Summary: We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form \[ (-\Delta)^{s/2}u=u-u^3, \] with \(s\in(0,1)\). More generally, we can take into account equations like \[ Lu=f(u), \] where \(f\) is a bistable nonlinearity and \(L\) is an integro-differential operator, possibly of anisotropic type.
The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in theresearch line dictated by a classical conjecture of E. De Giorgi in [“Convergence problems for functionals and operators”, in: Proceedings of the Intern. Meeting on Recent Methods in Nonlinear Analysis, Rome 1978, 131–188 (1979; Zbl 0405.49001)].
Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results in [S. Dipierro et al., Am. J. Math. 142, No. 4, 1083–1160 (2020; Zbl 1445.35302)].