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

By Alessio Figalli, J. Serra. Inventiones Mathematicae, 2017

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We prove that every bounded stable solution of $$\begin{aligned} (-\Delta )^{1/2} u + f(u) =0 \qquad \text{ in } \mathbb {R}^3 \end{aligned}$$ ( - Δ ) 1 / 2 u + f ( u ) = 0 in R 3 is a 1D profile, i.e., $$u(x)= \phi (e\cdot x)$$ u ( x ) = ϕ ( e · x ) for some $$e\in {\mathbb {S}}^2$$ e ∈ S 2 , where $$\phi :\mathbb {R}\rightarrow \mathbb {R}$$ ϕ : R → R is a nondecreasing bounded stable solution in dimension one. Equivalently, stable critical points of boundary reaction problems in $$\mathbb {R}^{d+1}_+=\mathbb {R}^{d+1}\cap \{x_{d+1}\ge 0\}$$ R + d + 1 = R d + 1 ∩ { x d + 1 ≥ 0 } of the form $$\begin{aligned} \int _{\{x_{d+1\ge 0}\}} \frac{1}{2} |\nabla U|^2 \,dx\, dx_{d+1} + \int _{\{x_{d+1}=0\}} F(U) \,dx \end{aligned}$$ ∫ { x d + 1 ≥ 0 } 1 2 | ∇ U | 2 d x d x d + 1 + ∫ { x d + 1 = 0 } F ( U ) d x are 1D when $$d=3.$$ 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 $$\Gamma $$ Γ -convergence result to the perimeter functional. In particular, when $$f(u)=u^3-u$$ f ( u ) = u 3 - u (or equivalently when $$F(U)=\frac{1}{4} (1-U^2)^2 $$ F ( U ) = 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 $$\mathbb {R}^3$$ 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 $$d=3$$ d = 3 .

Subfields: Applied Mathematics, Computational Theory and Mathematics

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