Opuscula Math. 44, no. 2 (2024), 249-265

Opuscula Mathematica

Anisotropic p-Laplace Equations on long cylindrical domain

Purbita Jana

Abstract. The main aim of this article is to study the Poisson type problem for anisotropic \(p\)-Laplace type equation on long cylindrical domains. The rate of convergence is shown to be exponential, thereby improving earlier known results for similar type of operators. The Poincaré inequality for a pseudo \(p\)-Laplace operator on an infinite strip-like domain is also studied and the best constant, like in many other situations in literature for other operators, is shown to be the same with the best Poincaré constant of an analogous problem set on a lower dimension.

Keywords: pseudo \(p\)-Laplace equation, cylindrical domains, asymptotic analysis.

Mathematics Subject Classification: 35P15, 35P30, 35B38.

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  1. K. Bal, K. Mohanta, P. Roy, F. Sk, Hardy and Poincaré inequalities in fractional Orlicz-Sobolev spaces, Nonlinear Anal. 216 (2022), 112697. https://doi.org/10.1016/j.na.2021.112697
  2. C. Bandle, M. Chipot, Large solutions in cylindrical domains, Adv. Math. Sci. Appl. 23 (2013), no. 2, 461-476.
  3. T. Boudjeriou, Asymptotic behavior of parabolic nonlocal equations in cylinders becoming unbounded, Bull. Malays. Math. Sci. Soc. 46 (2023), Article no. 19. https://doi.org/10.1007/s40840-022-01426-6
  4. P. Bousquet, L. Brasco, Lipschitz regularity for orthotropic functionals with nonstandard growth conditions, Rev. Mat. Iberoam. 36 (2020), no. 7, 1989-2032. https://doi.org/10.4171/rmi/1189
  5. A. Brada, Comportement asymptotique de solutions d’équations elliptiques semi-linéares dans un cylindre, Asymptot. Anal. 10 (1995), no. 4, 335-366.
  6. L. Brasco, E. Cinti, On fractional Hardy inequalities in convex sets, Discrete Contin. Dyn. Syst. 38 (2018), no. 8, 4019-4040. https://doi.org/10.3934/dcds.2018175
  7. B. Brighi, S. Guesmia, Asymptotic behavior of solution of hyperbolic problems on a~cylindrical domain, Discrete Contin. Dyn. Syst. 2007 Suppl. (2007), 160-169.
  8. A. Ceccaldi, S. Mardare, On correctors to elliptic problems in long cylinders, J. Elliptic Parabol. Equ. 5 (2019), no. 2, 473-491. https://doi.org/10.1007/s41808-019-00047-8
  9. M. Chipot, \(l\) goes to plus infinity: an update, J. Korean Soc. Ind. Appl. Math. 18 (2014), no. 2, 107-127. https://doi.org/10.12941/jksiam.2014.18.107
  10. M. Chipot, On the asymptotic behaviour of some problems of the calculus of variations, J. Elliptic Parabol. Equ. 1 (2015), no. 2, 307-323. https://doi.org/10.1007/BF03377383
  11. M. Chipot, S. Mardare, The Neumann problem in cylinders becoming unbounded in one direction, J. Math. Pures Appl. (9), 104 (2015), no. 5, 921-941. https://doi.org/10.1016/j.matpur.2015.05.008
  12. M. Chipot, A. Rougirel, On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded, Commun. Contemp. Math. 4 (2002), no. 1, 15-44. https://doi.org/10.1142/S0219199702000555
  13. M. Chipot, A. Rougirel, On the asymptotic behaviour of the eigenmodes for elliptic problems in domains becoming unbounded, Trans. Amer. Math. Soc. 360 (2008), no. 7, 3579-3602. https://doi.org/10.1090/S0002-9947-08-04361-4
  14. M. Chipot, Y. Xie, Asymptotic behavior of nonlinear parabolic problems with periodic data, [in:] Progress in Nonlinear Differential Equations and Their Applications, vol. 63, Birkhäuser Basel, 2005, 147-156. https://doi.org/10.1007/3-7643-7384-9_17
  15. M. Chipot, Y. Xie, Some issues on the \(p\)-Laplace equation in cylindrical domains, Tr. Mat. Inst. Steklova 261 (2008), 287-294. https://doi.org/10.1134/S0081543808020235
  16. M. Chipot, P. Roy, I. Shafrir, Asymptotics of eigenstates of elliptic problems with mixed boundary data on domains tending to infinity, Asymptot. Anal. 85 (2013), no. 3-4, 199-227. https://doi.org/10.3233/ASY-131182
  17. M. Chipot, A. Mojsic, P. Roy, On some variational problems set on domains tending to infinity, Discrete Contin. Dyn. Syst. 36 (2016), no. 7, 3603-3621. https://doi.org/10.3934/dcds.2016.36.3603
  18. M. Chipot, J. Dávila, M. del Pino, On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains, J. Fixed Point Theory Appl. 19 (2017), no. 1, 205-213. https://doi.org/10.1007/s11784-016-0349-1
  19. M. Chipot, W. Hackbusch, S. Sauter, A. Veit, Numerical approximation of Poisson problems in long domains, Vietnam J. Math. 50 (2022), no. 2, 375-393. https://doi.org/10.1007/s10013-021-00512-9
  20. I. Chowdhury, P. Roy, On the asymptotic analysis of problems involving fractional Laplacian in cylindrical domains tending to infinity, Commun. Contemp. Math. 19 (2017), no. 5, Article no. 1650035. https://doi.org/10.1142/S0219199716500358
  21. I. Chowdhury, P. Roy, On fractional Poincaré inequality for unbounded domains with finite ball conditions: Counter example, Nonlinear Anal. 228 (2023), Article no. 113189. https://doi.org/10.1016/j.na.2022.113189
  22. I. Chowdhury, G. Csató, P. Roy, F. Sk, Study of fractional Poincaré inequalities on unbounded domains, Discrete Contin. Dyn. Syst. 41 (2021), no. 6, 2993-3020. https://doi.org/10.3934/dcds.2020394
  23. J.I. Diaz, O.A. Oleinik, Nonlinear elliptic boundary value problems in unbounded domains and the asymptotic behaviour of its solutions, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 7, 787-792.
  24. L. Djilali, A. Rougirel, Galerkin method for time fractional diffusion equations, J. Elliptic Parabol. Equ. 4 (2018), no. 2, 349-368. https://doi.org/10.1007/s41808-018-0022-5
  25. Y. Dolak, C. Schmeiser, The Keller-Segel model with logistic sensitivity function and small diffusivity, SIAM J. Appl. Math. 66 (2005), no. 1, 286-308. https://doi.org/10.1137/040612841
  26. L. Esposito, P. Roy, F. Sk, On the asymptotic behavior of the eigenvalues of nonlinear elliptic problems in domains becoming unbounded, Asymptot. Anal. 123 (2021), no. 1-2, 79-94. https://doi.org/10.3233/asy-201626
  27. L.C. Evans, Partial Differential Equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. https://doi.org/10.1090/gsm/019
  28. S. Guesmia, Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size, Nonlinear Anal. 70 (2009), no. 9, 3320-3331. https://doi.org/10.1016/j.na.2008.04.036
  29. J. Haškovec, C. Schmeiser, A note on the anisotropic generalizations of the Sobolev and Morrey embedding theorems, Monatsh. Math. 158 (2009), no. 1, 71-79. https://doi.org/10.1007/s00605-008-0059-x
  30. J.-L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications: Volume II, Grundlehren der mathematischen Wissenschaften, vol. 182, Springer-Verlag, Softcover reprint of the original 1st ed. 1972.
  31. K. Mohanta, F. Sk, On the best constant in fractional \(p\)-Poincaré inequalities on cylindrical domains, Differential Integral Equations 34 (2021), no. 11-12, 691-712. https://doi.org/10.57262/die034-1112-691
  32. R. Rawat, H. Roy, P. Roy, Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction, (2023) arXiv:2307.09622. https://doi.org/10.48550/arXiv.2307.09622
  33. J. Weickert, Anisotropic Diffusion in Image Processing, Consortium for Mathematics in Industry, B.G. Teubner, Stuttgart, 1998.
  34. K. Yeressian, Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89 (2014), no. 1-2, 21-35. https://doi.org/10.3233/asy-141224
  • Communicated by J.I. Díaz.
  • Received: 2023-05-07.
  • Revised: 2023-08-18.
  • Accepted: 2023-08-23.
  • Published online: 2024-01-15.
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Cite this article as:
Purbita Jana, Anisotropic p-Laplace Equations on long cylindrical domain, Opuscula Math. 44, no. 2 (2024), 249-265, https://doi.org/10.7494/OpMath.2024.44.2.249

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