Infimum of an integral functional involving a symmetric matrix I have a symmetric $d \times d$ matrix $A$ and I have the following functional:
$$
\mathcal J(h) := \int_{B_1(0)} \vert \langle Au,u \rangle\vert \frac{\vert h'(\vert u \vert)\vert}{\vert u \vert} du,
$$
where $B_1(0)$ is the unit ball in $\mathbb R^d$ and $h \in C_c^\infty(\mathbb R)$ with $\text{supp}\, h\subset [-1,1]$ and $\int h = 1$. Let me denote by $H$ the set of these functions $h$. I would like to find/estimate 
$$
\inf_{h \in H} \mathcal J(h).
$$
Do you have any ideas on how to approach this kind of problems? I have never heard nor read about such a problem. I have been playing with some toy models in $\mathbb R^2$ without success, I do not even manage to guess who is the infimum.
 A: $\newcommand{\al}{\alpha}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\om}{\omega}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathcal{F}}
\newcommand{\E}{\operatorname{\mathsf E}} 
\newcommand{\PP}{\operatorname{\mathsf P}}
\newcommand{\ii}[1]{\operatorname{\mathsf I}\{#1\}} 
\newcommand{\tf}{\widetilde{f}}$ 
The infimum is $0$ for $d\ge2$. Indeed, take any $h \in H$. For any real $\ep\in(0,1)$ and all real $r>0$, let $h_\ep(r):=\frac1\ep\,h(\frac r\ep)$, so that $h_\ep\in H$, $\text{supp}\,h_\ep\subset[-\ep,\ep]$, and
$$|h'_\ep(r)|=\frac1{\ep^2}\,\Big|h'\Big(\frac r\ep\Big)\Big|\le \frac M{\ep^2}$$
for some real $M>0$ and all real $r>0$, whence 
\begin{equation}
 0\le\mathcal J(h_\ep)
 \ll\int_0^\ep r^2\frac{|h'_\ep(r)|}{r}\,r^{d-1}\,dr
 \ll\int_0^\ep r^2\frac{1/\ep^2}{r}\,r^{d-1}\,dr
 \ll\ep^{d+1-2}\to0
\end{equation}
as $\ep\downarrow0$.
