Homogenisation of phase-field functionals with linear growth
Francesco Colasanto, Matteo Focardi, Caterina Ida Zeppieri · Calculus of Variations and Partial Differential Equations · 2026
Abstract We propose a first rigorous homogenisation procedure in image-segmentation models by analysing the relative impact of (possibly random) fine-scale oscillations and phase-field regularisations for a family of elliptic functionals of Ambrosio and Tortorelli type, when the regularised volume term grows linearly in the gradient variable. In contrast to the more classical case of superlinear growth, we show that our functionals homogenise to a free-discontinuity energy whose surface term explicitly depends on the jump amplitude of the limit variable. The convergence result as above is obtained under very mild assumptions which allow us to treat, among other, the case of stationary random integrands .