*Published Paper*

**Inserted:** 27 jun 2019

**Last Updated:** 10 feb 2023

**Journal:** Invent. Math.

**Year:** 2020

**Abstract:**

We prove that every bounded stable solution of
\[ (-\Delta)^{1/2} u + f(u) =0 \qquad \mbox{in }\mathbb{R}^3\]
is a 1D profile, i.e., $u(x)= \phi(e\cdot x)$ for some $e\in \mathbb S^2$, where $\phi:\mathbb{R}\to \mathbb{R}$ is a nondecreasing bounded stable solution in dimension one.
Equivalently, stable critical points of boundary reaction problems in $\mathbb{R}^{d+1}_+=\mathbb{R}^{d+1}\cap \{x_{d+1}\geq 0\}$ of the form
\[
\int_{\{x_{d+1\geq 0}\}} \frac{1}{2}

{\nabla U}

^2 \,dx\, dx_{d+1} + \int_{\{x_{d+1}=0\}} F(U) \,dx
\]
are 1D when $d=3.$

These equations have been studied since the 1940’s in crystal dislocations. Also, as it happens for the Allen-Cahn equation, the associated energies enjoy a $\Gamma$-convergence result to the perimeter functional. In particular, when $f(u)=u^3-u$ (or equivalently when $F(U)=\frac14 (1-U^2)^2 $), our result implies the analogue of the De Giorgi conjecture for the half-Laplacian in dimension $4$, namely that monotone solutions are 1D.

Note that our result is a PDE version of the fact that stable embedded minimal surfaces in $\mathbb{R}^3$ are planes. It is interesting to observe that the corresponding statement about stable solutions to the Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still unknown for $d=3$.

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