Extensions
For a function f ∈ Lp(Ω) on an open subset Ω of ℝn, its extension by zero
is an element of Lp(ℝn). Furthermore,
In the case of the Sobolev space W1,p(Ω), extending a function u by zero will not necessarily yield an element of W1,p(ℝn). But for Ω bounded with Lipschitz boundary, there exists for every 1 ≤ p ≤ ∞ a bounded extension operator
such that
- Eu = u on Ω,
- Eu has compact support and
- there exists a constant c depending only on Ω and the dimension n, such that
Read more about this topic: Sobolev Space
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