PDES II, Math 8142 (old 562)
Prof. Guti´errez Homework 2, Dirichlet integral (Due on 2/24/09)
1. Prove that the Laplacian in polar coordinates is given by
urr +1
rur+1
r2uθθ.
2. Let Ωbe the unit disc given in polar coordinates r, θ, and suppose u(x,y)
is harmonic in Ωwith continuous boundary values f(θ) and assume u∈
C2(Ω)∩C(¯
Ω).
1. Let vbe defined by
v(r, θ)=a0
2+
∞
X
k=1
rk(akcos kθ+bksin kθ),
where ak,bkare the Fourier coefficients of f, that is,
ak=1
πZπ
−π
f(θ) cos kθdθ, bk=1
πZπ
−π
f(θ) sin kθdθ,
k=0,1,···.
Prove that vis well defined for all 0 <r<1 and all θand is a C2function
for r<1 and is harmonic for r<1.
2. Prove that
v(r, θ)=Z2π
0
f(φ)P(r, θ −φ)dφ, r<1,
where P(r, ξ)=1
2π1+2P∞
k=1rkcos kξ. HINT: insert the definition of
ak,bkin the series and interchange the sum and the integral (justify).
3. Prove using complex exponentials that
P(r, ξ)=1
2π
1−r2
1+r2−2rcos ξ,r<1.
Notice this is the Poisson kernel for the disc defined in class written in
polar coordinates.
4. The function vhas boundary values f. Therefore u(rcos θ, rsin θ)=
v(r, θ).
5. Prove that the Dirichlet integral of uwritten in polar coordinates is
D(u)=Z2π
0Z1
0
(v2
θ+r2v2
r)1
rdrdθ.
HINT: write the gradient of uin polar coordinates.
