Existence and relaxation results in special classes of deformations

Mikhail A. Sytchev

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Submission date: 29. Feb. 2000
Pages: 36
published in: Equadiff 99 : International Conference on Differential Equations ; Berlin, Germany ; 1 - 7 August 1999. Vol. 1 / B. Fiedler ... (eds.)
Singapore : World Scientific, 2000. - P. 460 - 462 
Bibtex
with the following different title: Existence and relaxation results in the class of anti-plane shear deformations
Keywords and phrases: existence and relaxation, mathematical theory of elasticity, weak convergence, young measure
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Abstract:
In this paper we deal with the existence and relaxation issues in variational problems from the mathematical theory of elasticity. We consider minimization of the energy functional in those classes of deformations which make the problem essentially scalar.

It turns out that in these cases the relaxation theorem holds for integrands that are bounded from below by a power function with power exceeding the dimension of the space of independent variables. The bound from below can be relaxed in the homogeneous case. The same bounds were used previously to rule out cavitation and other essential discontinuites in admissible deformations. In the homogeneous case we can also indicate a condition which is both necessary and sufficient for solvability of all boundary value minimization problems of the Dirichlet type.