Problem Set 3

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  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).
  3. A radial harmonic function on \mathbb B is constant.
  4. A positive harmonic function on \R^d is constant.
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