A smooth weight function of the desired property

Question

Let us fix some $C>0$. I would like to construct a smooth weight function $w(x): \mathbb{R} \rightarrow \mathbb{R}_{\geq 0}$ such that it is supported on $[-1/2, 1/2]$ and $\int_{- \infty}^{\infty} |w'(x)| dx < C$ while trying to keep $\int_{- \infty}^{\infty} w(x) dx$ as large as possible.

I would appreciate any examples and references. Thank you very much!

Answer 1

You can take $W$ even. It is cleat that $$ \int_0^{1/2}|w'|\ge\Bigl|\int_0^{1/2}w'\Bigr|=|w(0)|. $$ We want then $|w(0)|0$ close to $1/2$ and define $$ v(x)=\begin{cases} 1&0\le x\le a,\\ 1-\dfrac{(x-a)^2}{(1/2-a)^2} & a