Preprint

Inserted: 2 oct 2023
Last Updated: 2 oct 2023

Year: 2023

Abstract:

Given a bounded open connected set $\Omega \subset \mathbb R^2$ with Lipschitz boundary, we consider the class of piecewise constant maps $u$ taking three fixed values $\alpha,\beta,\gamma\in\mathbb R^2$, vertices of an equilateral triangle; for any $u$ in this class, using a weak notion of Jacobian determinant valid for $BV$ functions, we give a precise description of $\textrm{Det}(\nabla u)$ and show that the relaxed graph area of $u$ is bounded from above by a quantity related to the flat norm of $\textrm{Det}(\nabla u)$. The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions.

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