Upper bounds for the relaxed area of S 1 -valued Sobolev maps and its countably subadditive interior envelope

Giovanni Bellettini ^[1] ; Riccardo Scala ^[2] ; Giuseppe Scianna ^[2]
1. [1] Università di Siena, Siena, Italy; The Abdus Salam International Centre for Theoretical Physics ICTP, Trieste, Italy
2. [2] Università di Siena, Siena, Italy
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 6, 2024, págs. 2135-2178
Idioma: inglés
DOI: 10.4171/RMI/1475
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Resumen
- Given a connected bounded open Lipschitz set Ω⊂R 2 , we show that the relaxed Cartesian area functional A (u,Ω) of a map u∈W 1,1(Ω;S 1) is finite, and we provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture adapted to W 1,1(Ω;S 1), on the largest countably subadditive set function Au,⋅) smaller than or equal to A (u,⋅).