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.