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I am trying to follow a calculation shown in Chapter 1.4 of Han and Lin's Elliptic PDE textbook.

Let $u$ be a harmonic function on the unit ball $B_1$ in $\mathbb{R}^n$, and let $\eta \in C^\infty_c (B_1)$ with $\eta = 1$ on $B_{\frac{1}{2}}$.

First, one can show that $\Delta(|Du|^2) = 2|D^2 u|^2 + 2\langle Du, D(\Delta u) \rangle = 2|D^2 u|^2 \geq 0$. In other words, $|Du|^2$ is subharmonic.

Next, observe that

\begin{align*} \Delta(\eta^2|Du|^2) &= \Delta(\eta^2)|Du|^2 + 2 \langle D\eta^2, D(|Du|^2) \rangle + \eta^2 \Delta(|Du|^2) \\\\ &= 2(|D\eta|^2 + \eta \Delta \eta)|Du|^2 + 8 \langle \eta D\eta, D^2u \cdot Du \rangle + 2\eta^2|D^2u|^2. \end{align*}

In the text, the authors claim we can then "use Hölder's inequality" to bound the above quantity from below by $(2\eta \Delta \eta - 6|D\eta|^2)|Du|^2 \geq -C|Du|^2)$, where $C>0$ depends only on $\eta$. I've tried to apply various inequalities (I think the authors were referring to the Cauchy-Schwarz inequality $|D\eta|^2|Du|^2 \geq |\langle D\eta, Du \rangle|^2$) but I still haven't gotten the desired bound. I think the method involves completing a square I'm not seeing.

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  • $\begingroup$ When you write $|\mathrm D^2 u|$ which matrix norm are you using? $\endgroup$
    – K.defaoite
    May 15 at 15:50
  • $\begingroup$ Frobenius norm. $\endgroup$ May 15 at 16:26
  • $\begingroup$ I found a copy of the original text and I think you are missing a prefactor of $\eta^2$ in front of the last term. $\endgroup$
    – K.defaoite
    May 15 at 19:25
  • $\begingroup$ Also, it should be noted that the Cauchy-Schwartz inequality is a special case of Holder's inequality, so they are technically not incorrect. $\endgroup$
    – K.defaoite
    May 15 at 19:30
  • $\begingroup$ I just fixed that. $\endgroup$ May 16 at 18:25

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