Lower bounds for the two well problem with surface energy I:
Reduction to finite elements
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Submission date: 30. Apr. 2002
published in: Control, optimisation and calculus of variations, 11 (2005) 3, p. 310-356
DOI number (of the published article): 10.1051/cocv:2005009
with the following different title: A two well Liouville theorem
Keywords and phrases: non-convex functional, lower bounds, finite elements
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Let be a bounded domain in , let H be a matrix with . Let and consider the functional over the class of Lipschitz functions from satisfying affine boundary condition F. It can be shown by convex integration that there exists and with . In this paper we begin the study of the asymptotics of for such F. This is the simplest minimisation problem involving surface energy in which we can hope to see the effects of convex integration solutions. The only known lower bounds are . In this paper we link the behavior of to the minimum of over a suitable class of piecewise affine functions. Let be a triangulation of by triangles of diameter less than h and let denote the class of continuous functions that are piecewise affine on a triangulation . For function let be the interpolant, i.e. the function we obtain by defining to be the affine interpolation of u on the corners of . We show that if for some small there exists with
then for the interpolant satisfies .
Note that it is conjectured that and it is trivial that so we reduce the problem of non-trivial lower bounds on to the problem of non-trivial lower bounds on . This latter point will be addressed in a forthcoming paper.