= Failure of L2 closure of the global BV domain
For $1<a\le2$, the radial <function> $f(x)=(1+|x|)^{-a}$ belongs to $L^2(\mathbb R^2)$ and has finite distributional <total variation seminorm on a domain>, but is not in $L^1(\mathbb R^2)$. Cutting it off outside radius $T$ gives <bounded-variation space> <functions> tending to $f$ in $L^2$, with variation tending to that of $f$: the added jump costs $2\pi T(1+T)^{-a}\to0$. Thus the penalty equal to TV on $BV\cap L^2$ and infinity elsewhere is not <sequentially lower semicontinuous> in $L^2$. The closed <homogeneous bounded-variation space> extension avoids this domain issue.
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