= Solution
The <bounded-variation space> carries the <norm> $\|u\|_{BV}=\|u\|_{L^1(\Omega)}+|Du|(\Omega)$. Its <weak-star convergence in BV> is characterized by
$$
\boxed{u_k\to u\text{ in }L^1(\Omega),\qquad Du_k\overset{*}{\rightharpoonup}Du\text{ in }\mathcal M(\Omega;\mathbb R^n).}
$$
The second condition means convergence of the <vector measure> pairings against every $\varphi\in C_0(\Omega;\mathbb R^n)$. Equivalently, <strong convergence> in $L^1$ together with $\sup_k|Du_k|(\Omega)<\infty$ suffices: <integration by parts> identifies the limit on smooth compactly supported tests, and <uniform approximation> extends this to $C_0$ tests.
The <bounded-variation compactness> theorem says that
$$
\boxed{\sup_k\bigl(\|u_k\|_{L^1(\Omega)}+|Du_k|(\Omega)\bigr)<\infty}
$$
guarantees a <subsequence> convergent in <weak-star convergence in BV> on a bounded <Lipschitz domain>. This is the uniform criterion for relative sequential compactness. If asking only for the existence of one convergent <subsequence>, the exact condition is the existence of a BV-bounded <subsequence>, equivalently $\liminf_k\|u_k\|_{BV}<\infty$. The entire sequence need not be bounded: alternating zero functions and constants tending to infinity gives a simple example. Conversely, a convergent <subsequence> has bounded $L^1$ <norm> and bounded <total variation seminorm> by the <Uniform boundedness principle> for its derivative-measure pairings.
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