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Improvement of flatness for nonlocal phase transitions. (English) Zbl 1445.35302

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)].

MSC:

35R11 Fractional partial differential equations
35J20 Variational methods for second-order elliptic equations
35Q56 Ginzburg-Landau equations

Citations:

Zbl 1238.35019
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