Stable solutions to some elliptic problems: Minimal cones, the Allen-Cahn equation, and blow-up solutions

Xavier Cabré, Giorgio Poggesi

Research output: Chapter in Book/Conference paperChapterpeer-review

6 Citations (Scopus)


These notes record the lectures for the CIME Summer Course taught by the first author in Cetraro during the week of June 19–23, 2017. The notes contain the proofs of several results on the classification of stable solutions to some nonlinear elliptic equations. The results are crucial steps within the regularity theory of minimizers to such problems. We focus our attention on three different equations, emphasizing that the techniques and ideas in the three settings are quite similar. The first topic is the stability of minimal cones. We prove the minimality of the Simons cone in high dimensions, and we give almost all details in the proof of J. Simons on the flatness of stable minimal cones in low dimensions. Its semilinear analogue is a conjecture on the Allen-Cahn equation posed by E. De Giorgi in 1978. This is our second problem, for which we discuss some results, as well as an open problem in high dimensions on the saddle-shaped solution vanishing on the Simons cone. The third problem was raised by H. Brezis around 1996 and concerns the boundedness of stable solutions to reaction-diffusion equations in bounded domains. We present proofs on their regularity in low dimensions and discuss the main open problem in this topic. Moreover, we briefly comment on related results for harmonic maps, free boundary problems, and nonlocal minimal surfaces.

Original languageEnglish
Title of host publicationGeometry of PDEs and Related Problems
Subtitle of host publicationCetraro, Italy 2017
EditorsChiara Bianchini, Antoine Henrot, Rolando Magnanini
Place of PublicationAustria
PublisherSpringer-Verlag Wien
Number of pages45
ISBN (Electronic)9783319951867
ISBN (Print)9783319951850
Publication statusPublished - 4 Oct 2018
Externally publishedYes

Publication series

NameLecture Notes in Mathematics
ISSN (Print)0075-8434


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