Problem Set 10

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  1. Show that ||\cdot||_{H^1} is a norm, by showing it is induced by an inner product.
  2. Consider, for a connected domain \Omega, the energy form \mathscr E(u,v) = \int_\Omega \nabla u\cdot \nabla v.
    1. \mathscr E(u,v) is an inner product on H^1 modulo constants.
    2. \mathscr E(u,v) is an inner product on H_0^1.
  3. Show the equivalences of the Dirichlet principle.
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