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I am reading the proof of Proposition 1.31 in "Elliptic Partial Differential Equations" by Han & Lin.

It must be an easy application of the discrete Hölder inequality, but I don't get it.

Let $u$ be an harmonic function defined in the unitary ball $B_1 \subset \mathbb{R}^n$ and let $\eta \in C^1_0(B_1)$ be a cutoff function such that $\eta \equiv 1$ in $B_{\frac{1}{2}}$. Then by a direct calculation we have:

\begin{align} \Delta ( \eta^2 |Du|^2) &= 2 \eta \Delta \eta|Du|^2 + 2 |D\eta|^2|Du|^2 + 8\eta \sum_{i, j = 1}^n D_i \eta D_j u D_{ij }u + 2\eta^2 \sum_{i,j = 1}^n(D_{ij}u)^2. \end{align}

Then the book says that from the Hölder inequality we have:

\begin{align} \Delta ( \eta^2 |Du|^2) &\ge (2 \eta \Delta \eta -6 |D\eta|^2)|Du|^2. \end{align}

Probably it's trivial, but I can't understand why.

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    I know nothing about elliptical PDEs, not even sure what some of the notation means, so this is a wild guess. Take $|D\eta|^2|Du|^2$ and apply simply $a^2+b^2>2ab$. Isn't this enough to compare to the third term in your expression and the last one is obviously positive? – 2017-01-23

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I found the solution to my problem. Indeed it was just an application of the Hölder inequality and of the inequality $a^2 + b^2 \ge 2ab$ as suggested in the comment.

I showed that: $$ 8\eta \sum_{ij=1}^n D_i \eta D_j u D_{ij}u \ge - \Big( 8 |D\eta|^2|Du|^2 + 2 \eta^2 \sum_{ij=1}^n( D_{ij} u)^2 \Big) $$ and this is sufficient to get the statement of the book.

Here is the computation.

I actually proved the inequality above multiplied by $-1$.

\begin{align} -8 \eta \sum_{ij=1}^n D_i \eta D_j u D_{ij} u &\le 8 \sum_{ij=1}^n | \eta D_i \eta D_j u D_{ij}u| \\ &\le 8\sum_{i = 1}^n \big(|D_i \eta| \sum_{j=1}^n | \eta D_j u D_{ij} u| \big)\\ &\overset{Hölder}{\le} 8|D\eta|\Big( \sum_{i=1}^n\Big(\sum_{j=1}^n | \eta D_j u D_{ij} u|\Big)^2\Big)^{\frac{1}{2}} \\ &\overset{Hölder}{\le} 8|D\eta||Du| |\eta|\Big(\sum_{ij=1}^n(D_{ij}u)^2\Big)^{\frac{1}{2}} \end{align} Now we use the inequality $ab \le \frac{a^2}{2} + \frac{b^2}{2}$ with $a = 4|D\eta||Du|$ and $b = 2 |\eta| \sum_{ij=1}^n(D_{ij}u)^2$ and we get the conclusion.