Problem set 7, PDE

Problem 1

  1. If u is harmonic in the connected domain \Omega and is not constant, then u(\Omega) is open in \R.
  2. Suppose \Omega is bounded and that its boundary \partial\Omega is connected. If u is harmonic in \Omega, then u(\Omega)\subset u(\partial\Omega).

Problem 2

  1. A radial harmonic function on \mathbb B is constant.
  2. A positive harmonic function on \R^d is constant.

Problem 3

Suppose u(x) is harmonic in some domain in \R^n. Then

v(x) = |x|^{2-n} u\Big(\dfrac{x}{|x|^2}\Big)

is also harmonic in a suitable domain.

Problem 4

For n=2, find the Green’s function for the Laplace operator on the first quadrant.

 

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