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Symmetry and Attractors: Arbitrary Dimension

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Symmetrization and Stabilization of Solutions of Nonlinear Elliptic Equations

Part of the book series: Fields Institute Monographs ((FIM,volume 36))

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Abstract

Let Ω be the domain of \(\mathbb {R}^N\) (N ≥ 2) defined by

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Notes

  1. 1.

    In [13], the function f was assumed to be globally Lipschitz-continuous. Here, f is just assumed to be locally Lipschitz-continuous. However, since u is bounded, it is always possible to find a Lipschitz-continuous function \(\tilde {f}:\mathbb {R}^+\to \mathbb {R}\) satisfying (6.3) and such that \(\tilde {f}\) and f coincide on the range of u.

  2. 2.

    This property is obvious here due to (6.30). See Remark 6.4 for a comment about the other situations, corresponding to Theorems 6.9 , 6.10 and 6.11.

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Authors and Affiliations

  1. Institute of Computational Biology, Helmholtz Center Munich, Neuherberg, Bayern, Germany

    Messoud Efendiev

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Efendiev, M. (2018). Symmetry and Attractors: Arbitrary Dimension. 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_6

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