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

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.


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
Full Text: DOI


[1] H. W. Alt, L. A. Caffarelli, A. Friedman, A free boundary problem for quasi-linear elliptic equations, Ann. Scoula Norm Sup. Pisa 6(4) (1984), 1-44. · Zbl 0554.35129
[2] I. Ekeland, R. Temam, Convex Analysis and Variational Problems, North-Holland, Amsterdam, 1976. · Zbl 0322.90046
[3] R. L. Fosdick, G. Macsithigh, Helical shear of an elastic circular tube with a stored energy, Arch. Rational Mech. Anal. 84 (1983), 33-53. · Zbl 0529.73007
[4] D. Gilbarg, N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edn., Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[5] M. E. Gurtin, R. Temam, On the antiplane shear problem in finite elasticity, J. Elasticity 11(2) (1981), 197-206. · Zbl 0496.73036
[6] M. E. Gurtin, Two phase deformations of elastic solids, Arch Rational Mech. Anal. 84(1) (1983), 1-29. · Zbl 0525.73054
[7] D. Kinderlehrer, L. Nirenberg, Regularity in free boundary problems, Ann. Scoula Norm Sup. Pisa 4(2) (1977), 373-391. · Zbl 0352.35023
[8] J. L. Lewis, Capacity functions in convex rings, Arch Rational Mech. Anal. 66 (1977), 201-224. · Zbl 0393.46028
[9] P. Marcellini, A relation between existence and uniqueness for non-strictly convex integrals of the calculus of variations, in Mathematical Theories of Optimization, A. Dold and B. Eckmann, eds, Lecture Notes in Mathematics, Vol. 979, Springer-Verlag, Berlin, 1983, pp. 216-230. · Zbl 0505.49009
[10] P. Marcellini, Some remarks on uniqueness in the calculus of variations, in Nonlinear P.D.E.’s and Their Applications (College de France Seminar), H. Brezis and J. L. Lions, eds, Research Notes in Mathematics, No. 84, Pitman, Boston, 1984, pp. 148-153.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.