A nonconvex variational problem related to change of phase. (English) Zbl 0686.73018

Bauman, Patricia; Phillips, Daniel

Appl. Math. Optimization 21, No. 2, 113-138 (1990).

Summary: We investigate the elastostatic deformation of a tube whose cross-section is a convex ring \(\Omega\). The outer lateral surface is assumed to be held fixed and the inner surface is displaced in the axial direction at a uniform distance h. The problem becomes one of seeking minimizers for a functional \(J(u)=\int_{\Omega}\omega (| \nabla u|)dx\), where u(x) is the axial displacement and \(\omega\) (\(\cdot)\) is nonconvex. When \(\Omega\) is an annulus minimizers are known to exist. We prove existence and nonexistence results by studying a relaxed problem obtained by replacing \(\omega\) (\(| \cdot |)\) with its lower convex envelope, \(\omega^{**}(| \cdot |)\). If a minimizer for J(\(\cdot)\) exists it is also a solution to the relaxed problem and this leads to an overdetermined problem in some cases. When J(\(\cdot)\) 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.

MSC:

74S30	Other numerical methods in solid mechanics (MSC2010)
49J45	Methods involving semicontinuity and convergence; relaxation
74B20	Nonlinear elasticity
49J20	Existence theories for optimal control problems involving partial differential equations
49J27	Existence theories for problems in abstract spaces

Keywords:

elastostatic deformation; existence; nonexistence results; relaxed problem

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