Calibrations and null-Lagrangians for nonlocal perimeters and an application to the viscosity theory
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Calibrations and null‑Lagrangians for nonlocal perimeters and an application to the viscosity theory Xavier Cabré1,2,3 Received: 11 July 2019 / Accepted: 25 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 Ω ⊂ ℝn , we built a calibration for the nonlocal perimeter inside Ω ⊂ ℝn . The calibrating functional is a nonlocal null-Lagrangian. As a consequence, we conclude the minimality in Ω 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. Keywords Nonlocal perimeter · Calibration · Null-Lagrangian · Nonlocal minimal surfaces · Viscosity solutions Mathematics Subject Classification Primary 53C38 · Secondary 49Q10 · 49Q20 · 47G20 · 53A10
1 Introduction Given a measurable function K in ℝn , a bounded open set Ω ⊂ ℝn , and a measurable set E ⊂ ℝn , the K-nonlocal perimeter of Ω (in all of ℝn ) is defined by
P(Ω) ∶= L(Ω, Ωc ), while the K-nonlocal perimeter of E inside Ω is
* Xavier Cabré [email protected] 1
ICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain
2
Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
3
BGSMath, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Spain
13
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X. Cabré
PΩ (E) ∶= L(E ∩ Ω, Ec ∩ Ω) + L(E ∩ Ω, Ec ∩ Ωc ) + L(Ec ∩ Ω, E ∩ Ωc ),
(1.1)
where Ac = ℝn ⧵A denotes the complement of a set and
L(A, B) ∶=
∫A ∫B
K(x − y) dy dx
(1.2)
for any two disjoint measurable sets A, B ⊂ ℝn . For the kernel K, we assume that
K(z) ≥ 0,
K(−z) = K(z),
and
�ℝn
min(1, |z|)K(z) dz < ∞.
(1.3)
This is an extension of the fractional perimeter introduced by Caffarelli, Roquejoffre, and Savin [3], in which K(z) = |z|−n−𝛼 for some 𝛼 ∈ (0, 1). If it happens that PΩ (E) ≤ PΩ (F) for every measurable set F ⊂ ℝn satisfying F⧵Ω = E⧵Ω , we then call E a minimizer of the K-nonlocal perimeter in Ω. Given the outside datum E⧵Ω , if a minimizer E of the K-nonlocal perimeter inside Ω exists and if it is regular enough, then it is easy to verify that it satisfies the Euler–Lagrange equation
HK [E](x) = 0
(1.4)
at any point x ∈ 𝜕E ∩ Ω . Here, HK [E](x) denotes the so-called K-nonlocal mean curvature of E at a point x of its boundary and is defined by
HK [E](x) ∶=
∫ℝn
= lim 𝜀↓0
( ) 𝟙Ec (y) − 𝟙E (y) K(x − y) dy
∫ℝn ⧵B𝜀 (x)
( ) 𝟙Ec (y) − 𝟙E (y) K(x − y) dy,
(1.5)
where 𝟙 denotes the characteristic function and the integral is meant in the Cauchy principal v
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