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

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