Math 554
Homework 3
Due October 16, 2008, before class
Problem I 1. State and prove a trace operator theorem of the type
T:Wm,p(U)7→ W1,p(∂ U).
Try to minimize mand the assumptions on the open set U⊂Rn.
2. Consider the trace operator:
T:W1,2(U)7→ L2(∂U ),
defined in lecture 7 for U⊂Rnopen, with a bounded, piecewise
C1boundary satisfying the segment property and such that any
intersection of two C1pieces is transversal. Show that if g:∂U 7→
Ris continuous and piecewise C1on the boundary then there
exists a f∈W1,2(U) such that T f =g. (suggested) Show that it
suffices to have g∈C0,α(∂U ), α > 1+√5
4,for the above conclusion
to still hold. This means that gis continuous on the boundary
and there exists a constant K > such that for any x, y on the same
C1piece of the boundary we have: |g(x)−g(y)| ≤ K|x−y|α.
Problem II Prove that an extension operator E:Wm,p(U)7→ Wm,p(Rn)
exists for any m∈Nand 1 ≤p < ∞when :
1. n= 2 and U={(x, y)∈R2|0< y < 1};
2. n= 3 and U={x, y, z )∈R3|x > 0, y > 0, z > 0}.
Problem III H¨older spaces. Here Uis an open set in Rn.
kgkC0,γ(¯
U)= sup
x∈¯
U
|g(x)|+ sup
x,y∈¯
U,x6=y
|g(x)−g(y)|
|x−y|γ
C0,γ(¯
U) = {g∈C(U)| kgkC0,γ(¯
U)<∞}.
Show that (C0,γ(¯
U),kgkC0,γ(¯
U)) is a Banach space. (Suggested) Gen-
eralize to Cj,γ(¯
U), j = 0,1,....