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

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. 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 $\mathbb R^{d+1}_+=\mathbb R^{d+1}\cap \{x_{d+1}\geq 0\}$ when $d = 4$, by proving that all critical points of $$ \int_{\{x_{d+1\geq 0}\}} \frac12 |\nabla U|^2 \,dx\, dx_{d+1} + \int_{\{x_{d+1}=0\}} \frac 1 4 (1-U^2)^2 \,dx $$ that are monotone in $\mathbb R^d$ (that is, up to a rotation, $\partial_{x_d} U>0$) 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.