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doi: 10.3934/dcdsb.2019079

Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary

1. 

Dpto. de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

2. 

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

3. 

Depto. de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil

4. 

Depto. Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil

Dedicated to Peter Kloeden on his 70th aniversary

Received  April 2018 Revised  October 2018 Published  April 2019

In this paper we analyze the asymptotic behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating region with reaction terms concentrated in a neighborhood of the oscillatory boundary $\theta_\varepsilon \subset\Omega_{\varepsilon }\subset \mathbb{R}^2$ when a small parameter $\varepsilon >0$ goes to zero. Our main result is concerned with the upper and lower semicontinuity of the set of solutions in $H^1$. We show that the solutions of our perturbed equation can be approximated with one defined in a fixed limit domain, which also captures the effects of reaction terms that take place in the original problem as a flux condition on the boundary of the limit domain.

Citation: José M. Arrieta, Ariadne Nogueira, Marcone C. Pereira. Nonlinear elliptic equations with concentrating reaction terms at an oscillatory boundary. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019079
References:
[1]

G. S. Aragão and S. M. Bruschi, Limit of nonlinear elliptic equations with concentrated terms and varying domains: the non uniformly Lipschitz case, Electron. J. Differential Equations, 217 (2015), 1-14.

[2]

G. S. Aragão and S. M. Bruschi, Concentrated terms and varying domains in elliptic equations: Lipschitz case, Math. Methods Appl. Sci., 39 (2016), 3450-3460. doi: 10.1002/mma.3791.

[3]

G. S. Aragão and S. M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. of Diff. Equations, 253 (2012), 2573-2592. doi: 10.1016/j.jde.2012.07.008.

[4]

G. S. Aragão and S. M. Oliva, Asymptotic behavior of a reaction-diffusion problem with delay and reaction term concentrated in the boundary, São Paulo J. Math. Sci., 5 (2011), 347-376. doi: 10.11606/issn.2316-9028.v5i2p347-376.

[5]

G. S. AragãoA. L. Pereira and M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Mathematical Methods in the Applied Sciences, 35 (2012), 1110-1116. doi: 10.1002/mma.2525.

[6]

G. S. AragãoA. L. Pereira and M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differential Equations, 26 (2014), 871-888. doi: 10.1007/s10884-014-9412-z.

[7]

J. M. Arrieta and S. M. Bruschi, Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation, Math. Models and Meth. in Appl. Sciences, 17 (2007), 1555-1585. doi: 10.1142/S0218202507002388.

[8]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, Journal of Differential Equations, 199 (2004), 143-178. doi: 10.1016/j.jde.2003.09.004.

[9]

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

[10]

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

[11]

J. M. ArrietaA. Jiménez-Casas and A. Rodríguez-Bernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Revista Matemática Iberoamericana, 24 (2008), 183-211. doi: 10.4171/RMI/533.

[12]

J. M. ArrietaA. Nogueira and M. C. Pereira, Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries, Comput. Math. Appl., 77 (2019), 536-554. doi: 10.1016/j.camwa.2018.09.056.

[13]

R. Brizzi and J. P. Chalot, Boundary homogenization and Neumann boundary problem, Ric. Mat. XLVI, 46 (1997), 341-387.

[14]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829. doi: 10.1080/01630560600882723.

[15]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Vol. 13. Oxford Univ. Press on Demand, 1998.

[16]

S. N. Chandler-WildeD. P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: Quantitative estimates and counterexamples, Mathematika, 61 (2015), 414-443. doi: 10.1112/S0025579314000278.

[17] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications vol.17. Oxford University Press, New York, 1999.
[18]

L. C. Evans, Partial Differential Equations, Graduate studies in mathematics. American Mathematical Society, 1998.

[19]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011. doi: 10.1137/1.9781611972030.ch1.

[20]

A. Jiménez-Casas and A. Rodríguez-Bernal, Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 2377-2383. doi: 10.1016/j.na.2009.05.036.

[21]

A. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, Journal of Mathematical Analysis and Applications, 379 (2011), 567-588. doi: 10.1016/j.jmaa.2011.01.051.

[22]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, 1968.

[23]

S. A. Meier and M. Böhm, A note on the construction of function spaces for distributed-microstructure models with spatially varying cell geometry, Int. J. Numer. Anal. Model, 5 (2008), 109-125.

[24]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[25]

M. C. Pereira, Asymptotic analysis of a semilinear elliptic equation in highly oscillating thin domains, Zeitschrift fur Angewandte Mathematik und Physik, 67 (2016), 1-14. doi: 10.1007/s00033-016-0727-y.

[26]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

show all references

References:
[1]

G. S. Aragão and S. M. Bruschi, Limit of nonlinear elliptic equations with concentrated terms and varying domains: the non uniformly Lipschitz case, Electron. J. Differential Equations, 217 (2015), 1-14.

[2]

G. S. Aragão and S. M. Bruschi, Concentrated terms and varying domains in elliptic equations: Lipschitz case, Math. Methods Appl. Sci., 39 (2016), 3450-3460. doi: 10.1002/mma.3791.

[3]

G. S. Aragão and S. M. Oliva, Delay nonlinear boundary conditions as limit of reactions concentrating in the boundary, J. of Diff. Equations, 253 (2012), 2573-2592. doi: 10.1016/j.jde.2012.07.008.

[4]

G. S. Aragão and S. M. Oliva, Asymptotic behavior of a reaction-diffusion problem with delay and reaction term concentrated in the boundary, São Paulo J. Math. Sci., 5 (2011), 347-376. doi: 10.11606/issn.2316-9028.v5i2p347-376.

[5]

G. S. AragãoA. L. Pereira and M. C. Pereira, A nonlinear elliptic problem with terms concentrating in the boundary, Mathematical Methods in the Applied Sciences, 35 (2012), 1110-1116. doi: 10.1002/mma.2525.

[6]

G. S. AragãoA. L. Pereira and M. C. Pereira, Attractors for a nonlinear parabolic problem with terms concentrating on the boundary, J. Dynam. Differential Equations, 26 (2014), 871-888. doi: 10.1007/s10884-014-9412-z.

[7]

J. M. Arrieta and S. M. Bruschi, Rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a Lipschitz deformation, Math. Models and Meth. in Appl. Sciences, 17 (2007), 1555-1585. doi: 10.1142/S0218202507002388.

[8]

J. M. Arrieta and A. N. Carvalho, Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain, Journal of Differential Equations, 199 (2004), 143-178. doi: 10.1016/j.jde.2003.09.004.

[9]

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

[10]

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

[11]

J. M. ArrietaA. Jiménez-Casas and A. Rodríguez-Bernal, Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary, Revista Matemática Iberoamericana, 24 (2008), 183-211. doi: 10.4171/RMI/533.

[12]

J. M. ArrietaA. Nogueira and M. C. Pereira, Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries, Comput. Math. Appl., 77 (2019), 536-554. doi: 10.1016/j.camwa.2018.09.056.

[13]

R. Brizzi and J. P. Chalot, Boundary homogenization and Neumann boundary problem, Ric. Mat. XLVI, 46 (1997), 341-387.

[14]

A. N. Carvalho and S. Piskarev, A general approximation scheme for attractors of abstract parabolic problems, Numer. Funct. Anal. Optim., 27 (2006), 785-829. doi: 10.1080/01630560600882723.

[15]

T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Vol. 13. Oxford Univ. Press on Demand, 1998.

[16]

S. N. Chandler-WildeD. P. Hewett and A. Moiola, Interpolation of Hilbert and Sobolev spaces: Quantitative estimates and counterexamples, Mathematika, 61 (2015), 414-443. doi: 10.1112/S0025579314000278.

[17] D. Cioranescu and P. Donato, An Introduction to Homogenization, Oxford Lecture Series in Mathematics and its Applications vol.17. Oxford University Press, New York, 1999.
[18]

L. C. Evans, Partial Differential Equations, Graduate studies in mathematics. American Mathematical Society, 1998.

[19]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011. doi: 10.1137/1.9781611972030.ch1.

[20]

A. Jiménez-Casas and A. Rodríguez-Bernal, Asymptotic behaviour of a parabolic problem with terms concentrated in the boundary, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 2377-2383. doi: 10.1016/j.na.2009.05.036.

[21]

A. Jiménez-Casas and A. Rodríguez-Bernal, Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary, Journal of Mathematical Analysis and Applications, 379 (2011), 567-588. doi: 10.1016/j.jmaa.2011.01.051.

[22]

O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, 1968.

[23]

S. A. Meier and M. Böhm, A note on the construction of function spaces for distributed-microstructure models with spatially varying cell geometry, Int. J. Numer. Anal. Model, 5 (2008), 109-125.

[24]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[25]

M. C. Pereira, Asymptotic analysis of a semilinear elliptic equation in highly oscillating thin domains, Zeitschrift fur Angewandte Mathematik und Physik, 67 (2016), 1-14. doi: 10.1007/s00033-016-0727-y.

[26]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

Figure 1.  The oscillatory domain $ \Omega_ \varepsilon $ and strip $ \theta_ \varepsilon $ where reactions take place
Figure 2.  Fixed $ x_1\in(0,1) $ and $ \varepsilon>0 $, we get a fiber of the oscillatory domain for $ \varepsilon<\varepsilon_0 $
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