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Suppose $\Omega:=B(0,1)\subset \mathbb R^2$. Let $u\in BV(\Omega)$ be a radially symmetric, i.e., $u(x)=u(Rx)$ for all $R\in SO(2)$. In addition, suppose $w$ to be an affine function,. i.e., $\partial_1 w=c_1$, $\partial_2w=c_2$ where $c_1$ $c_2$ are two constant.

I am wondering could we obtain a sequence of function $(u_n)\subset C^\infty(\bar\Omega)$ with $u_n$ is radially symmetric and $$ \|\nabla u_n\|_{L^1(\Omega)}\to \|Du\|_{\mathcal M(\Omega)}$$ as well as $$ \|\nabla u_n+\nabla w\|_{L^1(\Omega)}\to \|Du+\nabla w\|_{\mathcal M(\Omega)}$$ where $\|\cdot\|_{\mathcal M(\Omega)}$ means the total variation of radon measure.\

Thank you!

PS: I also put this question on MSE. Please feel to ask me to delete one of them.

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  • $\begingroup$ I'm not sure the requirements you asked for amounts to an approximation, which I would reserve for the following: $\|\nabla u_n-Du\|_{M(\Omega)}\to 0$. THen the second requirement is identical to the first, which I guess is easily met if you extend $u_n$ a little beyond the boundary and convolute it with a radial bump function. $\endgroup$ – Fan Zheng Mar 5 '15 at 2:35
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    $\begingroup$ @FanZheng But usually for BV approximation we can not have $\|\nabla u_n-Di\|_{\mathcal M}\to 0$ as you said. This is one of disadvantage of BV function compare with Sobolev functions... $\endgroup$ – JumpJump Mar 5 '15 at 2:44

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