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<p>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 <i>linearly</i> 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 <i>stationary random integrands</i>.</p>

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Homogenisation of phase-field functionals with linear growth

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月3日

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  • Francesco Colasanto,
  • Matteo Focardi,
  • Caterina Ida Zeppieri

摘要

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.