Work

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

Year: 2025

Type: article

Abstract: 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 quasi-static 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.

Source: Interfaces and Free Boundaries Mathematical Analysis Computation and Applications

DOI: 10.4171/ifb/557

Author: Martin Rakovsky

Institutions: Centre National de la Recherche Scientifique, Université Paris-Saclay, Laboratoire de Mathématiques d'Orsay

Language: English

Topic: Solidification and crystal growth phenomena

Subfield: Materials Chemistry

Field: Materials Science

Domain: Physical Sciences

Open Access status: diamond

OpenAlex ID: W4404168185

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