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Milena Chermisi, Carnegie Mellon UniversityRigidity estimates in nonlinear elasticity Abstract: In the study of solid-solid phase transitions and in particular of shape-memory alloys one is interested in studying variational models of the form
where the energy density is invariant under rotations and it is minimized by several copies of the set of the proper rotations, i.e., by sets of the form
While the set (case ) is rigid, in the sense that there are no nontrivial gradient fields taking values in , the set is in general not rigid. After a brief review of known rigidity results, we present a quantitative rigidity estimate for a multiwell problem () in dimension . Precisely, we show that if a gradient field is -close to the set , a set of the form , and and an appropriate bound on the length of the interfaces holds, then the gradient field is actually close to only one of the wells . The estimate holds for any connected subdomain, and has the optimal scaling. Results of these kind have several applications, e.g., in studying the scaling of singularly perturbed problem under Dirichlet boundary conditions or in proving compactness and -convergence for a sequence of singularly perturbed functionals of the kind
THURSDAY, January 22, 2009 |