Existence, uniqueness and L2 t (Hx2) ∩ L∞ t (Hx1) ∩ Ht1(L2 x ) regularity of the gradient flow of the Ambrosio-Tortorelli functional

Abstract

We consider the gradient flow of the Ambrosio-Tortorelli functional, proving existence, uniqueness and L2t (Hx2) ∩ L∞t (Hx1) ∩ Ht1(L2x) regularity of the solution in dimension 2. Such functional is an approximation in 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 [9] but the crucial estimate is attained employing a different technique, and in the end it allows us to prove better estimates than the ones obtained in [9]. In particular we prove that if U ⊂ R2 is a bounded Lipshitz domain, the initial data (u0,z0) ∈ [H1(U)]2 and 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.