Hand in y our solutions un til W ednesda y , No v em b er 15, 2.15 pm (PO b o x of y our T A in V3-128)
total points: 20
Prof. Dr. Moritz Kaßmann Fakultät für Mathematik
Wintersemester 17/18 Universität Bielefeld
Partial Differential Equations II Exercise sheet V, November 11
The aim of this exercise sheet is to prove some technical results that are related to the lecture on Friday, November 10. Every exercise is worth 10 points. You receive the maximal points (20 points) if you solve two exercises correctly.
Exercise V.1 (7 + 3 points)
Assume H is a d×dmatrix of the form H =QD, where Q is orthogonal andD is a diagonal matrix with positive entriesµ1, . . . , µd.
a) Prove that, fork∈ {0,1,2}and0< α <1, the normskukCk,α(Rd) and keukCk,α(Rd)
are equivalent, whereu:Rd→Rand ue=u◦H−1.
b) Prove that, for k∈ {0,1,2}and0< α <1, the normskukCk,α(Rd+) andkuke Ck,α(Rd+)
are equivalent, where the linear map associated withH is assumed to map Rd+ to Rd+ andu:Rd+→R,eu=u◦H−1.
Exercise V.2 (10 points)
AssumeΩ0 bΩ⊂Rd, where Ω0 andΩare open, connected. Letε >0. Show that there is c(ε)>0 such that for u∈Ck,α(Ω),k∈ {0,1,2},0≤β < α <1,
[u]Ck,β(Ω0)≤ε[u]Ck,α(Ω)+c(ε) sup
x∈Ω
|u(x)|.
Exercise V.3 (10 points)
Assume Ω⊂Rd is an open convex cone. Assume 0 ≤β ≤α < 1. Show that there is a constant c=c(d, α,Ω)>0such that for u∈Cα(Ω)
[u]Cβ(Ω) ≤cεα−β[u]Cα(Ω)+cε−βsup
x∈Ω
|u(x)|.
Exercise V.4 (10 points)
Assume Ω0 ⊂Ω⊂Rd, where Ω0 and Ω are open and convex. Assumeu ∈Ck,α(Ω)for some k∈N0,0< α <1, andu= 0 on Ω\Ω0. Prove
[u]Ck,α(Ω)= [u]Ck,α(Ω0).
Hint: You might want to consider the one-dimensional case first.
