  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.

Analysis of PDEs