Traces
Sobolev spaces are often considered when investigating partial differential equations. It is essential to consider boundary values of Sobolev functions. If u ∈ C(Ω), those boundary values are described by the restriction . However, it is not clear how to describe values at the boundary for u ∈ Wk,p(Ω), as the n-dimensional measure of the boundary is zero. The following theorem resolves the problem:
- Trace Theorem. Assume Ω is bounded with Lipschitz boundary. Then there exists a bounded linear operator such that
- and
Tu is called the trace of u. Roughly speaking, this theorem extends the restriction operator to the Sobolev space W1,p(Ω) for well-behaved Ω. Note that the trace operator T is in general not surjective, but maps for p ∈ (1,∞) onto the Sobolev-Slobodeckij space .
Intuitively, taking the trace costs 1/p of a derivative. The functions u in W1,p(Ω) with zero trace, i.e. Tu = 0, can be characterized by the equality
where
In other words, for Ω bounded with Lipschitz boundary, trace-zero functions in W1,p(Ω) can be approximated by smooth functions with compact support.
Read more about this topic: Sobolev Space
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