Abstract
For nonnegative even kernels K, we consider the K-nonlocal perimeter functional acting on sets. Assuming the existence of a foliation of space made of solutions of the associated K-nonlocal mean curvature equation in an open set \(\Omega \subset \mathbb {R}^n\), we built a calibration for the nonlocal perimeter inside \(\Omega \subset \mathbb {R}^n\). The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in \(\Omega \) of each leaf of the foliation. As an application, we prove the minimality of K-nonlocal minimal graphs and that they are the unique minimizers subject to their own exterior data. As a second application of the calibration, we give a simple proof of an important result from the seminal paper of Caffarelli, Roquejoffre, and Savin, stating that minimizers of the fractional perimeter are viscosity solutions.
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Notes
However, a remarkable existence result of minimizers for very general kernels has been proved in [6] as a consequence of a delicate a priori BV-estimate established in the same paper.
When \(K\in L^1(\mathbb {R}^n)\), minimizers satisfy the Euler–Lagrange equation pointwise, as it was shown by Mazón, Rossi, and Toledo [13].
When \(K\not \in L^1(\mathbb {R}^n)\), this hypothesis imposes some regularity on the level sets of \(\phi _E\). It is needed for the proofs in this section to work. An analogue regularity assumption on the “nonlocal field” \(\zeta (x,y)\) is also needed in the article [14], even if not explicitly stated there, to ensure that all quantities in the Proof of Theorem 2 in [14]—such as the second expression for a(v)— are well defined.
However, (3.4) shows that we will necessarily have \(\int _{E{\setminus } F} \widetilde{H}_K [\phi _E](x)\, \mathrm{d}x >-\infty \), since \(\mathcal {P}_\Omega (E)\ge 0\) and \(\mathcal {P}_\Omega (F)<+\infty \).
This last assumption is also made in Theorem 5.1 of [3], as mentioned in the beginning of Section 4 of that paper.
This continuity result has also been proved in other articles, but we find the proof in [10] the most natural and direct —it contains, though, a couple of “easy-to-correct” typos.
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Funding
X. C. is supported by grants MTM2017-84214-C2-1-P and MdM-2014-0445 (Government of Spain), and is a member of the research group 2017SGR1392 (Government of Catalonia).
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Cabré, X. Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory. Annali di Matematica 199, 1979–1995 (2020). https://doi.org/10.1007/s10231-020-00952-z
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DOI: https://doi.org/10.1007/s10231-020-00952-z