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Following the Dynamics of Structural Variants in Experimentally Evolved Populations
Published on: February 3, 2023
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A global method for deterministic and stochastic homogenisation in BV.
Filippo Cagnetti1, Gianni Dal Maso2, Lucia Scardia3
1Department of Mathematics, University of Sussex, Brighton, UK.
Annals of PDE
|April 25, 2022
Summary
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.
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.
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