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For a given symmetric semi-definite matrix $2\times2 $ (or $n\times n$) $A=(a_{ij})$ and a smooth function $u:\mathbb R^2(\mathbb R^n)\to\mathbb R$.

Q Can we find some constant $k$ such that the following holds

$$\left(\sum_{ij}a_{ij}u_iu_j\right)^2\leq k\sum_k\sum_{ij}a_{ij}u_{ik}u_{jk},$$ where $u_i=\frac{\partial u}{\partial x_i}$ and $u_{ik}=\frac{\partial^2 u}{\partial x_i\partial x_k}$ .

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    What is the relation between $u_i$ and $u_{ik}$? – 2017-01-12
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    How is this related to [tag:pde]? Are $u_i, u_{ij}, ...$ partial derivatives of $u$? – 2017-01-12
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    @MartinR the inequality comes from the pde, the big goal is to deduce $L\leq k$, some one used such an inequality. – 2017-01-12

1 Answers 1

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In general, no. Consider $u$ linear, i.e. $u(x) = c + b\cdot x$ for $c \in \mathbb{R}$ and $b \in \mathbb{R}^n$. Then $\nabla u = b$ and $D^2 u =0$. The right side of your inequality is $0$, but the left is $$ \sum_{ij} a_{ij} b_i b_j, $$ which will be positive for some choice of $b$ unless $A=0$ since $A$ is positive semidefinite. This gives a contradiction.