer Science & Business Media, 2013. Google Scholar

[40]

V. V. Yurinskiĭ, Averaging of symmetric diffusion in a random medium, Sibirsk. Mat. Zh., 27 (1986), 167–180,215.  Google Scholar

Figure 1.  The error $ |A^{R,*}-A^*| $ for approximations to homogenized tensor using periodic correctors in log-log scale for the functions $ A_1 $ and $ A_2 $ with respect to $ R $
Figure 2.  The error $ |A^{R,*}-A^*| $ for approximations to homogenized tensor using periodic correctors in log-log scale for the function $ A_3 $ with respect to $ R $
Figure 3.  The error $ |A^{R,D,*}-A^*| $ for Dirichlet approximations in log-log scale for the functions $ A_1 $ and $ A_2 $ with respect to $ R $
Figure 4.  The error $ |A^{R,D,*}-A^*| $ for Dirichlet approximations in log-log scale for the function $ A_3 $ with respect to $ R $
Figure 5.  The averaged $ L^2 $ norm of the difference of the gradients $ E(R) = \left(\mathit{{\rlap{-} \smallint }}_{Y_{R}}|\nabla w^{R,D,e_1}(y)-\nabla w^{R,e_1}(y)|^2\,dy\right)^{1/2} $ in log-log scale for the correctors corresponding to the periodic matrices $ A_1 $ and $ A_2 $ plotted as a function of $ R $
Figure 6.  The averaged $ L^2 $ norm of the difference of the gradients $ E(R) = \left(\mathit{{\rlap{-} \smallint }}_{Y_{R}}|\nabla w^{R,D,e_1}(y)-\nabla w^{R,e_1}(y)|^2\,dy\right)^{1/2} $ in log-log scale for the correctors corresponding to the quasiperiodic matrix $ A_3 $ plotted as a function of $ R $
Figure 7.  The absolute error $ |A^{R,D,*}-A^{R,*}| $ in log-log scale for the periodic matrices $ A_1 $ and $ A_2 $ plotted as a function of $ R $
Figure 8.  The absolute error $ |A^{R,D,*}-A^{R,*}| $ in log-log scale for the quasiperiodic matrix $ A_3 $ plotted as a function of $ R $
[1]

Andriy Bondarenko, Guy Bouchitté, Luísa Mascarenhas, Rajesh Mahadevan. Rate of convergence for correctors in almost periodic homogenization. Discrete & Continuous Dynamical Systems, 2005, 13 (2) : 503-514. doi: 10.3934/dcds.2005.13.503

[2]

Grégoire Allaire, Tuhin Ghosh, Muthusamy Vanninathan. Homogenization of stokes system using bloch waves. Networks & Heterogeneous Media, 2017, 12 (4) : 525-550. doi: 10.3934/nhm.2017022

[3]

Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711

[4]

Vivek Tewary. Combined effects of homogenization and singular perturbations: A bloch wave approach. Networks & Heterogeneous Media, 2021, 16 (3) : 427-458. doi: 10.3934/nhm.2021012

[5]

Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181

[6]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Computational networks and systems-homogenization of self-adjoint differential operators in variational form on periodic networks and micro-architectured systems. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 139-169. doi: 10.3934/naco.2017010

[7]

François Hamel, Emmanuel Russ, Nikolai Nadirashvili. Comparisons of eigenvalues of second order elliptic operators. Conference Publications, 2007, 2007 (Special) : 477-486. doi: 10.3934/proc.2007.2007.477

[8]

Miklós Horváth, Márton Kiss. A bound for ratios of eigenvalues of Schrodinger operators on the real line. Conference Publications, 2005, 2005 (Special) : 403-409. doi: 10.3934/proc.2005.2005.403

[9]

Carlos Conca, Luis Friz, Jaime H. Ortega. Direct integral decomposition for periodic function spaces and application to Bloch waves. Networks & Heterogeneous Media, 2008, 3 (3) : 555-566. doi: 10.3934/nhm.2008.3.555

[10]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703

[11]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026

[12]

Laura Sigalotti. Homogenization of pinning conditions on periodic networks. Networks & Heterogeneous Media, 2012, 7 (3) : 543-582. doi: 10.3934/nhm.2012.7.543

[13]

Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure & Applied Analysis, 2011, 10 (3) : 983-994. doi: 10.3934/cpaa.2011.10.983

[14]

Marko Kostić. Almost periodic type functions and densities. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021008

[15]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677

[16]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753

[17]

Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent. Networks & Heterogeneous Media, 2020, 15 (2) : 281-305. doi: 10.3934/nhm.2020013

[18]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857

[19]

Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012

[20]

Wenxian Shen, Xiaoxia Xie. On principal spectrum points/principal eigenvalues of nonlocal dispersal operators and applications. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1665-1696. doi: 10.3934/dcds.2015.35.1665

2020 Impact Factor: 1.327

Article outline

Figures and Tables

[Back to Top]