Introduction to Partial Differential Equations
Exercise Sheet 9
Ross Ogilvie 28th October,
2024
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Consider the Dirichlet problem for the Laplace equation on with on , where is an open and bounded subset and is a continuous function. We know from the weak maximum principle that there is at most one solution. In this question we see that for some domains, existence is not guaranteed.
Hint. Use Lemma 3.23, even if we haven’t reached it in lectures yet. (3 points)
Let be the upper half-space and the dividing hyperplane. We call reflection in the plane .
is harmonic. (3 points)
(2 points)
Here, ‘formal’ means that you do not need to prove that the integrals are finite/well-defined.
(3 points)