Hand in y our solutions un til W ednesda y , No v em b er 1 , 2.15 p m (PO b o x of y our T A in V3-128)

Hand in y our solutions un til W ednesda y , No v em b er 1 , 2.15 p m (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 III, October 25

The aim of this exercise sheet is to get useful insights into Hölder spaces.

Exercise III.1 (5 points)

Show that the Banach spacesC^α([a, b]),0< α <1, and C^0,1([a, b]) are not separable.

Exercise III.2 (5 points)

Let us begin with some definitions.

(1) Givenk∈N0, u∈C^k(R^d), andy∈R^d, define the Taylor-Polynomial T_y^ku:R^d→R as follows:

T_y^ku(x) = X

|γ|≤k

1

γ!∂^γu(y)(x−y)^γ.

(2) LetP_k denote the set of all polynomialsR^d→Rof degree less or equal to k.

(3) For a ball B_r(x₀)⊂R^d, a functionu:R^d→Randk∈N0, define Ek(u;Br(x0)) = inf

p∈P_k sup

x∈Br(x0)

|u(x)−p(x)|.

Assume 0< α≤1 andk∈N0. Show that the quantity [u]⁰_Ck,α = sup

r>0,x0∈R^d

r^−k−αE_k(u;B_r(x₀))

is comparable to the seminorm[u]_C^k,α for sufficiently regular functions u.

Exercise III.3 (5 points)

For bounded domains Ωwith a nice boundary, the inclusion

C^m,α(Ω)⊂C^k,β(Ω) (1)

holds true ifm+α > k+β. This exercise shows that this natural inclusion might fail if the domain has a ”bad“ boundary. Let Ω⊂R² be defined by

Ω ={(x, y)∈R²|y <p

|x|, x²+y²<1}. Define a function u: Ω→Rby

u(x, y) =

(sgn(x)y^γ if y >0,

0 if y≤0,

for some γ∈(1,2). Use this function in order to show that (1) fails in general.

Exercise III.4 (5 points)

Given f ∈ C^α(Ω), where Ω ⊂ R^d is some domain, we have seen that solutions u to the equation ∆u = f in Ω satisfy u ∈ C^2,α(Ω₀) for Ω₀ b Ω¹. This is called interior regularity. Let us prove a simple first result on boundary regularity. Recall the definition M⁺=M∩R^d₊={x∈M|x_d>0} for any setM ⊂R^d.

Let’s writeBr instead ofBr(0)for r >0. Assumef ∈C^α(B₂⁺) for someα ∈(0,1). Show

N_B⁺

2 f ∈C^2,α(B₁⁺), [D²N_B+

2f]_Cα(B₁⁺)≤c[f]_Cα(B⁺₂), wherec is some positive constant independent of f.

1Note that this implication is false forα= 0, cf. Exercise XII.3 from the first PDE lecture last term.

2