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arXiv · 2312.11636

Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory

Xavier Cabre·Iñigo U. Erneta·Juan-Carlos Felipe-Navarro
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Abstract

In this article we build a null-Lagrangian and a calibration for general nonlocal elliptic functionals in the presence of a field of extremals. Thus, our construction assumes the existence of a family of solutions to the Euler-Lagrange equation whose graphs produce a foliation. Then, as a consequence of the calibration, we show the minimality of each leaf in the foliation. Our model case is the energy functional for the fractional Laplacian, for which such a null-Lagrangian was recently discovered by us. As a first application of our calibration, we show that monotone solutions to translation invariant nonlocal equations are minimizers. Our second application is somehow surprising, since here ``minimality'' is assumed instead of being concluded. We will see that the foliation framework is broad enough to provide a proof which establishes that minimizers of nonlocal elliptic functionals are viscosity solutions.

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Xavier Cabre, Iñigo U. Erneta, Juan-Carlos Felipe-Navarro. 2023-12-18. Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory. https://arxiv.org/abs/2312.11636

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