Dipierro, Serena; Serra, Joaquim; Valdinoci, Enrico Improvement of flatness for nonlocal phase transitions. (English) Zbl 1445.35302 Am. J. Math. 142, No. 4, 1083-1160 (2020). Summary: We establish an improvement of flatness result for critical points of Ginzburg-Landau energies with long-range interactions. It applies in particular to solutions of \((-\Delta)^{s/2} u = u - u^3\) in \(\mathbb{R}^n\) with \(s \in (0, 1)\). As a corollary, we establish that solutions with asymptotically flat level sets are \(1\) D and prove the analogue of the De Giorgi conjecture (in the setting of minimizers) in dimension \(n = 3\) for all \(s \in (0, 1)\) and in dimensions \(4 \le n \le 8\) for \(s \in (0, 1)\) sufficiently close to \(1\).The robustness of the proofs, which do not rely on the extension of Caffarelli and Silvestre, allows us to include anisotropic functionals in our analysis.Our improvement of flatness result holds for all solutions, and not only minimizers. This cannot be achieved in the classical case \(-\Delta u = u - u^3\) (in view of the solutions bifurcating from catenoids constructed by M. Del Pino et al. [Ann. Math. (2) 174, No. 3, 1485–1569 (2011; Zbl 1238.35019)]. Cited in 2 ReviewsCited in 19 Documents MSC: 35R11 Fractional partial differential equations 35J20 Variational methods for second-order elliptic equations 35Q56 Ginzburg-Landau equations Keywords:critical points of Ginzburg-Landau energies; long-range interactions Citations:Zbl 1238.35019 PDFBibTeX XMLCite \textit{S. Dipierro} et al., Am. J. Math. 142, No. 4, 1083--1160 (2020; Zbl 1445.35302) Full Text: DOI arXiv