11
$\begingroup$

For any real-valued smooth function $u$, we have the Kato inequality

$|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$,

which holds when $|\nabla u|\neq0$.

If moreover $u$ is harmonic (in an open set in $\mathbb{R}^n$), how would the Kato inequality be improved to

$|\nabla|\nabla u||^2\leq\frac{nāˆ’1}{n}(\operatorname{trace}(\operatorname{Hess}(u)))^2$ ?

  • 3
    This question has been asked on MO and has answers there. – 2014-12-14

0 Answers 0