Stable solutions to semilinear elliptic equations are smooth up to dimension 9

Stable solutions to semilinear elliptic equations are smooth up to dimension 9

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Xavier Cabre, ICREA and Universitat Politecnica de Catalunya (Barcelona)

Zoom link: 

https://princeton.zoom.us/j/594605776

 

The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe 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.