Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2015 / Paper 2 / 3 / c

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- Solution c

Solution

Impose the homogeneous Dirichlet boundary condition on the boundary variable:

G (x, x^{'}) = 0 for x \in S, x^{'} \in V .

This removes the term containing the unknown normal derivative of

ϕ

. The prescribed boundary values alone then determine the solution through

ϕ (x^{'}) = - \int_{S} f (x) \partial_{n_{x}} G (x, x^{'}) d S .

For an unbounded domain one also imposes the appropriate decay condition at infinity. This defines the required Dirichlet Green function.

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