On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1

We prove that every bounded stable solution of $$(-\mathrm{\Delta }{)}^{1\mathrm{/}2}u+f(u)=0in{\mathbb{R}}^3$$ is a 1D profile, i.e., $u(x)=\varphi (e\cdot x)$ for some $e\in {\mathbb{S}}^2$, where $\varphi :\mathbb{R}\to \mathbb{R}$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in ${\mathbb{R}}_{+}^{d+1}={\mathbb{R}}^{d+1}\cap \{x_{d+1}\ge 0\}$ when $d=4$, by proving that all critical points of $${\int }_{\{x_{d+1\ge 0}\}}\frac{1}{2}\mathrm{\mid }\mathrm{\nabla }U{\mathrm{\mid }}^{2}dxd{x}_{d+1}+{\int }_{\{x_{d+1}=0\}}\frac{1}{4}(1-U^2{)}^{2}dx$$ that are monotone in ${\mathbb{R}}^d$ (that is, up to a rotation, ${\mathrm{\partial }}_{x_d}U>0$) are one dimensional. Our result is analogue to the fact that stable embedded minimal surfaces in ${\mathbb{R}}^3$ are planes. Note that the corresponding result about stable solutions to the classical Allen-Cahn equation (namely, when the half-Laplacian is replaced by the classical Laplacian) is still open.