How to use Holder inequality in PDE?

Question

I am reading Qing Han,& Fanghua Lin’s elliptic partial differential equations . I can’t understand this step using Holder inequality. Could anyone help me explain how this inequality applied here?

Answer 1

Actually, it’s Cauchy-Schwarz inequality used here. \begin{align} 8|\eta D_i\eta D_j u D_{ij}u|&=|4 D_i\eta D_j u\cdot 2 \eta D_{ij}u|\\&\leq\frac{1}{2}\left(16 {D_i\eta}^2 {D_j u}^2+4{\eta}^2{D_{ij}u}^2\right)\\&=8 {D_i\eta}^2 {D_j u}^2+2 {\eta}^2{D_{ij}u}^2. \end{align} Hence we know $$8\eta\sum_{i,j=1}^n D_i\eta D_j u D_{ij}u\geq -8|D\eta|^2|D u|^2-2\eta^2\sum_{i,j=1}^n(D_{ij}u)^2.$$