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This study explores homogenization for free-discontinuity functionals under linear conditions. It introduces a novel deterministic approach with general assumptions, leading to a stochastic homogenization result for stationary random integrands.

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Area of Science:

  • Mathematical Analysis
  • Partial Differential Equations
  • Calculus of Variations

Background:

  • Homogenization theory studies the behavior of composite materials with microstructures.
  • Free-discontinuity functionals are crucial in modeling phenomena with interfaces or cracks.
  • Existing methods often rely on periodicity assumptions for integrands.

Purpose of the Study:

  • To investigate deterministic and stochastic homogenization of free-discontinuity functionals.
  • To develop a generalized approach not requiring periodic integrands.
  • To analyze the characterization of limit integrands in stochastic homogenization.

Main Methods:

  • Deterministic homogenization under linear growth and coercivity conditions.
  • Application of the pointwise Subadditive Ergodic Theorem (by Akcoglu and Krengel).
  • Analysis of stationary random integrands for stochastic homogenization.

Main Results:

  • A novel deterministic homogenization result under general assumptions on integrands.
  • Proof of stochastic homogenization for stationary random integrands.
  • Characterization of limit integrands using asymptotic cell formulas.

Conclusions:

  • The study extends homogenization theory to more general cases, including non-periodic integrands.
  • The findings provide a unified framework for both deterministic and stochastic homogenization.
  • The results are applicable to various fields involving heterogeneous media and free discontinuities.