R. Continolo - V. Lorenzini - R. Scala - G. Scianna
Regularity and convergence of critical points of an Ambrosio-Tortorelli functional with linear growth and of its $\Gamma$-limit
Published Paper
Inserted: 16 mar 2025
Last Updated: 26 sep 2025
Journal: Journal of Differential Equations
Year: 2025
Abstract:
In the one-dimensional setting we consider an Ambrosio-Tortorelli functional $F_\varepsilon(u,v)$ which has linear growth with respect to $u'$. We prove that under suitable conditions on the fidelity term, minimizers and critical points of $F_\varepsilon$ are Sobolev regular, and that the same is true for the $\Gamma$-limit $F$ of $F_\varepsilon$. As a corollary, we obtain that the functional $A_w(u)$ computing the length of the generalized graph of a function of bounded variation $u$, under the same conditions on the fidelity term, admits a unique minimizer of class $C^1$. This solves a conjecture by De Giorgi \cite{DG} in the one-dimensional case.
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