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Solutions to Laplace's equation in Sobolev spave (existence of)

  1. Mar 15, 2013 #1
    1. The problem statement, all variables and given/known data
    Let ##U\subset\mathbb{R}^m## be a bounded set with smooth boundary ##\partial U##.
    Consider a boundary value problem $$-\bigtriangleup u=f,\quad u|_{\partial U}=0.$$with ##f\in L^2(U)##.
    Use the Riesz representation Theorem that the problem has a weak solution ##u\in W_0^{1,2}(U).##

    2. Relevant equations
    ##u\in W_0^{1,2}(U)## is a weak solution if ##\int_U\sum_{k=1}^m u_{x_k}v_{x_k}=\int_U fv##.
    It is not allowed to use Lax-Milgram Theorem, but it is hinted that Poincare inequality might be useful.

    From what I can see, the exercise will involve various norm estimates, but what I can't see is how these estimates can yield the existence.
     
  2. jcsd
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