Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 107 / 3 / c

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- Solution c

Solution

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.

The strong minimum principle for elliptic operators forces

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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