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Zentrum Mathematik
Tehnishe Universität Münhen
Prof. Dr. Bernd Shmidt
27. Otober 2009
Dr. Johannes Giannoulis
Sheet 2
Partial Dierential Equations I
Winter semester 2009/10
Exerise 5: Banah spae of harmoni funtions
Let U ⊂ Rn be open and bounded. Show that ( u ∈ C(U ) | u harmoni in U , k·k∞ )
(with kuk∞ := sup |u(x)|) is a Banah spae.
x∈U
Hint:
You may use the fat that (C(U ), k·k∞ ) is a Banah spae.
Exerise 6: Neumann boundary-value problem for the Poisson equation
Let U ⊂ Rn be open and bounded with a C 1 -boundary and outer unit normal ν , as well
as f ∈ C(U ), g ∈ C(∂U). Show that if the Poisson equation with Neumann boundary
onditions,
−∆u = f in U
,
∂ν u = g on ∂U
has a solution u ∈ C 2 (U), then
Z
f dx +
U
Z
g dS = 0.
∂U
Exerise 7: Harnak's inequality
Let U ⊂ Rn be open. Show that for every onneted V ⊂⊂ U there exists a C > 0 suh
that
sup u ≤ C inf u
V
V
for all funtions u ≥ 0, whih are harmoni in U .
Exerise 8: Faraday age
The Poisson equation of eletrostatis
ρ = −∆ϕ
(1)
desribes the relation between the eletri harge density ρ and the eletrostati potential
ϕ. Suppose U ⊂ R3 , U open with C 1 -boundary, is a bounded region, surrounded by a
onduting material ∂U . Let ρ ∈ Cc (R3 ) be a ompatly supported harge density, whih
vanishes on U , i.e. supp (ρ) ⊂ R3 \U , and let ϕ ∈ C 2 (R3 ) be a orresponding potential,
whih satises (1). Aording to the theory of eletrostatis, the potential ϕ has to be
onstant on ∂U :
ϕ|∂U = const.
Show that this implies that the eletri eld vanishes in U :
∇ϕ|U = 0.