Hand in y our solutions un til W ednesda y , Octob er 27, 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 II, October 19
We have seen that the exterior ball condition for a domain implies the existence of a barrier at every point of the boundary. The first two exercises deal with barriers for domains with non-smooth boundaries.
Exercise II.1 (5 points)
Givenλ >0, define a coneV ⊂Rd byV ={(x, h)|x∈Rd−1, h >0,|x| ≤λh}.
SetΩ =B1(0)\V. Defineg:∂Ω→Rby g(x) =|x|andw: Ω→Rby
w(x) = sup{v(x)|v∈Sg}, where Sg is the set of allv∈C(Ω) that are subharmonic inΩ withv≤g on∂Ω. Show thatb=−w is a barrier forΩat 0.
Exercise II.2 (5 points)
Using the previous exercise, formulate and prove an existence result for the Dirichlet problem in bounded domains, which are as general as possible.
The next exercise deals with the Dirichlet problem in an unbounded domain.
Exercise II.3 (5 points)
Assume d≥ 3. Let Ω be a bounded domain with a smooth boundary and g ∈C(∂Ω).
Prove that there exists a unique solutionu∈C∞(Rd\Ω)∩C(Rd\Ω) of the following problem
∆u= 0 inRd\Ω, u=g on ∂Ω,
|x|→∞lim u(x) = 0 .
Show that the functionu satisfies
|u(x)| ≤c|x|2−d (x∈Rd)
for some constantc≥1.
Hint: Approach the question of uniqueness and existence separately. Study auxiliary problems in domains of the formBR(0)\Ωfor large radii R.1
1If you have a lot of energy, you may want to formulate and prove an analogous result ford= 2.
The aim of the next exercise is to recall Hölder spaces.
Exercise II.4 (5 points)
Assume 0< α≤1. Given a function C(Rd), define
kfk1 = sup
x∈R|f(x)|+ sup
x,h∈R
f(x+h)−f(x)
|h|α
, kfk2 = sup
x∈R|f(x)|+ sup
x,h∈R
f(x+h)−2f(x)+f(x−h)
|h|α
.
Note that these quantities are not finite in general.
(a) Prove that for everyα∈(0,1)there is a constantc≥1such that
c−1kfk2 ≤ kfk1 ≤ckfk2 (f ∈Cc∞(Rd)). (b) Discuss the same question forα = 1.
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