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

  • Alessio FigalliEmail author
  • Joaquim Serra
Article
  • 113 Downloads

Abstract

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 \(d=3\).

Notes

Acknowledgements

This project has received funding from the European Research Council under the Grant Agreement No. 721675 “Regularity and Stability in Partial Differential Equations (RSPDE)”.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsETH ZürichZurichSwitzerland

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