Introduction to Partial Differential Equations
Exercise Sheet 7
Ross Ogilvie 14th October,
2024
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Let be the standard mollifier. Let be any distribution, not necessarily regular.
then .
Prove that
for
the constant function. (3 points)
Hint. Define .
Let be a harmonic function. Show that the following functions are also harmonic.
Together these show that the Laplacian is invariant under similarities (Euclidean motions, reflection and rescaling). (6 points)