Problem set 14, Harmonic Analysis

Due May 24th

Problem 1

The minimum of

$latex f(x,y,z) =$

$latex (a-x)^2 + (x-y)^2 + (y-a)^2 + (x-b)^2 + (b-z)^2 + (z-x)^2 + (y-z)^2 + (z-c)^2 + (c-y)^2$

is attained at

$latex \displaystyle x^* = \frac{2a+2b+c}{5},\; y^* = \frac{2a+b+2c}{5},\; z^* = \frac{a+2b+2c}{5},$

with

$latex f(x^*, y^*, z^*) = \dfrac{3}{5}\big((a-b)^2 + (b-c)^2 + (c-a)^2\big).$

Problem 2

One can obtain the values $latex u(f_2(q_1)) = x, u(f_3(q_1)) = y$ of a harmonic function in terms of the values $latex a,b,c$ at the points $latex p_2, q_1, p_3$, respectively  (as in the figure below).

Problem 3

If $latex u$ is a harmonic function with boundary values $latex u(p_1) = u(p_2) = 0$ and $latex u(p_3) = 1$, then its restriction to the bottom side of the Sierpinski triangle is an increasing function on $latex [0,1]$. (Hint: Use the previous problem.)

Problem 4

If $latex u$ is harmonic on the Sierpinski gasket $latex S$, then there exists a constant $latex c>0$ such that

$latex |u(x) - u(y)| \le c \Big(\dfrac{3}{5}\Big)^m$

if $latex x\sim y$ in $latex V_m$.

Problem 5

If $latex u$ is harmonic on the Sierpinski gasket $latex S$, then it is uniformly continuous on $latex V_*$ and thus extends to a continuous function on $latex S$.