Strong approximation in h-mass of rectifiable currents under homological constraint
Let h : R $\rightarrow$ R+ be a lower semi-continuous subbadditive and even function such that h(0) = 0 and h($θ$) $\ge$ $α$|$θ$| for some $α$ > 0. The h-mass of a k-polyhedral chain P =$\sum$j $θ$j$σ$j in R n (0 $\le$ k $\le$ n) is defined as M h (P) := j h($θ$j) H k ($σ$j). If T = $τ$ (M, $θ$, $ξ$) is a k-rectifiable chain, the definition extends to M h (T) := M h($θ$) dH k. Given such a rectifiable flat chain T with M h (T) < $\infty$ and $\partial$T polyhedral, we prove that for every $η$ > 0, it decomposes as T = P + $\partial$V with P polyhedral, V rectifiable, M h (V) < $η$ and M h (P) < M h (T) + $η$. In short, we have a polyhedral chain P which strongly approximates T in h-mass and preserves the homological constraint $\partial$P = $\partial$T. These results are motivated by the study of approximations of M h by smoother functionals but they also provide explicit formulas for the lower semicontinuous envelope of T $\rightarrow$ M h (T) + I $\partial$S ($\partial$T) with respect to the topology of the flat norm.