Hand in y our solutions un til W ednesda y , No v em b er 22, 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 VI, November 16
Every exercise is worth 10 points. You receive the maximal points (20 points) if you solve two exercises correctly.
Exercise VI.1 (8 + 2 points)
(a) LetBR(x0)⊂Rdandg∈Ck,α(∂BR(x0))for some k∈ {0,1,2},0< α <1. Letλ >1.
Provide a functioneg∈Cck,α(BλR(x0))such that
eg=g on∂BR(x0).
(b) Formulate an analogous result for BR(x0) replaced by a general domainΩ⊂Rd.
Exercise VI.2 (10 points)
Assume Ω⊂Rd is a domain with ∂Ω∈Ck,α for somek∈N, 0< α <1. LetΩe be an open set with ΩbΩ. Show that there is a continuous extension operator frome Ck,α(Ω) toCck,α(eΩ), i.e., for every functionu∈Ck,α(Ω)there is a function ue∈Ck,α(eΩ)such that u=ueon Ωand
kuke Ck,α(eΩ) ≤ckukCk,α(Ω)
withc >0 is independent ofu.
Hints: First, prove the result for d = 1, Ω =R+, Ω =e R. To this end, prove that for f ∈Ck,α(R+)the extensionfe:R→R, given by
fe(x) =
f(x) for x >0,
k+1
P
i=1
λif(−x/i) for x≤0,
with appropriate λ1, . . . , λk+1 > 0, provides an extension fe ∈ Ck,α(R). Second, ex- tend this observation from R+ toRd+. Third, use a partition of unity and appropriate diffeomorphisms.
Exercise VI.3 (10 points)
Use polar coordinates to find a nontrivial function u:{(x, y)|y≥0} →Rsatisfying
∆u= 0 in{(x, y)|y >0}, u= 0 on{(x,0)|x >0},
∂
∂yu= 0 on{(x,0)|x≤0}. Determine the largest value for α >0 such thatu∈Cα(B+1).
