Abstract
In the previous chapter, we showed that nonnegative solutions of elliptic equations in “asymptotically symmetric” domains are “asymptotically symmetric” as well (see Theorem 3.3 ). However, in order to prove Theorem 3.3 , we imposed a restriction on the dimension (less or equal 3) of the underlying domain, which was crucial for our proof. The goal of this chapter is to extend Theorem 3.3 for higher dimensions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. V. Babin and M. I. Vishik. Attractors of evolution equations. Transl. from the Russian by A. V. Babin. English. Amsterdam etc.: North-Holland, 1992, pp. x + 532.
L. Ambrosio and X. Cabré. “Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi.” English. In: J. Am. Math. Soc. 13.4 (2000), pp. 725–739.
A. V. Babin. “Attractor of the generalized semigroup generated by an elliptic equation in a cylindrical domain”. In: Russian Acad. Sci. Izv. Math. 44 (1995), pp. 207–223.
A. V. Babin and G. R. Sell. “Attractors of non-autonomous parabolic equations and their symmetry properties.” English. In: J. Differ. Equations 160.1 (2000), 1–50, art. no. jdeq.1999.3654.
J. Ball. “Continuity properties and global attractors of generalized semi-flows and the Navier-Stokes equations.” English. In: J. Nonlinear Sci. 7.5 (1997), pp. 475–502.
H. Berestycki, F. Hamel, and R. Monneau. “One-dimensional symmetry of bounded entire solutions of some elliptic equations.” English. In: Duke Math. J. 103.3 (2000), pp. 375–396.
H. Berestycki, L. Caffarelli, and L. Nirenberg. “Further qualitative properties for elliptic equations in unbounded domains.” English. In: Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25.1-2 (1997), pp. 69–94.
H. Berestycki, F. Hamel, and L. Rossi. “Liouville-type results for semi-linear elliptic equations in unbounded domains.” English. In: Ann. Mat. Pura Appl. (4) 186.3 (2007), pp. 469–507.
A. Calsina, X. Mora, and J. Solà-Morales. “The dynamical approach to elliptic problems in cylindrical domains, and a study of their parabolic singular limit.” English. In: J. Differ. Equations 102.2 (1993), pp. 244–304.
V. V. Chepyzhov and M. I. Vishik. “Evolution equations and their trajectory attractors.” English. In: J. Math. Pures Appl. (9) 76.10 (1997), pp. 913–964.
M. A. Efendiev. “On an elliptic attractor in an asymptotically symmetric unbounded domain in \(\mathbb R_+^4\)”. In: Bull. Lond. Math. Soc. 39.6 (2007), pp. 911–920.
M. A. Efendiev and S. V. Zelik. “The attractor for a nonlinear reaction-diffusion system in an unbounded domain.” English. In: Commun. Pure Appl. Math. 54.6 (2001), pp. 625–688.
P. Hess and P. Poláčik. “Symmetry and convergence properties for non- negative solutions of nonautonomous reaction-diffusion problems.” English. In: Proc. R. Soc. Edinb., Sect. A, Math. 124.3 (1994), pp. 573–587.
K. Kirchgaessner. “Wave-solutions of reversible systems and applications.” English. In: J. Differ. Equations 45 (1982), pp. 113–127.
G. R. Sell. “Global attractors for the three-dimensional Navier-Stokes equations.” English. In: J. Dyn. Differ. Equations 8.1 (1996), pp. 1–33.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Efendiev, M. (2018). Symmetry and Attractors: The Case N ≤ 4. In: Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations. Fields Institute Monographs, vol 36. Springer, Cham. https://doi.org/10.1007/978-3-319-98407-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-98407-0_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-98406-3
Online ISBN: 978-3-319-98407-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)