A Weierstrass extremal field theory for the fractional Laplacian
In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We buil...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/404687 |
| Acceso en línea: | https://hdl.handle.net/2117/404687 https://dx.doi.org/10.1515/acv-2022-0099 |
| Access Level: | acceso abierto |
| Palabra clave: | Weierstrass field theory Field of extremals Fractional Laplacian Calibration Null-Lagrangian Minimality Classificació AMS::53 Differential geometry::53C Global differential geometry Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type Classificació AMS::47 Operator theory::47G Integral, integro-differential, and pseudodifferential operators Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | In this paper, we extend, for the first time, part of the Weierstrass extremal field theory in the Calculus of Variations to a nonlocal framework. Our model case is the energy functional for the fractional Laplacian (the Gagliardo–Sobolev seminorm), for which such a theory was still unknown. We build a null-Lagrangian and a calibration for nonlinear equations involving the fractional Laplacian 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 the minimality of each leaf in the foliation follows from the existence of the calibration. As an application, we show that monotone solutions to fractional semilinear equations are minimizers. In a forthcoming work, we generalize the theory to a wide class of nonlocal elliptic functionals and give an application to the viscosity theory. |
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