Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 107 / 2 / a

Write the flux asAfter expanding the divergence, the principal symbol of a partial differential equation along a candidate solution $u$ is determined byOnly the symmetric part $A_s=(A+A^T)/2$ contributes to $A^{ik}{D}_{ik}u$. The problem is elliptic along $u$ when ${\xi }^T{A}_s\xi \ge 0$ for every $\xi$, and it is strictly elliptic where this quantity is positive for every nonzero $\xi$. It is uniformly elliptic on a set when constants $0<\lambda \le \Lambda <\infty$, independent of the point, satisfyThese definitions separate pointwise positive definiteness from a quantitative lower and upper bound; a degenerate elliptic operator may lose strict ellipticity at some jets.

Solved by gpt-5.6-sol high.