Restrictions of Harmonic Functions on the Sierpinski Gasket to Segments

Abstract

The restrictions of a harmonic function on the Sierpinski Gasket (SG) to the segments in SG have been of some interest. We show that the sufficient conditions for the monotonicity of these restrictions given by Dalrymple, Strichartz and Vinson are also necessary. We then prove that the normal derivative of a harmonic function on SG on the junction points of the contour of a triangle in SG is always nonzero with at most a single exception. We finally give an explicit derivative computation for the restriction of a harmonic function on SG to segments at specific points of the segments: The derivative is zero at points dividing the segment in ratio 1:3. This shows that the restriction of a harmonic function to a segment of SG has the following curious property: The restriction has infinite derivatives on a dense set of the segment (at junction points) and vanishing derivatives on another dense set.