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Space-time integral currents of bounded variation.

Filip Rindler1

  • 1Mathematics Institute, University of Warwick, Coventry, CV4 7AL UK.

Calculus of Variations and Partial Differential Equations
|December 29, 2022
PubMed
Summary

This study introduces a new mathematical framework for analyzing geometric evolutions, developing a Lipschitz deformation distance that simplifies physical dissipation measurements in elasto-plasticity.

Area of Science:

  • Geometric analysis
  • Continuum mechanics
  • Materials science

Background:

  • Recent elasto-plastic evolution models driven by dislocation flow.
  • Need for variational approaches to rate-independent geometric evolutions.

Purpose of the Study:

  • Develop a theory of space-time integral currents with bounded variation.
  • Introduce a Lipschitz deformation distance for integral currents.
  • Analyze rate-independent geometric evolutions.

Main Methods:

  • Theory of space-time integral currents with bounded variation in time.
  • Introduction of Lipschitz deformation distance.
  • Application of variational principles.

Main Results:

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  • Helly-type compactness theorem.
  • Deformation theorem and isoperimetric inequality.
  • Equivalence of deformation distance convergence with weak* convergence.

Conclusions:

  • The Lipschitz deformation distance provides a natural variational approach to geometric evolutions.
  • This distance is physically relevant as a simplified dissipation measure.
  • The Lipschitz deformation distance coincides with the Whitney flat metric for boundaryless currents, unifying dissipation measurement concepts.