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Problem Set 10

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  1. Show that $latex ||\cdot||_{H^1}$ is a norm, by showing it is induced by an inner product.

  2. Consider, for a connected domain $latex \Omega$, the energy form $latex \mathscr E(u,v) = \int_\Omega \nabla u\cdot \nabla v$.

    1. $latex \mathscr E(u,v)$ is an inner product on $latex H^1$ modulo constants.

    2. $latex \mathscr E(u,v)$ is an inner product on $latex H_0^1$.

  3. Show the equivalences of the Dirichlet principle.