Problem 1. Let u : Ω → R be an harmonic function on an open set Ω ⊂ Rn . Show
that, for every x ∈ Ω and 0 < r < R with BR (x) ⊂ Ω, “Almgren’s frequency function”
ˆ
r
|∇u(y)|2 dy
B (x)
I(x, r) = ˆ r
(u(y))2 dS(y)
∂Br (x)
is monotone in r.
Problem 2. Assume u : Rn → Rn is a C 2 map such that its differential du is an
orthogonal transformation at every point. Prove that u is affine.
Problem 3. Consider the initial value problems
(
∂t u − ∂x2 u = 0 ,
u(0, x) = u0 (x) ,
(0.1)
and
(
(??)
∂t u − ∂x2 u = u2 ,
u(0, x) = u0 (x) .
(0.2)
Show that
• For every C 2 periodic u0 there is a periodic solution u ∈ C 2 ([0, ∞) × R) of
(0.1).
• There is a C 2 periodic u0 for which there are no periodic solutions u ∈ C 2 ([0, ∞)×
R) of (0.2).
Problem 4. Give a continuous function f : R → R which is nowhere differentiable.
Problem 5. Give a continuous function f : [−π, π] → R whose Fourier series diverges
at a point x0 ∈] − π, π[.
Problem 6. Let Ω+ be an open subset of the halfspace R2 + = {(x1 , x2 ) ∈ R2 : x2 > 0}
with the property that Γ = Ω+ ∩ {x2 = 0} is an interval. Let u be an harmonic function
on Ω+ which extends continuously to Ω+ ∪ Γ if we set u = 0 on Γ. Prove that the
following function

u(x , x ) ,
für x2 ≥ 0,
1
2
U (x1 , x2 ) =
−u(x1 , −x2 ) ,
für x2 < 0,
is harmnoic on Ω+ ∪ Γ ∪ Ω− , where Ω− denotes the reflection of Ω+ throguh x2 = 0,
i.e. Ω− = {(x1 , x2 ) ∈ R2 : (x1 , −x2 ) ∈ Ω+ }.
1
2
Problem 7. Let Ω be a bounded open subset of R2 . For every open Ω0 ⊂⊂ Ω there is
a constant C, depending only on Ω0 and Ω, such that the following inequality holds for
every positive harmonic function on Ω:
sup u ≤ C inf0 u .
Ω
Ω0
Problem 8. Let S be a formal power series with
S(0) = 0
und
S 0 (0) 6= 0 ,
and denote by T its formal inverse, i.e. the formal power series such that
T (0) = 0
und
S ◦ T = Id .
Prove that, if the convergence radius of S is positive, so is the convergence radius of T .
Problem 9. Let D1 = x ∈ R2 |x| ≤ 1 . Prove that every continuous map h : D1 →
D1 has necessarily a fixed point (i.e. there is x ∈ D1 with h(x) = x).
Problem 10. Let A ∈ Cn×n be a given complex matrix and σ ⊂ C the set of its
eigenvalues. Let C be a contour C, Ω the domain enclosed by C and assume that
C ∩ σ = ∅. Consider the following matrix
˛
1
(ζ id −A)−1 dζ.
(0.3)
P =
2πi C
Prove that
(1) P is a projection, i.e. P P = P ;
(2) the subspace V = P Cn is A-invariant, i.e. AP = P A;
(3) if v is an eigenvector of A with eigenvalue λ, then

v , if λ ∈ Ω,
Pv =
0 , if λ ∈
/ Ω.