DOI: 10.68381/jca13003 ISSN: 0944-6532
Estimates of Quasiconvex Polytopes in the Calculus of Variations
Kewei Zhang
We give direct estimates for the quasiconvex polytopes
Q(K)
Q
(
K
)
generated by a finite set
K\subset M^{N\times n}
K
⊂
M
N
×
n
. More precisely, we bound the quasiconvex envelope
Q\operatorname{dist}(\cdot,K)
Q
dist
(
⋅
,
K
)
near a convex exposed face of
C(X)
C
(
X
)
which does not have rank-one connections. Our estimates depend on the weak-(1,1) bounds for certain singular integral operators and the geometric features of the convex polytope
C(K)
C
(
K
)
. We show by an example that our estimate is ‘local’ and independent of the ‘size’ of
K
K
, hence it is a better estimate than the polyconvex hull
P(K)
P
(
K
)
which is ‘size’ dependent.