Some Examples of Crinkles
Part of the
The IMA Volumes in Mathematics and its Applications
book series (IMA, volume 1)
Many physical problems can be phrased as the problem of minimizing some energy functional E[f] over a given class of admissible functions f. It can happen that there is a minimizing sequence fn that approaches a limit f̄ but f̄ does not minimize E, either because f̄ is not in the admissible class or because E is not lower semicontinuous. In the examples that I discuss here, this happens because the derivatives <Inline>#</Inline> are highly discontinuous and do not approach f̄’ in the limit. I call such sequences crinkles, and call the limiting function f̄ the carrier of the crinkle. Young  has written a book on the subject; he calls such sequences generalized curves. In control theory the same sort of thing is also called a chattering state.
KeywordsBody Shape Deformation Gradient Completion Process Boundary Load Inhomogeneous Boundary Condition
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