Preprint

Inserted: 6 jul 2023
Last Updated: 10 jul 2023

Year: 2023

Abstract:

Given a bounded open connected Lipschitz set $\Omega\subset\mathbb R^2$, we show that the relaxed Cartesian area functional $\mathcal A(u,\Omega)$ of a map $u\in W^{1,1}(\Omega;\mathbb S^1)$ is finite, and provide a useful upper bound for its value. Using this estimate, we prove a modified version of a De Giorgi conjecture 17 adapted to $W^{1,1}(\Omega;\mathbb S^1)$, on the largest countably subadditive set function smaller than or equal to $\mathcal A(u,\cdot)$.

Download:

• BSS.pdf