April  2020, 19(4): 1891-1914. doi: 10.3934/cpaa.2020083

Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary

1. 

Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid

2. 

Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, C/Nicolás Cabrera 13-15, Cantoblanco, 28049 Madrid, Spain

3. 

Grupo de Investigación Dinámica No Lineal, ICAI, Universidad Pontificia de Comillas, Madrid, Spain

* Corresponding author

Dedicated to Tomás Caraballo on the occasion of his 60-th birthday

Received  July 2019 Revised  October 2019 Published  January 2020

Fund Project: This first author (JMA) has been partially supported by grants MTM2016-75465-P, ICMAT Severo Ochoa project SEV-2015-0554, MICINN, Spain and Grupo de Investigación CADEDIF, UCM. The second author (MVP) has been partially supported by grant MTM2016-75465-P MICINN, Spain and Grupo de Investigación CADEDIF, UCM.

In this work we consider higher dimensional thin domains with the property that both boundaries, bottom and top, present oscillations of weak type. We consider the Laplace operator with Neumann boundary conditions and analyze the behavior of the solutions as the thin domain shrinks to a fixed domain $ \omega\subset \mathbb{R}^n $. We obtain the convergence of the resolvent of the elliptic operators in the sense of compact convergence of operators, which in particular implies the convergence of the spectra. This convergence of the resolvent operators will allow us to conclude the global dynamics, in terms of the global attractors of a reaction diffusion equation in the thin domains. In particular, we show the upper semicontinuity of the attractors and stationary states. An important case treated is the case of a quasiperiodic situation, where the bottom and top oscillations are periodic but with period rationally independent.

Citation: José M. Arrieta, Manuel Villanueva-Pesqueira. Elliptic and parabolic problems in thin domains with doubly weak oscillatory boundary. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1891-1914. doi: 10.3934/cpaa.2020083
References:
[1]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.

[2]

N. Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films, J. Anal. Math, 83 (2001), 151-182.  doi: 10.1007/BF02790260.

[3]

J. M. Arrieta, Spectral Properties of Schrödinger Operators under Perturbations of the Domain, Ph.D. Thesis, Georgia Institute of Technology, 1991.

[4]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, Journal of Diff Equations, 199 (2004), 143-178.  doi: 10.1007/BF02790260.

[5]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, Journal of Diff. Equations, 231 (2006), 551-597.  doi: 10.1016/j.jde.2006.06.002.

[6]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, Journal of Diff. Equations, 247 (2009), 174-202.  doi: 10.1016/j.jde.2009.03.014.

[7]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, Journal of Diff. Equations, 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.

[8]

J. M. ArrietaM. C. CarvalhoPereira and R. P. Da Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 5111-5132.  doi: 10.1016/j.na.2011.05.006.

[9]

J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary, Journal de Mathematiques Pures et Apliquees, 96 (2011), 29–57. doi: 10.1016/j.matpur.2011.02.003.

[10]

J. M. Arrieta and M. C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, Journal of Mathematical Analysis and Applications, 444 (2013), 86–104. doi: 10.1016/j.jmaa.2013.02.061.

[11]

J. M. Arrieta and E. Santamaría, Distance of attractors of reaction-diffusion equations in thin domains, Journal of Differential Equations, 263 (2017), 5459-5506.  doi: 10.1016/j.jde.2017.06.023.

[12]

J. M. Arrieta and M. Villanueva-Pesqueira, Thin domains with doubly oscillatory boundary, Mathematical Methods in Applied Science, 37 (2014), 158-166.  doi: 10.1002/mma.2875.

[13]

M. Arrieta and M. Villanueva-Pesqueira, Thin domains with non-smooth periodic oscillatory boundaries, Journal of Mathematical Analysis and Applications, 446 (2017), 30-164.  doi: 10.1016/j.jmaa.2016.08.039.

[14]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[15]

A. G. BelyaevA. L. Pyatnitskii and G. A. Chechkin, Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary, Siberian Math. J., 39 (1998), 621-644.  doi: 10.1007/BF02673049.

[16]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publ. Company, 1978.

[17]

A. S. Besicovitch, Almost Periodic Functions, Dover, 1954.

[18]

D. BlanchardA. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a thin plate, Part Ⅱ, J. Math. Pures Appl, 88 (2007), 149-190.  doi: 10.1016/j.matpur.2007.04.004.

[19]

H. Bohr, Almost Periodic Functions, New York: Chelsea, 1947.

[20]

A. N. Carvalho, J. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical-Systems, Applied Mathematical Sciences, Vol. 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[21]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829.  doi: 10.1080/01630560600882723.

[22]

J. Casado-DíazM. Luna-Laynez and F. J. Suárez-Grau, Asymptotic behavior of the Navier-Stokes system in a thin domain with Navier condition on a slightly rough boundary, SIAM J. Math. Anal., 45 (2013), 1641-1674.  doi: 10.1137/120873479.

[23]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer Verlag, 1999. doi: 10.1007/978-1-4612-2158-6.

[24]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, 1999.

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathe. Surveys and Monographs 25 MAS, 1998.

[26]

J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures and Appl., 71 (1992), 33-95. 

[27]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Math., 840, Springer-Verlag, 1981.

[28]

O. V. Kapustyan and J. Valero, On the connecgtedness and asymptotic behaviour of solutions of reaction-diffusion systems, Journal of Math. Anal. and Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.

[29]

O. V. KapustyanP. O Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, 323 (2006), 614-633.  doi: 10.3934/dcds.2014.34.4155.

[30]

C. Komo, Influence of surface roughness to solutions of the Boussinesq equations with Robin boundary condition, Rev Mat Complut., 28 (2015), 123-155.  doi: 10.1007/s13163-014-0152-8.

[31]

S. M. Kozlov, Averaging differential operators with almost-periodic rapidly oscillating coefficients, Math. USSR-Sb., 35 (1979), 481-498. 

[32]

T. A. Mel'nyk and A. V. Popov, Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness, Nonlinear Oscil., 13 (2010), 57-84.  doi: 10.1007/s11072-010-0101-5.

[33]

N. Meunier and J. Van Schaftingen, Periodic reiterated homogenization for elliptic functions, J. Math. Pures Appl., 84 (2005), 1716-1743.  doi: 10.1016/j.matpur.2005.08.003.

[34]

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

[35]

M. C. Pereira, Parabolic problems in highly oscillating thin domains, Ann. Mat. Pura Appl., 194 (2015), 1203-1244.  doi: 10.1007/s10231-014-0421-7.

[36]

G. Raugel, Dynamics of partial differential equations on thin domains, in Dynamical Systems Montecatini Terme, (1994), 208–315, Lecture Notes in Math., 1609, Springer, Berlin, 1995., doi: 10.1007/BFb0095241.

[37]

E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer Verlag, 1980.

[38]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 2002. doi: 10.1007/978-1-4757-5037-9.

[39]

F. Stummel, Diskrete Konvergenz linearer Operatoren Ⅰ, Math. Ann., 190 (1970), 45-92.  doi: 10.1007/BF01349967.

[40]

F. Stummel, Diskrete Konvergenz linearer Operatoren Ⅱ, Math. Z., 120 (1971), 231-264.  doi: 10.1007/BF01117498.

[41]

F. Stummel, Diskrete Konvergenz linearer Operatoren Ⅲ, Linear Operators and Approximation, (Proc. Conf., Oberwolfach, 1971), 196–216, Internat. Ser. Numer. Math., Vol. 20, Birkhuser, (1972).

[42]

R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8.

[43]

G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Analysis, Theory, Methods and Applications, 2 (1978), 647-687.  doi: 10.1016/0362-546X(78)90013-5.

[44]

G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner-Texte zur Mathematik. B. G. Teubner Verlag, Leipzig, 1976.

show all references

References:
[1]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.

[2]

N. Ansini and A. Braides, Homogenization of oscillating boundaries and applications to thin films, J. Anal. Math, 83 (2001), 151-182.  doi: 10.1007/BF02790260.

[3]

J. M. Arrieta, Spectral Properties of Schrödinger Operators under Perturbations of the Domain, Ph.D. Thesis, Georgia Institute of Technology, 1991.

[4]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, Journal of Diff Equations, 199 (2004), 143-178.  doi: 10.1007/BF02790260.

[5]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅰ. Continuity of the set of equilibria, Journal of Diff. Equations, 231 (2006), 551-597.  doi: 10.1016/j.jde.2006.06.002.

[6]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅱ. The limiting problem, Journal of Diff. Equations, 247 (2009), 174-202.  doi: 10.1016/j.jde.2009.03.014.

[7]

J. M. ArrietaA. N. Carvalho and G. Lozada-Cruz, Dynamics in dumbbell domains Ⅲ. Continuity of attractors, Journal of Diff. Equations, 247 (2009), 225-259.  doi: 10.1016/j.jde.2008.12.014.

[8]

J. M. ArrietaM. C. CarvalhoPereira and R. P. Da Silva, Semilinear parabolic problems in thin domains with a highly oscillatory boundary, Nonlinear Analysis: Theory, Methods and Applications, 74 (2011), 5111-5132.  doi: 10.1016/j.na.2011.05.006.

[9]

J. M. Arrieta and M. C. Pereira, Homogenization in a thin domain with an oscillatory boundary, Journal de Mathematiques Pures et Apliquees, 96 (2011), 29–57. doi: 10.1016/j.matpur.2011.02.003.

[10]

J. M. Arrieta and M. C. Pereira, The Neumann problem in thin domains with very highly oscillatory boundaries, Journal of Mathematical Analysis and Applications, 444 (2013), 86–104. doi: 10.1016/j.jmaa.2013.02.061.

[11]

J. M. Arrieta and E. Santamaría, Distance of attractors of reaction-diffusion equations in thin domains, Journal of Differential Equations, 263 (2017), 5459-5506.  doi: 10.1016/j.jde.2017.06.023.

[12]

J. M. Arrieta and M. Villanueva-Pesqueira, Thin domains with doubly oscillatory boundary, Mathematical Methods in Applied Science, 37 (2014), 158-166.  doi: 10.1002/mma.2875.

[13]

M. Arrieta and M. Villanueva-Pesqueira, Thin domains with non-smooth periodic oscillatory boundaries, Journal of Mathematical Analysis and Applications, 446 (2017), 30-164.  doi: 10.1016/j.jmaa.2016.08.039.

[14]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, Studies in Mathematics and its Applications, 25, North-Holland Publishing Co., Amsterdam, 1992.

[15]

A. G. BelyaevA. L. Pyatnitskii and G. A. Chechkin, Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary, Siberian Math. J., 39 (1998), 621-644.  doi: 10.1007/BF02673049.

[16]

A. Bensoussan, J. L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland Publ. Company, 1978.

[17]

A. S. Besicovitch, Almost Periodic Functions, Dover, 1954.

[18]

D. BlanchardA. Gaudiello and G. Griso, Junction of a periodic family of elastic rods with a thin plate, Part Ⅱ, J. Math. Pures Appl, 88 (2007), 149-190.  doi: 10.1016/j.matpur.2007.04.004.

[19]

H. Bohr, Almost Periodic Functions, New York: Chelsea, 1947.

[20]

A. N. Carvalho, J. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical-Systems, Applied Mathematical Sciences, Vol. 182, Springer, 2012. doi: 10.1007/978-1-4614-4581-4.

[21]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numerical Functional Analysis and Optimization, 27 (2006), 785-829.  doi: 10.1080/01630560600882723.

[22]

J. Casado-DíazM. Luna-Laynez and F. J. Suárez-Grau, Asymptotic behavior of the Navier-Stokes system in a thin domain with Navier condition on a slightly rough boundary, SIAM J. Math. Anal., 45 (2013), 1641-1674.  doi: 10.1137/120873479.

[23]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer Verlag, 1999. doi: 10.1007/978-1-4612-2158-6.

[24]

D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford University Press, 1999.

[25]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathe. Surveys and Monographs 25 MAS, 1998.

[26]

J. K. Hale and G. Raugel, Reaction-diffusion equation on thin domains, J. Math. Pures and Appl., 71 (1992), 33-95. 

[27]

D. B. Henry, Geometric Theory of Semilinear Parabolic Equations, Lectures Notes in Math., 840, Springer-Verlag, 1981.

[28]

O. V. Kapustyan and J. Valero, On the connecgtedness and asymptotic behaviour of solutions of reaction-diffusion systems, Journal of Math. Anal. and Appl., 323 (2006), 614-633.  doi: 10.1016/j.jmaa.2005.10.042.

[29]

O. V. KapustyanP. O Kasyanov and J. Valero, Structure and regularity of the global attractor of a reaction-diffusion equations with Caratheodory's nonlinearity, Nonlinear Analysis, 323 (2006), 614-633.  doi: 10.3934/dcds.2014.34.4155.

[30]

C. Komo, Influence of surface roughness to solutions of the Boussinesq equations with Robin boundary condition, Rev Mat Complut., 28 (2015), 123-155.  doi: 10.1007/s13163-014-0152-8.

[31]

S. M. Kozlov, Averaging differential operators with almost-periodic rapidly oscillating coefficients, Math. USSR-Sb., 35 (1979), 481-498. 

[32]

T. A. Mel'nyk and A. V. Popov, Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness, Nonlinear Oscil., 13 (2010), 57-84.  doi: 10.1007/s11072-010-0101-5.

[33]

N. Meunier and J. Van Schaftingen, Periodic reiterated homogenization for elliptic functions, J. Math. Pures Appl., 84 (2005), 1716-1743.  doi: 10.1016/j.matpur.2005.08.003.

[34]

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

[35]

M. C. Pereira, Parabolic problems in highly oscillating thin domains, Ann. Mat. Pura Appl., 194 (2015), 1203-1244.  doi: 10.1007/s10231-014-0421-7.

[36]

G. Raugel, Dynamics of partial differential equations on thin domains, in Dynamical Systems Montecatini Terme, (1994), 208–315, Lecture Notes in Math., 1609, Springer, Berlin, 1995., doi: 10.1007/BFb0095241.

[37]

E. Sánchez-Palencia, Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer Verlag, 1980.

[38]

G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences, 143, Springer, 2002. doi: 10.1007/978-1-4757-5037-9.

[39]

F. Stummel, Diskrete Konvergenz linearer Operatoren Ⅰ, Math. Ann., 190 (1970), 45-92.  doi: 10.1007/BF01349967.

[40]

F. Stummel, Diskrete Konvergenz linearer Operatoren Ⅱ, Math. Z., 120 (1971), 231-264.  doi: 10.1007/BF01117498.

[41]

F. Stummel, Diskrete Konvergenz linearer Operatoren Ⅲ, Linear Operators and Approximation, (Proc. Conf., Oberwolfach, 1971), 196–216, Internat. Ser. Numer. Math., Vol. 20, Birkhuser, (1972).

[42]

R. Teman, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68, Springer-Verlag, 1988. doi: 10.1007/978-1-4684-0313-8.

[43]

G. Vainikko, Approximative methods for nonlinear equations (two approaches to the convergence problem), Nonlinear Analysis, Theory, Methods and Applications, 2 (1978), 647-687.  doi: 10.1016/0362-546X(78)90013-5.

[44]

G. Vainikko, Funktionalanalysis der Diskretisierungsmethoden, Teubner-Texte zur Mathematik. B. G. Teubner Verlag, Leipzig, 1976.

Figure 1.  Thin domain $ R^ \epsilon $ with doubly weak oscillatory boundary
Figure 2.  Thin domain $ R_a^\epsilon $ obtained from $ R^ \epsilon $ of Figure 1
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