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Rate-independent evolution of sets

  • * Corresponding author: Riccarda Rossi

    * Corresponding author: Riccarda Rossi 

Dedicated to Alexander Mielke on the occasion of his 60th birthday

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  • The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of (the complement of) a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes.

    In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

    Mathematics Subject Classification: Primary: 35A15, 35R37, 74R10; Secondary: 49Q10.


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  • Figure 1.  An example for nonconnectedness: A needle-like forcing F.

    Figure 3.  Solutions of the minimization problem (54) with forcing $ F^c $ being an equilateral triangle, a square, and a regular hexagon, respectively

    Figure 8.  Partial $ C^1 $ regularity. The two solutions correspond to $ v(x) = 3/4 - |x{-}1/2| $ (left) and $ v(x) = 1/4 + |x{-}1/2| $ (right)

    Figure 2.  The C1 competitor profile

    Figure 7.  Convex forcing $ F^{c} $. The two solutions correspond to $ v(x) = 3/4 - \beta(x{-}1/2)^2 $ for $ \beta = 2 $ (left) and $ \beta = 1/5 $ (right). The minimal set $ Z $ is convex

    Figure 4.  The evolution from (60) for $ M = 5 $ and time $ t = 2 $.

    Figure 5.  An evolution of connected sets fulfilling the compatibility condition (50), time flows from left to right

    Figure 6.  The effect of changing the parameter $ a $. The two solutions correspond to $ v(x) = (x{-}1/2)^2+1/2 $ for $ a = 7 $ (left) and $ a = 3 $ (right). The top adhesion zone is smaller for smaller $ a $. Note that the parts of the boundary of $ Z $ which are not in contact with $ F^c $ are arcs of circles with radius $ 1/a $ (recall that $ a $ is different in the two figures), as predicted in Subsection 4.1

    Figure 9.  Extreme configurations. The solution for $ v(x) = \max\{1-5|x{-}1/2|,1/2\} $ (left) and $ v(x) = \lfloor 5x\rfloor/5+1/5 $ (right)

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