Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 1 / a / Solution

The operator is strictly elliptic when its symmetric principal matrix $A(x)=(a^{ij}(x))$ is positive definite at every point:It is a uniformly elliptic operator when some $\lambda >0$ satisfiesfor every $x$ and $\xi$. Here the least eigenvalue of the continuous matrix $A(x)$ is a positive continuous function on the compact set ${\overline{B}}_1$. It therefore has a positive minimum, which supplies $\lambda$ and proves uniform ellipticity.