Regularity of stable solutions to reaction-diffusion elliptic equations

Xavier Cabre

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arXiv · 2112.08277

Regularity of stable solutions to reaction-diffusion elliptic equations

Xavier Cabre

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Abstract

The boundedness of stable solutions to semilinear (or reaction-diffusion) elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are unbounded (or singular). This note describes, for non-expert readers, a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems. We also describe, briefly, a famous analogue question in differential geometry: the regularity of stable minimal surfaces.

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Xavier Cabre. 2021-12-15. Regularity of stable solutions to reaction-diffusion elliptic equations. https://arxiv.org/abs/2112.08277

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