Weak variations of Lipschitz graphs and stability of phase boundaries |
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Authors: | Yury Grabovsky Vladislav A Kucher Lev Truskinovsky |
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Institution: | 1. Department of Mathematics, Temple University, Philadelphia, PA, USA 2. Institute of Mechanical Engineering, Russian Academy of Sciences, St. Petersburg, Russia 3. Laboratoire de M??canique des Solides, Ecole Polytechnique, Palaiseau Cedex, France
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Abstract: | In the case of Lipschitz extremals of vectorial variational problems, an important class of strong variations originates from
smooth deformations of the corresponding non-smooth graphs. These seemingly singular variations, which can be viewed as combinations
of weak inner and outer variations, produce directions of differentiability of the functional and lead to singularity-centered
necessary conditions on strong local minima: an equality, arising from stationarity, and an inequality, implying configurational
stability of the singularity set. To illustrate the underlying coupling between inner and outer variations, we study in detail
the case of smooth surfaces of gradient discontinuity representing, for instance, martensitic phase boundaries in non-linear
elasticity. |
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