Existence, uniqueness and L2t(H2x) ∩ L∞t(H1x) ∩ H1t(L2x) regularity of the gradient flow of the Ambrosio–Tortorelli functional

We consider the gradient flow of the Ambrosio-Tortorelli functional, proving existence, uniqueness and L-t(2)(H-x(2)) boolean AND L-t(infinity)(H-x(1)) boolean AND H-t(1)(L-x(2)) regularity of the solution in dimension 2. Such functional is an approximation in the sense of Gamma-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 subset of R-2 is a bounded C2 domain, the initial data (u(0), z(0)) is an element of [H-1(U)](2) with 0 <= z(0) <= 1, then for every T > 0 there exists a unique gradient flow (u(t), z(t)) of the Ambrosio-Tortorelli functional such that...