Bounded-variation compactness
= Bounded-variation compactness
On a bounded <Lipschitz domain>, a sequence with bounded $L^1$ <norms> and bounded <total variation seminorm> has a subsequence converging strongly in $L^1$. The limit is in the <BV space>, and its variation does not exceed the limit inferior. This supplies the <compactness> step of the <direct method in the calculus of variations> for linearly growing <gradient> energies.