Preprint 37/2002

Lower bounds for the two well problem with surface energy I:
Reduction to finite elements

Andrew Lorent

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Submission date: 30. Apr. 2002
Pages: 93
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
MSC-Numbers: 49M25
Keywords and phrases: non-convex functional, lower bounds, finite elements
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Abstract:
Let formula55 be a bounded domain in formula57, let H be a formula61 matrix with formula63. Let formula65 and consider the functional formula67 over the class formula69 of Lipschitz functions from formula55 satisfying affine boundary condition F. It can be shown by convex integration that there exists formula75 and formula77 with formula79. In this paper we begin the study of the asymptotics of formula81 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 formula85. In this paper we link the behavior of formula87 to the minimum of formula89 over a suitable class of piecewise affine functions. Let formula91 be a triangulation of formula55 by triangles of diameter less than h and let formula97 denote the class of continuous functions that are piecewise affine on a triangulation formula91. For function formula101 let formula103 be the interpolant, i.e. the function we obtain by defining formula105 to be the affine interpolation of u on the corners of formula109. We show that if for some small formula111 there exists formula113 with
displaymath115
then for formula117 the interpolant formula119 satisfies formula121.

Note that it is conjectured that formula123 and it is trivial that formula125 so we reduce the problem of non-trivial lower bounds on formula127 to the problem of non-trivial lower bounds on formula129. This latter point will be addressed in a forthcoming paper.

20.02.2013, 14:48