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First Sobolev Embedding Theorem

  1. Oct 23, 2011 #1
    Hello there,

    I have a question about Sobolev's first theorem.

    Essentially it is said that for p = 1 the space [tax] W^{l}_{p} [\tex] belongs to the space of continuous if l ≥ n.
    So, in 1D I just need the finction and its first derivative to be integrable, which makes perfect sense.
    I can not get why though for n > 1 I need information on derivatives of higher order to conclude the finction is continuous. In other words, can anyone show me a counterexample of a function [tex] W^{1}_{1} [\tex] not continuous over a 2D domain?

    The second curiosity concerns the following statement I have found on the web, http://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/
    "
    (One can already see this with the most basic example of Sobolev embedding, coming from the fundamental theorem of calculus. If a (continuously differentiable) function f has f' in L1 then we of course have f belongs to L∞ ; but the converse is far from true.) "

    Now the Sobolev theroem tell us that in order for a function to be L∞, in 1D, we just need it to belong to [tex] W^{1}_{1} [\tex], which seems to me larger then the class of continuoulsy differentiable function, am I missing something?
     
  2. jcsd
  3. Oct 24, 2011 #2
    Hi there,

    I am starting to wonder if my question was unclear, or meaningless..
    If so please let me know and I will reformulate it.

    Many thanks
     
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