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Pré-Publication, Document De Travail Année : 2024

Phase-field approximation for 1-dimensional shape optimization problems

Résumé

In this paper we propose an unified framework for the phase field approximation of 1-dimensional shape optimization problems with connectedness constraints in any dimension. In particular, we focus on the average distance minimizers problem and the Wasserstein-$\H^1$ problem recently introduced in~\cite{chambolle20231d}. The scheme relies on the $p$-Ambrosio-Tortorelli energy and the diffuse connectedness functional proposed in~\cite{dondl2021keeping} that penalizes how disconnected the level sets of phase fields are. We argue that choosing $p>d$, not only the optimal profiles coming from the Ambrosio Tortorelli term present sharper transitions, but it also allows us to control the level sets of phase fields, enabling the analysis of the connectedness functional. This leads to general $\Gamma-\liminf$ and $\limsup$ inequalities that are easily adaptable to prove $\Gamma$-convergence results for the average distance and Wasserstein-$\H^1$ problems.
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Dates et versions

hal-04620380 , version 1 (21-06-2024)

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  • HAL Id : hal-04620380 , version 1

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Joao Miguel Machado. Phase-field approximation for 1-dimensional shape optimization problems. 2024. ⟨hal-04620380⟩
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