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Published online by Cambridge University Press: 18 December 2009
Sobolev Spaces
Sobolev spaces are, roughly speaking, spaces of p-integrable functions whose derivatives are also p-integrable. There are two basic types of Sobolev spaces we wish to consider. In the first situation, we have a bounded domain Ω ⊂ ℝN and we consider functions u : Ω → ℝ for which u ≡ 0 on ∂Ω. In the second, we look at functions u : [0, T] → ℝN satisfying u(0) = u(T). The only problem in straightforwardly defining these spaces is that p-integrable functions need not be differentiate and restricting our attention to those that are differentiate does not provide us with what we want — the resulting spaces are not complete. We must weaken our notion of differentiability.
Definition A.1. In the following two cases we define an appropriate notion of weak differentiability.
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