Does anybody know an example of a sequence $f_k \in BV(\mathbb{R^n}) \ $ where $n>1$ and $f \in L^{1}_{\operatorname{loc}}(\mathbb{R^n})$ such that $ f_k \rightarrow f \ $ in $L^{1}_{\operatorname{loc}}(\mathbb{R^n}) \ $ but we have strictly: $ \|Df\|(\mathbb{R^n}) < \liminf \|Df_k\|(\mathbb{R^n}) \ $?
Thanks a lot for any input.
Let $$f_k(x)=k^{-1}\sin (kx_1) \phi(x) $$ where $\phi$ is smooth, nonnegative and compactly supported. Then $f_k\to 0$ in $L^1$, but the integrals of $|Df_k|$ are bounded from below: $$\int |\cos (kx_1)|\phi(x) \ge \int \cos^2 (kx_1)\phi(x)=\frac12\int \phi(x)+\frac12 \int \cos (2kx_1)\phi(x) $$ where $$\int \cos (2kx_1)\phi(x) = - \frac{1}{2k} \int \sin (2kx_1)\frac{\partial \phi}{\partial x_1} \to 0 $$ (or just use the Riemann-Lebesgue lemma).