Problem Set 11

Park City Mathematics Institute
Undergraduate Summer School 2018

Introduction to Harmonic Analysis

  1. Prove by induction, for the interval case, that
    \min\{ \mathscr E_m(u): u|_{\{0,1\}}=v\} = \mathscr E_0(v) = (v(0)-v(1))^2,
    with the minimizer satisfying
    \displaystyle u\Big(\frac{2k+1}{2^m}\Big) = \frac{1}{2}\Big(u\Big(\frac{k}{2^{m-1}}\Big) + u\Big(\frac{k+1}{2^{m-1}}\Big)\Big).
  2. The minimum of
    f(x,y,z) = (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)^2MinAlg.png
    is attained at
    \displaystyle x^* = \frac{2a+2b+c}{5},\; y^* = \frac{2a+b+2c}{5},\; z^* = \frac{a+2b+2c}{5},
    with f(x^*, y^*, z^*) = \dfrac{3}{5}\big((a-b)^2 + (b-c)^2 + (c-a)^2\big).
  3. Prove that one can obtain the values u(F_2(q_1)) = x, u(F_3(q_1)) = y of a harmonic function in terms of the values a,b,c at the points p_2, q_1, p_3, respectively  (as in the figure below).ResAlg.png
  4. Use the previous problem to show that, if u is a harmonic function with boundary values u(p_1) = u(p_2) = 0 and u(p_3) = 1, then its restriction to the bottom side of the Sierpinski triangle is increasing.
  5. If u is harmonic, then it is uniformly continuous (use the fact |u(x) - u(y)| \le c \Big(\dfrac{3}{5}\Big)^m if x\sim y in V_m.)
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