I was working on the following problem in PDE (Evans Chapter 6 Problem 5).
I already know how to deduce the final statement. What I'm stuck on is in proving that $Lv \leq 0$, for $\lambda$ large enough. I currently have $Lv \leq (-2 \theta + 1) |\nabla^2 u|^2 + (-2\lambda\theta + \sum_{i,j = 1}^{n} |\nabla a_{ij}|^2) |\nabla u|^2$, where $\theta > 0$ is the constant associated with the property of uniform ellipticity. The following two references:
- Two exercises on Evans PDE book.
- http://homepage.ntu.edu.tw/~d04221001/Notes/Problems%20and%20Solutions%20Section/Evans%20PDE/Evans%20PDE%20Solution%20Chapter%206%20Second-Order%20Elliptic%20Equations.pdf
allude to a bound on $\nabla a_{ij}$ which I don't understand how to justify. After bounding that term they also claim that there must be some $\lambda$ such that $Lv \leq (-2 \theta + 1) |\nabla^2 u|^2 + (-2\lambda\theta + A) |\nabla u|^2 \leq 0$ holds, where A is the constant bounding the sum over the $\nabla a_{ij}$ terms. I don't see why there should be such a relationship between $|\nabla u|^2$ and $|\nabla^2 u|^2$.
Any nudge in the right direction would be greatly appreciated.