Math 509, Spring 2007 Jerry L. Kazdan
Math 509: Problem Set 10 (due Thurs. April 12, 2007)
1. Let u(θ,t)be the temperature at a point θon the circle, S1= [−π,π]at time tand assume that
u(θ,t)satisfies the heat equation ∂u
∂t=∂2u
∂θ2
and is of course periodic in θwith period 2π. If the initial temperature is u(θ,0) = f(θ)∈
C(S1), show that
lim
t→∞u(θ,t) = constant
and determine this constant in terms of f.
2. Solve the Laplace equation ∆u=0 on the outside of the unit disk, so r=px2+y2>1 with
u(r,θ)|r=1=2+cosθ−3 sinθon the unit circle (here θis the angle in polar coordinates). In
you solution, assume that u(r,θ)is bounded as r→∞.
3. Suppose that uis a harmonic function in the disk D={x2+y2<4}with boundary condition
(in polar coordinates) u(2,θ) = 5+2cos 3θ. Without computing the solution:
a) Find the maximum value of the solution in D.
b) Find the minimum value of the solution in D.
c) Find the value of uat the origin.
4. Let uand vbe harmonic functions in a bounded (connected) region Dwith u=fand v=gon
the boundary of D. If f≤g, show that u≤vwith strict inequality everywhere unless f≡g,
5. Say ∆u≥0 in a bounded (connected) region D.
a) Prove that in any disk Q∈Dthe value of uat the center of the disk is at most the average
of its value on the boundary of the disk.
b) Show that if utakes its maximum value at an interior point of Dthen umust be a constant.
c) Assume ∆v=Fand ∆w=Gin a bounded (connected) region Dwith v=0 and w=0
on the boundary of D. If F>Gin D, show that v<win D.
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