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 (ESAIM-COCV), 11 (2005) 3, p. 310-356 
DOI number (of the published article): 10.1051/cocv:2005009
Bibtex
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 
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.