Construction of a sequence with special properties

2018-03-12 20:23:11

Define the space of functions $$X(Q):=\{u\in L^1(Q): \ u'\in L^2(Q)\},$$ where $Q=(0,1)$ is a unit interval.

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous, $f(0)=0$ and such that $\alpha t^2\leq f(t)\leq \beta (1+t^2)$ for all $t\in\mathbb{R}$ and some constants $0<\alpha\leq\beta$.

I want to construct a sequence of functions $(v_k)_k\subset X(Q)$, such that

$v_k(0)=0$, $v_k(1)=m$ for some $m\in\mathbb{R}$ and such that $$\int_Q f(v_k'(s))ds\leq \frac{C}{k}$$ for some $C>0$.

I tried to define it in a form $$v_k(s):=\begin{cases} 0 & for \ s\in[0,1-k^{-1}) \\ m(ks+1-k) & for \ s\in[1-k^{-1},1]\end{cases},$$ however I can't get the right decay of the integral this way. I also tried to interpolate between 0 and $m$ by polynomial of higher order, but it doesn't help. I have a gut-feeling that maybe the staircase-type construction for the distributional derivative $v_k'$ should be the starting point for finding $v_k$. Anyway, I'd appreciate any ideas.

Thank you.