Year: 2024
Type: article
Abstract: \nWe 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.\n
Author: Tommaso Cortopassi
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6940901597
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