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