Infimum of an integral functional involving a symmetric matrix

Question

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.

Answer 1

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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 $\ep\downarrow0$.