Abstract
Let
{h:\mathbb{R}\to\mathbb{R}_{+}}
be a lower semicontinuous subbadditive and even function such that
{h(0)=0}
and
{h(\theta)\geq\alpha|\theta|}
for some
{\alpha>0}
. If
{T=\tau(M,\theta,\xi)}
is a k-rectifiable chain, its h-mass is defined as
\mathbb{M}_{h}(T):=\int_{M}h(\theta)\,d\mathcal{H}^{k}.
Given such a rectifiable flat chain T with
{\mathbb{M}_{h}(T)<\infty}
and
{\partial T}
polyhedral, we prove that for every
{\eta>0}
, it decomposes as
{T=P+\partial V}
with P polyhedral, V rectifiable,
{\mathbb{M}_{h}(V)<\eta}
and
{\mathbb{M}_{h}(P)<\mathbb{M}_{h}(T)+\eta}
. In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint
{\partial P=\partial T}
.
When
{h^{\prime}(0^{+})}
is well defined and finite, the definition of the h-mass extends as a finite functional on the space of finite mass k-chains (not necessarily rectifiable). We prove in this case a similar approximation result for finite mass k-chains with polyhedral boundary.
These results are motivated by the study of approximations of
{\mathbb{M}_{h}}
by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of
{T\mapsto\mathbb{M}_{h}(T)+\mathbb{I}_{\partial S}(\partial T)}
with respect to the topology of the flat norm.
Radial representation of lower semicontinuous envelope
