*Published Paper*

**Inserted:** 13 sep 2012

**Last Updated:** 6 feb 2018

**Journal:** Discrete Contin. Dyn. Syst.

**Volume:** 28

**Number:** 3

**Pages:** 1179--1206

**Year:** 2010

**Abstract:**

We establish sharp energy estimates for some solutions, such as global minimizers, monotone solutions and saddle-shaped solutions, of the fractional nonlinear equation $(-\Delta)^{1/2} u=f(u)$ in $\mathbb R^n$. Our energy estimates hold for every nonlinearity $f$ and are sharp since they are optimal for one-dimensional solutions, that is, for solutions depending only on one Euclidian variable. As a consequence, in dimension $n=3$, we deduce the one-dimensional symmetry of every global minimizer and of every monotone solution. This result is the analog of a conjecture of De Giorgi on one-dimensional symmetry for the classical equation $-\Delta u=f(u)$ in $\mathbb R^n$.

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