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non-convex energy functional

ESAIM: Mathematical Modelling and Numerical Analysis (1)

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Analysis of a one-dimensional variational modelofthe equilibrium shapel of a deformable crystal
Eric Bonnetier, Richard S. Falk, Michael A. Grinfeld
Journal:

ESAIM: Mathematical Modelling and Numerical Analysis / Volume 33 / Issue 3 / May 1999

Published online by Cambridge University Press:

15 August 2002, pp. 573-591

Print publication:

May 1999
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The equilibrium configurations of a one-dimensional variational model thatcombines terms expressing the bulk energy of a deformable crystal and itssurface energy are studied. After elimination of the displacement, theproblem reduces to the minimization of a nonconvex and nonlocal functional ofa single function, the thickness. Depending on a parameter which strengthensone of the terms comprising the energy at the expense of the other, it isshown that this functional may have a stable absolute minimum or only aminimizing sequence in which the term corresponding to the bulk energy isforced to zero by the production of a crack in the material.