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On De Giorgi’s conjecture: Recent progress and open problems

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Abstract

In 1979, De Giorgi conjectured that the only bounded monotone solutions to the Allen-Cahn equation

$$\Delta{u}+u-{u^3}=0\;\text{in}\;\mathbb{R}^N$$

are one-dimensional. This conjecture and its connection with minimal surfaces and Toda systems are the subject of this survey article.

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Acknowledgements

This work was partially supported by Natural Sciences and Engineering Research Council of Canada.

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  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z 4, Canada

    Hardy Chan & Juncheng Wei

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Chan, H., Wei, J. On De Giorgi’s conjecture: Recent progress and open problems. Sci. China Math. 61, 1925–1946 (2018). https://doi.org/10.1007/s11425-017-9307-4

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