Hand in y our solutions un til W ednesda y , Octob er 18, 2.1 5 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 I, October 12
Here,Ωdenotes a domain inRd, i.e., an open connected subset ofRd.
Exercise I.1 (5 points)
Assumeun: Ω→Ris harmonic forn∈Nand the sequence(un)is bounded. Let K⊂Ω be any compact subdomain. Show that the sequence(un) contains a subsequence, which converges uniformly on K to a limit functionu, which itself is harmonic in Ω.
Exercise I.2 (5 points)
Recall that a function u∈C(Ω)is called subharmonic inΩif
u(x)≤ − Z
B
u (x∈Ω)
holds true for every ball B that is centered around x and satisfiesB⊂Ω. Show that this is equivalent to the following condition: For every ballB that satisfiesB ⊂Ωand every function h that is harmonic inB, the following implication holds true:
u≤h on ∂B =⇒ u≤hinB .
Exercise I.3 (5 points)
Give an example of a domainΩ⊂Rd with ∂Ω∈C1, which does not satisfy the exterior ball condition.
Exercise I.4 (5 points)
Let Ω be an arbitrary domain. Show that there exists a function u : Ω → R that is harmonic inΩ and satisfies
lim inf
x→z u(x) =−∞, lim sup
x→z
u(x) = +∞
for every pointz∈∂Ω.
Hint: You might want to consider a simple domain such as the unit ball first. More generally, you could assume that∂Ωdoes not contain isolated points. In any case, some tricky construction is needed. Try to constructu in the form
u=
∞
X
n=1
un,
whereun is singular at only one point.
