Example : sequence of functions $u_n$

Question

Looking for a sequence of functions $u_n$ in $U=B(0,1)\subset\mathbb{R}^2$ ($x=(x_1, x_2))$ satisfying $$||u_n||_{L^1(U)}\rightarrow0\quad \mbox{as $n\rightarrow \infty$ and}\quad||D_{x_1}u_n||_{L^1(U)}\geq||u_n||_{L^1(\partial U)}$$for any $n$. Moreover, $||u_n||_{L^1(\partial U)} \mbox{: constant}>0$

Any suggestions?

Answer 1

How about $u_n=x^{n-1}+\frac{2\pi n^2}{\mbox{volume}(B(0,1))}x_1\cdot I_{B(0,1/n)}$ ?