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Restrictions of Harmonic Functions on the Sierpinski Gasket to Segments

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.


Introduction
We will first briefly recall the rudiments of harmonic analysis on the Sierpinski Gasket. ( [1], [2], [3], [4]) Let K be the Sierpinski Gasket (SG) constructed on the unit equilateral triangle G 0 with vertices {p 0 , p 1 , p 2 } and G m be the graph in the m th step as in the following figure.
Let T m be a minimal triangle with vertices v i , v j and v k in G m . The sides of T m can be ordered by the values |f

Theorem 1 ([DSV]).
Let f be a harmonic function on SG, E an edge in G m with endpoints v 0 , v 1 and midpoint v 01 . Suppose and Then the restriction of f to E is strictly increasing.

Theorem 2 ([DSV]).
The restriction of f to the two largest edges of T m is monotone. On the smallest edge of T m , the restriction of f might be monotone or not; but if it is not monotone, then it has a unique extremum.
(We changed the wording of the Theorem 2 in [DSV] slightly.) In this paper we will show that the sufficient conditions in Theorem 1 are also necessary for the monotonicity of the restriction to an adge in G m . Then we will characterize when exactly simultaneous monotonicity of restrictions to all three edges of a triangle T m in G m occurs. Furthermore, we will prove that at the junction points of any segment E in G m , the derivatives of the restriction of a harmonic function exist improperly (possibly with exception at a single point), and we will also prove that on another dense subset of E the derivatives of the restriction are zero. Now we remark that it is enough to prove these statements for the triangle G 0 instead of considering an arbitrary triangle T m in G m , because the procedure of harmonic extension is the same for G 0 or T m . In this spirit and for simplicity we now reformulate the above two theorems of DSV (Dalrymple, Strichartz and Vinson): Let f be a harmonic function on SG and assume β = f (p 1 ) < γ = f (p 2 ). Let p 12 denote the midpoint of [p 1 , p 2 ]. Then, the restriction of f to the edge [p 1 , p 2 ] is strictly increasing, if the inequalities are satisfied. (We remark that by harmonicity f (p 12 ) = 2β+2γ+α Order the edges of G 0 by the values |f Then the restrictions of f to the two largest of [p 0 , p 1 ], [p 0 , p 2 ] and [p 1 , p 2 ] are monotone. On the smallest edge, the restriction of f might be monotone or not; but if it is not monotone, then it has a unique extremum.
Before proceeding further, we will recast the inequalities i) and ii) of Theorem 1 ′ in a more sympatetic form: (Proof is straightforward. )

Characterization of the Monotonicity of the Restrictions
Consider the side [p 1 , p 2 ] = [0, 1] of G 0 and the restriction of the harmonic function f defined by (2) to [p 1 , p 2 ]. The following lemma can be proved by induction on m.
(Actually, by symmetry, one of these equalities implies the other three).
Proof of Theorem 3. If β < γ and 2β − γ ≤ α ≤ 2γ − β, then by Lemma 1, the DSV inequalities are satisfied, and consequently the restriction of f to [p 1 , p 2 ] is strictly increasing by Theorem 1 ′ . Now we show the necessity of the inequalities β < γ and 2β − γ ≤ α ≤ 2γ − β. The inequality β < γ is obvious. For all m = 1, 2, ... we have by Lemma 2 and the inequality assumption From (10) and (11) we obtain Computing the supremum and infimum gives Now we will find conditions on α, β and γ which guarantee the monotonicity of restrictions of a harmonic function f on SG to all three edges of G 0 .
These three inequalities can only be satisfied if at least one of the numbers 3α − δ, 3β − δ or 3γ − δ vanishes. This is also sufficient as easily can be seen. For example, let 3α − δ = 0. Then the second and third inequalities are obviously satisfied and the first is also true, because then 2α = β + γ and this means that α = 1 2 [(2β − γ) + (2γ − β)] and this is enough for the first inequality to be satisfied ✷ Remark 3. Interestingly, the equations 2α = β + γ, 2β = α + γ, 2γ = α + β are equivalent to the vanishing of the normal derivatives at the vertices. [BST] Remark 4. We can express the content of Theorem 4 also in terms of side lengths of G 0 (with respect to f ) : By definition, the lengths of the edges of SG are |α − β|, |α − γ|, |β − γ|. By Theorem 2 ([DSV]) we know that the restriction of f to the two largest edges are monotone. After having characterized simultaneous strictly monotonicity on all three edges in Theorem 4, we can say that this is exactly the case if and only if the G 0 is isosceles, the third side being the largest: For example, if 2α = β +γ, then |α−β| = |α−γ| and |β − γ| = 2|α − β| > |α − β|. In other words, a non-monotone restriction occurs, iff the length of the smallest edge is strictly less than the lengths of the other two edges. So, there are two cases for non-monotonicity: Either G 0 is scalene, or , if it is isosceles, then the length of the third side is strictly less than the others. In the latter case, the third side of the isosceles must have necessarily side length zero. (See Fig. 2)

Derivatives of Restrictions at the Junction Points
We will show in this section that the derivatives of the restriction of a harmonic function on SG to any segment in SG exist and are infinite at all junction points with possibly a single exception. It is again enough to show this for the edges of G 0 by general reasons we indicated in the introduction.
We first need the following  x m − x 0 is defined and tends to 0 (or ±∞) as m −→ ∞. If a < 0 then the left derivative of g at x 0 exists and is 0 (or ±∞); If a > 0 then the right derivative of g at x 0 exists and is 0 (or ±∞).
Proof. We consider only the case, where g is monotone increasing and a > 0. Let x 0 ∈ [0, 1) and x > x 0 . Then there exists m ∈ N such that As x tends to x 0 , m tends to infinity and from the inequalities we get the result ✷ Remark 5. In the above lemma, one-sided monotonicity is obviously enough for one-sided derivative calculations. Now we will compute the derivative of the restriction at the point p = 1/2, for monotone restrictions. Then by Lemma 3 the left-hand derivative at p = 1 2 is +∞. Analogously, using (9), we obtain that the right-hand derivative at p = 1 2 is also +∞ ✷ Applying Lemma 4 to smaller triangles, we see that the derivatives exist improperly at all inner junction points of [p 1 , p 2 ] in whose vicinity the restriction is strictly monotone. Using Lemma 3 and Lemma 2, (6) and (7), we can compute the derivatives at p 1 and p 2 also. The derivatives of the restriction are related to normal derivatives (For normal derivatives see [BST]). It can be seen from the work of [BST] using monotonicity of restrictions of a harmonic function f on SG, that if the normal derivative at a junction point vanishes, then the derivative of the restriction to a segment containing the junction point also vanishes at that junction point. But we have seen above, that at junction points on the contour G 0 the derivative is infinite with possibly a single exception. This proves that the normal derivative is nonzero for all junction points on the contour G 0 with at most a single exception for a non-constant f .
Remark 6. It can be shown that, if the numbers α, β, γ (being not all equal) are linearly independent over the field of rational numbers, then the normal derivative of the harmonic function on SG determined by α, β, γ is never zero on any junction point on SG. More strictly, if there does not exist a relation nα + mβ + kγ = 0 between α, β, γ (being not all equal) with n, m, k integers and n + m + k = 0, then the normal derivative is never zero on any junction point on SG.
Summarizing the above considerations, we obtain Theorem 5. The normal derivative of a non-constant harmonic function f on SG is nonzero at all junction points on G 0 ⊂ SG, with at most a single exception. This exception occurs at a vertex of G 0 , iff all restrictions of f to the edges of G 0 are monotone.

Zero Derivatives
In this section we will show that the derivative of the restriction of a harmonic function f on SG to an edge of any G m is differentiable at a point dividing the edge in ratio 1:3 and the derivative there vanishes. It is again enough to show this for the edge [p 1 , p 2 ] = [0, 1] of G 0 as the extention rule for the harmonic function is the same at every scale.