 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-105/2/b/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-05

The principal coefficients satisfy the positive uniformly elliptic operator condition if there is

θ > 0

, independent of

x

, such that

i, j = 1 \sum n a^{ij} (x) ξ_{i} ξ_{j} \geq θ ∣ ξ ∣^{2} (x \in \overline{U}, ξ \in R^{n}) .

The smooth coefficients on the compact closure also give a uniform upper bound

\sum a^{ij} ξ_{i} ξ_{j} \leq Θ∣ ξ ∣^{2}

. The first-order and zeroth-order coefficients do not enter uniform ellipticity.

 New to topics? Read the docs here!