Matrix inequality in a paper by Piccinini-Spagnolo

In the paper 'On the Holder continuity of solutions of second order elliptic equations in two variables' by Piccinini and Spagnolo, they prove the following estimate: $$\begin{array}{ll} \left(\int_S p_{11} |u_T|^2 \right)^{\frac12}\left(\int_S \frac{\langle P(x) (u_N,u_T), (1,0) \rangle^2}{p_{11}} \right)^{\frac12} &\leq & \sqrt{\frac{\Lambda}{\lambda}} \left(\int_S \lambda |u_T|^2 \right)^{\frac12}\left(\int_S \frac{\langle P(x) (u_N,u_T), (1,0) \rangle^2}{p_{11}} \right)^{\frac12} \\ & \leq & \sqrt{\frac{\Lambda}{\lambda}}\frac12 \left(\int_S \lambda |u_T|^2 + \frac{(p_{11} u_N + p_{12}u_T)^2}{p_{11}} \right)\\ & \leq & \sqrt{\frac{\Lambda}{\lambda}}\frac12 \left(\int_S \left( p_{22} - \frac{p_{12}^2}{p_{11}}\right) |u_T|^2 + \frac{(p_{11} u_N + p_{12}u_T)^2}{p_{11}} \right)\\ & = & \sqrt{\frac{\Lambda}{\lambda}}\frac12 \left(\int_S \langle P(x) (u_N,u_T),(u_N,u_T)\rangle \right) \end{array}$$ Here $$P(x)$$ is a symmetric positive definite $$2\times2$$ matrix with entries $$(p_{ij})$$ and $$\lambda |\xi|^2 \leq \langle P(x)\xi,\xi\rangle \leq \Lambda |\xi|^2$$ and $$(u_N,u_T)$$ are the normal and tangential derivative with $$S$$ being the unit circle in $$\mathbb{R}^2$$.

The crucial part of the above calculation is the inequality $$\lambda \leq \left( p_{22} - \frac{p_{12}^2}{p_{11}}\right)$$.

My question is the following: Is there a different way to prove this estimate without explicitly writing down the expressions and instead proceed using only matrix inequalities?

Let $$0<\lambda_1\le\lambda_2$$ be the eigenvalues of the symmetric positive $$2\times2$$ matrix $$P$$. Then $$\lambda\le\lambda_1=\min_{|\xi|=1} \langle P\xi,\xi\rangle \le\langle Pe_1,e_1\rangle=p_{11}\le \lambda_2 =\max_{|\xi|=1} \langle P\xi,\xi\rangle,$$ so $$\lambda p_{11}\le \lambda_1\lambda_2=\det P=p_{11}p_{22}-p_{12}^2$$, whence the inequality, since $$\langle Pe_1,e_1\rangle=p_{11}>0$$.
• (actually the middle $\le$ in the displayed inequalities is not needed, but it's also obvious) Dec 20, 2022 at 20:45