, Volume 21, Issue 1, pp 113-138

A nonconvex variational problem related to change of phase

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Abstract

We investigate the elastostatic deformation of a tube whose crosssection is a convex ring Ω. The outer lateral surface is assumed to be held fixed and the inner surface is displaced in the axial direction a uniform distanceh. The problem becomes one of seeking minimizers for a functionalJ(u) = ∫ Ω ω(|∇u|) dx whereu(x) is the axial displacement andω(·) is nonconvex. When Ω is an annulus minimizers are known to exist. We prove existence and nonexistence results by studying a relaxed problem obtained by replacingω(|·|) with its lower convex envelope,ω **(|·|). If a minimizer forJ(·) exists it is also a solution to the relaxed problem and this leads to an overdetermined problem in some cases.

WhenJ(·) has no minimizer, solutions of the relaxed problem are of interest. We show that the relaxed problem has a unique solution and give detailed information on its structure.

This work was partially supported by National Science Foundation Grants DMS-8601515 and DMS-8704368.
Communicated by David Kinderlehrer