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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.11636 |
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| _version_ | 1866913860141711360 |
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| author | Cabre, Xavier Erneta, Iñigo U. Felipe-Navarro, Juan-Carlos |
| author_facet | Cabre, Xavier Erneta, Iñigo U. Felipe-Navarro, Juan-Carlos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_11636 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory Cabre, Xavier Erneta, Iñigo U. Felipe-Navarro, Juan-Carlos Analysis of PDEs 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. |
| title | Null-Lagrangians and calibrations for general nonlocal functionals and an application to the viscosity theory |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2312.11636 |