A sharp-interface limit for the geometrically linear two-well problem in the gradient theory of phase transitions in two dimensions
Sergio Conti and Ben Schweizer
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Submission date: 14. Oct. 2003
published in: Archive for rational mechanics and analysis, 179 (2006) 3, p. 413-452
DOI number (of the published article): 10.1007/s00205-005-0397-y
with the following different title: A sharp-interface limit for a two-well problem in geometrically linear elasticity
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We obtain a Gamma-convergence result for the gradient theory of solid-solid phase transitions, in the case of two geometrically linear wells in two dimensions. We consider the functionals
where , W depends only on the symmetric part of , and W(F)=0 for two distinct values of F, say A and B. We show that, under suitable growth assumptions on W and for star-shaped domains , as converges, in the sense of Gamma convergence, to a functional . The limit is finite only on functions u such that the symmetric part of is a function of bounded variation which takes only values A and B. On those functions, the energy concentrates on the jump set J of , with a surface energy depending on the normal to J, and is given by
The interfaces can have, in general, two orientations.