arXiv subjects

Martin Rakovsky

Publications and source records attributed to Martin Rakovsky.

3 recordsLinked to original sources

arXiv2026

Existence, uniqueness and regularity of solutions to the parabolic Ambrosio-Tortorelli system

We investigate the existence, uniqueness, and regularity of the gradient flow of the Ambrosio-Tortorelli functional, viewed as an initial-boundary value problem, in arbitrary dimension. For any initial data, using a time-discrete Euler scheme, we establish the existence of a weak gradient flow satisfying a maximum principle. We also identify a functional space in which uniqueness holds. We further show that such gradient flow is smooth in the interior of the space-time domain. Under additional assumptions on the initial data, the regularity of the boundary of the domain, we prove optimal regularity for the solution, up to the space-time boundary.

math.AP↗

arXiv2026

Critical points of the two-dimensional Ambrosio-Tortorelli functional with convergence of the phase-field energy

We consider a family $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ of critical points of the Ambrosio-Tortorelli functional. Assuming a uniform energy bound, the sequence $\{(u_\varepsilon, v_\varepsilon)\}_{\varepsilon>0}$ converges in $L^2(\Omega)$ to a limit $(u, 1)$ as $\varepsilon \to 0$, where $u$ is in $SBV^2(\Omega)$. It was previously shown that if the full Ambrosio-Tortorelli energy associated to $(u_\varepsilon,v_\varepsilon)$ converges to the Mumford-Shah energy of $u$, then the first inner variation converges as well. In particular, $u$ is a critical point of the Mumford-Shah functional in the sense of inner variations. In this work, focusing on the two-dimensional setting, we extend this result under the sole convergence of the phase-field energy to the length energy term in the Mumford-Shah functional.

math.AP↗

arXiv2024

Critical points of the one dimensional Ambrosio-Tortorelli functional with an obstacle condition

We consider a family of critical points of the Ambrosio-Tortorelli energy with an obstacle condition on the phase field variable. This problem can be interpreted as a time discretization of a quasistatic evolution problem where the obstacle at step $n$ is defined as the solution at step $n-1$. The obstacle condition now reads as an irreversibility condition (the crack can only increase in time). The questions tackled here are the regularity of the critical points, the properties inherited from the obstacle sequence, the position of the limit points and the equipartition of the phase field energy. The limits of such critical points turn out to be critical points of the Mumford-Shah energy that inherit the possible discontinuities induced by the obstacle sequence.

math.AP↗