Abstract
In this chapter we extend the density estimate in Theorem 5.2 by replacing the constant solution a with a symmetric, minimal, hyperbolic connection e, (and more generally with any equivariant minimal hyperbolic solution), and then derive Liouville theorems and asymptotic information for minimal solutions under symmetry hypotheses. Utilizing the extended density estimate we give a proof of a result of Alama et al. (Calc Var 5:359–390, 1997) on the existence of stationary layered solutions in \({\mathbb R}^2\). The Alama, Bronsard and Gui example is revisited in Chap. 9 under no symmetry hypotheses. Our results were originally obtained by a different method in Alikakos and Fusco (Annali della Scuola Normale Superiore di Pisa XV:809–836, 2016).
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Notes
- 1.
i.e., with positive definite Hessian at a ±.
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Acknowledgements
We would like to thank Mihalis Nikolouzos for useful discussions on the proof of Corollary 8.2.
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Alikakos, N.D., Fusco, G., Smyrnelis, P. (2018). Hierarchical Structure—Stratification. In: Elliptic Systems of Phase Transition Type. Progress in Nonlinear Differential Equations and Their Applications, vol 91. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-90572-3_8
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