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Roughening Instability of Broken Extremals

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Abstract

We derive a new general jump condition on a broken Weierstrass–Erdmann extremal of a vectorial variational problem. Such extremals, containing surfaces of gradient discontinuity, are ubiquitous in shape optimization and in the theory of elastic phase transformations. The new condition, which does not have a one dimensional analog, reflects the stationarity of the singular surface with respect to two-scale variations that are nontrivial generalizations of Weierstrass needles. The over-determinacy of the ensuing free boundary problem suggests that typical stable solutions must involve microstructures or chattering controls.

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  1. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Yury Grabovsky

  2. Ecole Polytechnique, Laboratoire de Mécanique des Solides, Palaiseau Cedex, France

    Lev Truskinovsky

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  1. Yury Grabovsky
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  2. Lev Truskinovsky
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Grabovsky, Y., Truskinovsky, L. Roughening Instability of Broken Extremals. Arch Rational Mech Anal 200, 183–202 (2011). https://doi.org/10.1007/s00205-010-0377-8

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