 Solution

ID: 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/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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