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A symmetry result in \({\mathbb {R}}^2\) for global minimizers of a general type of nonlocal energy

Abstract

In this paper, we are interested in a general type of nonlocal energy, defined on a ball \(B_R\subset {\mathbb {R}}^n\) for some \(R>0\) as

$$\begin{aligned} {\mathcal {E}}(u, B_R)= \iint _{{\mathbb {R}}^{2n}{\setminus } ({\mathcal {C}}B_R)^2} F( u(x)-u(y),x-y)dx dy+\int _{B_R} W(u)\, dx.\end{aligned}$$

We prove that in \({\mathbb {R}}^2\), under suitable assumptions on the functions F and W, bounded continuous global energy minimizers are one-dimensional. This proves a De Giorgi conjecture for minimizers in dimension two, for a general type of nonlocal energy.

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Correspondence to Claudia Bucur.

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The author is member of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and she is supported by the INdAM Starting Grant “PDEs, free boundaries, nonlocal equations and applications”. I sincerely thank Enrico Valdinoci and Luca Lombardini for their very useful suggestions.

Communicated by A. Malchiodi.

Appendix A. Some known results

Appendix A. Some known results

Proposition 14

Let \(\Omega \subset {\mathcal {O}} \subset {\mathbb {R}}^n\) be bounded, open sets such that \(|{\mathcal {O}} {\setminus } \Omega |>0\) and let \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be a measurable function. Then

$$\begin{aligned} \begin{aligned} \Vert u\Vert ^p_{L^p(\Omega )} \le&\; \frac{2^{p-1}}{|{\mathcal {O}} {\setminus } \Omega |} \left( d_{{\mathcal {O}}}^{n+sp} \int _{\Omega }\ \int _{{\mathcal {O}} {\setminus } \Omega } \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dx dy+ |\Omega |\, \Vert u\Vert ^p_{L^p({\mathcal {O}} {\setminus } \Omega )} \right) , \end{aligned} \end{aligned}$$

with \(d_{{\mathcal {O}}}=\text{ diam }({\mathcal {O}})\).

Proof

We have that

$$\begin{aligned} \begin{aligned} |u(x)|^p =&\; |u(x)-u(y)+u(y)|^p \\ =&\; \frac{1}{|{\mathcal {O}} {\setminus } \Omega |} \int _{{\mathcal {O}} {\setminus } \Omega } |u(x)-u(y)+u(y)|^p dy \\ \le&\; \frac{2^{p-1}}{|{\mathcal {O}} {\setminus } \Omega |} \left( \int _{{\mathcal {O}} {\setminus } \Omega } \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} |x-y|^{n+sp} + |u(y)|^p\, dy\right) \\ \le&\; \frac{2^{p-1}}{|{\mathcal {O}} {\setminus } \Omega |} \left( d_{{\mathcal {O}}}^{n+sp}\int _{{\mathcal {O}} {\setminus } \Omega } \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}} dy +\int _{{\mathcal {O}} {\setminus } \Omega } |u(y)|^p\, dy\right) . \end{aligned} \end{aligned}$$

The conclusion follows by integrating on \(\Omega \). \(\square \)

We recall also a fractional Poincaré inequality (see [25, Proposition 2.1] for the proof).

Proposition 15

(A fractional Poincaré inequality) Let \(\Omega \subset {\mathbb {R}}^n\) be bounded, open set and let \(u:{\mathbb {R}}^n\rightarrow {\mathbb {R}}\) be in \(L^1(\Omega )\). Then

$$\begin{aligned} \begin{aligned} \Vert u-u_{\Omega }\Vert _{L^p(\Omega )} \le \left( \frac{d_{\Omega }^{{n+sp}}}{|\Omega |}\right) ^{\frac{1}{p}} [u]_{W^{s,p}(\Omega )}, \end{aligned} \end{aligned}$$

where

$$\begin{aligned} u_\Omega =\frac{1}{|\Omega |} \int _{\Omega } u(x) dx \qquad \text{ and } \qquad d_{\Omega }=\text{ diam }(\Omega ). \end{aligned}$$

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Bucur, C. A symmetry result in \({\mathbb {R}}^2\) for global minimizers of a general type of nonlocal energy. Calc. Var. 59, 52 (2020). https://doi.org/10.1007/s00526-020-1698-6

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Mathematics Subject Classification

  • Primary 3547G10
  • 35R11
  • Secondary 35B08