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

2019; Springer Science+Business Media; Volume: 219; Issue: 1 Linguagem: Inglês

10.1007/s00222-019-00904-2

1432-1297

Alessio Figalli, Joaquim Serra,

Geometric Analysis and Curvature Flows

We prove that every bounded stable solution of $$\begin{aligned} (-\Delta )^{1/2} u + f(u) =0 \qquad \text{ in } \mathbb {R}^3 \end{aligned}$$is a 1D profile, i.e., $$u(x)= \phi (e\cdot x)$$ for some $$e\in {\mathbb {S}}^2$$, where $$\phi :\mathbb {R}\rightarrow \mathbb {R}$$ is a nondecreasing bounded stable solution in dimension one. Equivalently, stable critical points of boundary reaction problems in $$\mathbb {R}^{d+1}_+=\mathbb {R}^{d+1}\cap \{x_{d+1}\ge 0\}$$ of the form $$\begin{aligned} \int _{\{x_{d+1\ge 0}\}} \frac{1}{2} |\nabla U|^2 \,dx\, dx_{d+1} + \int _{\{x_{d+1}=0\}} F(U) \,dx \end{aligned}$$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 $$F(U)=\frac{1}{4} (1-U^2)^2 $$), 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 $$d=3$$.