Rate-independent evolution of sets

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)