American Journal of Mathematics

abstract:

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 $\Bbb{R}^n$ 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, M. Kowalczyk, and J. Wei ({\it J. Differential Geom.}, 2013)).