If \(E=\{x_d>h(x_1,\dots,x_{d-1});\ h:\mathbb{R}^{d-1}\to\mathbb{R}\}\), then \(\partial E\) is a minimal surface, and \(h\) is affine, or equivalently, \(E\) is a halfspace for \(d\le 8\). In [Proc. Int. Meet., Rome 1978, 131–188 (1979; Zbl 0405.49001)], E. De Giorgi published his famous conjecture stating that if \(u\in C^2(\mathbb{R}^d)\) is a solution of the Allen-Cahn equation \((*)\,-\Delta u=u-u^3\), \(|u|\le 1\), satisfying \(\partial_{x_d}u>0\), then all level sets \(\{u=\lambda\}\) of \(u\) must be hyperplanes for \(d\le 8\). A solution function \(u\) is said to be a 1D profile if \(u(x)=\varphi(e\cdot x)\) for some \(e\in\mathbb{S}^2\), where \(\varphi:\mathbb{R}\to\mathbb{R}\) is a nondecreasing bounded stable solution in dimension one. Conjecture was proved by N. Ghoussoub and C. Gui in [Math. Ann. 311, No. 3, 481–491 (1998; Zbl 0918.35046)] for \(d=2\), and by L. Ambrosio and X. Cabré in [J. Am. Math. Soc. 13, No. 4, 725–739 (2000; Zbl 0968.35041)] for \(d=3\). In [Ann. Math. (2) 169, No. 1, 41–78 (2009; Zbl 1180.35499)], O. Savin proved the conjecture for \(4\le d\le 8\) under the additional condition that \(\lim\limits_{x_d\to\pm\infty} u=\pm1\). It turns out that the De Giorgi conjuncture is still valid if \(-\Delta\) in \((*)\) is replaced with \((-\Delta)^{1/2}\) to form the fractional Allen-Cahn equation \((**)\,(-\Delta)^{1/2} u=u-u^3\), \(|u|\le 1\). More general form of De Giorgi conjuncture is SDG Conjuncture stating that if \(u\in C^2(\mathbb{R}^d)\) is a stable solution of \((*)\) or \((***)\, (-\Delta)^{1/2} u+f(u)=0\), then all level sets \(\{u=\lambda\}\) of \(u\) must be hyperplanes for \(d\le 7\). Also, the validity of this conjuncture implies both previous conjunctures. The SDG conjuncture was proven only for \(d=2\) by H. Berestycki et al. in [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25, No. 1–2, 69–94 (1997; Zbl 1079.35513)] for \((*)\), and by X. Cabré and J. Solá-Morales in [Commun. Pure Appl. Math. 58, No. 12, 1678–1732 (2005; Zbl 1102.35034)] for \((***)\).
The goal of this paper is to prove this conjuncture for \(d=3\). The author proves that if \(u\) is a bounded stable solution of \((***)\) with \(d=3\), and \(f\in C^{0,\alpha}\) for some \(\alpha>0\), then \(u\) is 1D profile. As a corollary of this result it is shown that if \(f(u)=u^3-u\), then the conjuncture with \((**)\) is true for \(d=4\). Also, as a critical ingredient of the proof is the general energy estimate, which holds in every dimension \(d\ge 2\). The author shows that if \(R\ge 1\), \(M_0\ge 2\), \(\alpha\in(0,1)\), \(u\) is a stable solution of \((***)\) in \(B_R\subset\mathbb{R}^d\), and \(f:[-1,1]\to\mathbb{R}\) satisfies \(\|f\|_{C^{0,\alpha}([-1,1])}\le M_0\), then there exists a constant \(C>0\), depending only on \(d\) and \(\alpha\), such that \[ \int\limits_{B_{R/2}}|\nabla u|\le CR^{d-1}\log(M_0R) \] and \[ \iint\limits_{\mathbb{R}^d\times\mathbb{R}^d\setminus B^c_{R/2}\times B^c_{R/2}}\frac{|u(x)-u(y)|^2}{|x-y|^{d+1}}dxdy\le CR^{d-1}\log^2(M_0R) \]