Chapter 45

Traces Of Sobolev Spaces45.1 Traces Of Sobolev Spaces, Half Space

In this section consider the trace of W m,p(Rn+

)onto a Sobolev space of functions defined

on Rn−1. This latter Sobolev space will be defined in terms of the following theory insuch a way that the trace map is continuous. The trace map is continuous as a map fromW m,p

(Rn+

)to W m−1,p

(Rn−1

)but here I will give a better conclusion using the above the-

ory.

Definition 45.1.1 Let θ ∈ (0,1) and let Ω be an open subset of Rm. We define

W θ ,p (Ω)≡ T(W 1,p (Ω) ,Lp (Ω) , p,1−θ

).

Thus, from the above general theory, W 1,p (Ω) ↪→ W θ ,p (Ω) ↪→ Lp (Ω) = Lp (Ω) +W 1,p (Ω) . Now we consider the trace map for Sobolev space.

Lemma 45.1.2 Let φ ∈C∞(Rn+

). Then γφ (x′)≡ φ (x′,0) . Then γ : C∞

(Rn+

)→ Lp

(Rn−1

)is continuous as a map from W 1,p

(Rn+

)to Lp

(Rn−1

).

Proof: We know

φ(x′,xn

)= γφ

(x′)+∫ xn

0

∂φ (x′, t)∂ t

dt

Then by Jensen’s inequality,∫Rn−1

∣∣γφ(x′)∣∣p dx′

=∫ 1

0

∫Rn−1

∣∣γφ(x′)∣∣p dx′dxn

≤ C∫ 1

0

∫Rn−1

∣∣φ (x′,xn)∣∣p dx′dxn

+C∫ 1

0

∫Rn−1

∣∣∣∣∫ xn

0

∂φ (x′, t)∂ t

dt∣∣∣∣p dx′dxn

≤ C ||φ ||p0,p,Rn++C

∫ 1

0xp−1

n

∫Rn−1

∫ xn

0

∣∣∣∣∂φ (x′, t)∂ t

∣∣∣∣p dtdx′dxn

≤ C ||φ ||p0,p,Rn++C

∫ 1

0xp−1

n

∫Rn−1

∫∞

0

∣∣∣∣∂φ (x′, t)∂ t

∣∣∣∣p dtdx′dxn

≤ C ||φ ||p0,p,Rn++

Cp

∫Rn−1

∫∞

0

∣∣∣∣∂φ (x′, t)∂ t

∣∣∣∣p dtdx′

≤ C ||φ ||p1,p,Rn+

This proves the lemma.

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