Park City Mathematics Institute
Undergraduate Summer School 2018
Introduction to Harmonic Analysis
- Prove by induction, for the interval case, that
with the minimizer satisfying
- The minimum of
is attained at
- Prove that one can obtain the values of a harmonic function in terms of the values at the points , respectively (as in the figure below).
- Use the previous problem to show that, if is a harmonic function with boundary values and , then its restriction to the bottom side of the Sierpinski triangle is increasing.
- If is harmonic, then it is uniformly continuous (use the fact if in .)
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