Some Techniques in W^{1,p}(Ω)

Daniel Spector

March 16, 2009

Abstract

For Ω ⊂ R^{N} open, the Sobolev spaces W^{1,p}(Ω) are defined by

W^{1,p}(Ω) := {u ∈ L^{p}(Ω) : ∀ i = 1 to N ∃ v_{i} ∈ L^{p}(Ω) s.t.

∫_{Ω} u\frac{∂ϕ}{∂x_{i}} dx = − ∫_{Ω} v_{i}ϕdx ∀ ϕ ∈ C_{c}^{∞}(Ω)\}

The Sobolev spaces W^{1,p}(Ω) can be equipped with a norm

||u||_{W^{1,p}(Ω)} = ||u||_{L^{p}(Ω)} + ∑_{i=1}^{N} ||v_{i}||_{L^{p}(Ω)}

the makes them complete, and for Ω sufficiently regular with 1 < p < ∞ are reflexive and separable.

This weaker notion of differentiability has applications to partial differential equations, where existence is easier to prove in a weak sense. Of course, then uniqueness and regularity of the solution then need to be demonstrated. We will examine a weak formulation of

−Δu + u = f in Ω
u = 0 on ∂Ω

where f is a fixed function defined in Ω, and use it to discuss some of the basic ideas of Sobolev spaces. Finally, we will use some of the techniques in Sobolev spaces (density results, mollification, Fubini and the Fundamental theorem) to prove the embedding of W^{1,p}(Ω) into the space of Holder continuous functions for p > N.