Preprint
Inserted: 25 sep 2026
Last Updated: 25 sep 2026
Year: 2026
Abstract:
Let $\Omega\subset\mathbb{R}^d$ be open. We consider functions
$u\in W^{1,1}_{\mathrm{loc}}(\Omega)$ whose gradient perturbations of the
identity $x\mapsto x+t\nabla u(x)$ are expanding for almost every $t>0$,
in the sense that they push the Lebesgue measure on $\Omega$ forward to a
measure dominated by Lebesgue measure. We prove that
$\nabla u\in\mathrm{BV}_{\mathrm{loc}}(\Omega;\mathbb{R}^d)$, that the
distributional Hessian $D^2u$ is a positive-semidefinite matrix-valued
measure satisfying $
D^2u
\leq\Delta u$, and consequently that $u$ has a
locally convex representative. This settles a conjecture of Bianchini and
Talamini (arXiv:2603.18819, 2026), which originates from a question of
Liu and Pego (Pure Appl. Anal., 2025).
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