Issue |
ESAIM: COCV
Volume 30, 2024
|
|
---|---|---|
Article Number | 70 | |
Number of page(s) | 14 | |
DOI | https://doi.org/10.1051/cocv/2024060 | |
Published online | 07 October 2024 |
Existence, uniqueness and L2t(H2x) ∩ L∞t(H1x) ∩ H1t(L2x) regularity of the gradient flow of the Ambrosio–Tortorelli functional
Piazza dei Cavalieri, 7, 56126, Pisa, Italy
* Corresponding author: tommaso.cortopassi@sns.it
Received:
8
May
2023
Accepted:
7
August
2024
We consider the gradient flow of the Ambrosio–Tortorelli functional, proving existence, uniqueness and L2t(H2x) ∩ L∞t(H1x) ∩ H1t(L2x) regularity of the solution in dimension 2. Such functional is an approximation in the sense of Γ-convergence of the Mumford–Shah functional often used in problems of image segmentation and fracture mechanics. The strategy of the proof essentially follows the one of []Feng and Prohl, M2AN Math. Model. Numer. Anal. 38 (2004) 291–320] but the crucial estimate is attained employing a different technique, and in the end it allows to prove better estimates than the ones obtained in [Feng and Prohl, M2AN Math. Model. Numer. Anal. 38 (2004) 291–320]. In particular we prove that if U ⊂ ℝ2 is a bounded C2 domain, the initial data (u0, z0) ∈ [H1(U)]2 with 0 ≤ z0 ≤ 1, then for every T > 0 there exists a unique gradient flow (u(t), z(t)) of the Ambrosio–Tortorelli functional such that (u, z) ∈ [L2(0, T; H2(U)) ∩ L∞(0,T;H1(U)) ∩H1(0,T;L2(U)]2 while previously such regularity was known only for short times.
Mathematics Subject Classification: 35B65 / 35K55 / 68U10 / 94A08
Key words: Mumford-Shah functional / gradient flow / a priori estimates / elliptic system
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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