Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 341 / 5 / a

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- Solution a

Solution

Write the operator in Sturm-Liouville form:

Lu = - ((1 + x^{2}) u^{'})^{'} + x^{2} u .

The natural solution space is the Sobolev space

H_{0}^{1} (0, 1)

. Multiplication by a test function

v \in H_{0}^{1} (0, 1)

and integration by parts gives the symmetric bilinear form

a (u, v) = \int_{0}^{1} [(1 + x^{2}) u^{'} v^{'} + x^{2} uv] d x .

It is bounded and

a (u, u) \geq \int_{0}^{1} ∣ u^{'} ∣^{2} d x \geq C ∥ u ∥_{H^{1}}^{2}

by the Poincare inequality. Thus

L

is positive definite and

a

is coercive. The Lax-Milgram theorem gives a unique weak solution of the variational problem

find u \in H_{0}^{1} (0, 1) such that a (u, v) = \int_{0}^{1} f v d x for every v \in H_{0}^{1} (0, 1) .

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