 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-107/3/c/solution

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

Nonnegativity lets us dispense with a sign restriction on

c

c_{0} = min (c, 0)

and

L_{0} = a^{ij} D_{ij} + b^{i} D_{i} + c_{0}

. Then

L_{0} u = f - max (c, 0) u \leq 0.

u

to vanish identically if it has an interior zero. Otherwise it is strictly positive:

u \equiv 0 or u > 0 in Ω.

Connectedness is part of the usual domain hypothesis; on a disconnected open set this conclusion applies separately to each component.

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