# Example : sequence of functions unu_n

Looking for a sequence of functions unu_n in U=B(0,1)⊂R2U=B(0,1)\subset\mathbb{R}^2 (x=(x1,x2))x=(x_1, x_2)) satisfying ||un||L1(U)→0as n→∞ and||Dx1un||L1(U)≥||un||L1(∂U)||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 nn.
Moreover, ||u_n||_{L^1(\partial U)} \mbox{: constant}>0||u_n||_{L^1(\partial U)} \mbox{: constant}>0

Any suggestions?

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Take u_n(x)=\frac{x_1^n}{2w_n}u_n(x)=\frac{x_1^n}{2w_n} for x_1\geq 0x_1\geq 0 and u_n(x)=0u_n(x)=0 for x_1\leq 0x_1\leq 0 where w_n=\int_0^{\pi/2}\cos^n(t)dtw_n=\int_0^{\pi/2}\cos^n(t)dt (see Wallis’ integral).

Then \|u_n\|_{L^1(\partial U)}=1\|u_n\|_{L^1(\partial U)}=1,
\|u_n\|_{L^1(U)}=\frac{1}{w_n}\int_{\rho=0}^1\int_{t=0}^{\pi/2}\rho^n\cos^n(t) (\rho d\rho dt)=\frac{1}{n+2}\to 0\|u_n\|_{L^1(U)}=\frac{1}{w_n}\int_{\rho=0}^1\int_{t=0}^{\pi/2}\rho^n\cos^n(t) (\rho d\rho dt)=\frac{1}{n+2}\to 0
and
\|D_{x_1}u_n\|_{L^1(U)}=\frac{1}{w_n}\int_{\rho=0}^1\int_{t=0}^{\pi/2}n\rho^{n-1}\cos^{n-1}(t) (\rho d\rho dt)=\frac{w_{n-1}}{w_n}\cdot \frac{n}{n+1}\to 1.\|D_{x_1}u_n\|_{L^1(U)}=\frac{1}{w_n}\int_{\rho=0}^1\int_{t=0}^{\pi/2}n\rho^{n-1}\cos^{n-1}(t) (\rho d\rho dt)=\frac{w_{n-1}}{w_n}\cdot \frac{n}{n+1}\to 1.

P.S. Ouch, it does not work because \frac{w_{n-1}}{w_n}\cdot \frac{n}{n+1}\leq 1\frac{w_{n-1}}{w_n}\cdot \frac{n}{n+1}\leq 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)}u_n=x^{n-1}+\frac{2\pi n^2}{\mbox{volume}(B(0,1))}x_1\cdot I_{B(0,1/n)} ?