Gradients of Laplacian Eigenfunctions on the Sierpinski Gasket

We use spectral decimation to provide formulae for computing the harmonic gradients of Laplacian eigenfunctions on the Sierpinski Gasket. These formulae are given in terms of special functions that are defined as infinite products.


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There are few functions more ubiquitous in Euclidean analysis than the sine, cosine and exponential, which are the eigenfunctions of the Laplacian on an interval in R. In the theory of analysis on fractals, the Laplacian eigenfunctions arguably have an even more prominent role, as the Laplacian is the fundamental differential operator on which the analysis is based. Despite this, there are a number of interesting open questions about the structure of such eigenfunctions. In this paper we consider the local behavior of Laplacian eigenfunctions on the Sierpinski Gasket (S G), in terms of the harmonic tangents and gradients introduced by Teplyaev in [6]. Using the spectral decimation method of Fukashima and Shima [1] (see also Chapter 3 of [5]) we give infinite product formulae for the tangents at boundary points, and use them to describe the one-sided tangents at junction points. These results may be seen as a Sierpinski Gasket version of the well known formulae for the derivatives of the sine and cosine functions on an interval, though the precise analogue on [0, 1] is more complicated (see . The Sierpinski Gasket is the simplest non-trivial example of a fractal to which the standard theory of analysis on fractals applies. We refer to the monographs [2,5] for detailed proofs of all results we use from this theory. Recall that S G ⊂ R 2 is the attractor of an iterated function system consisting of three maps F i (x) = (x+q i )/2, where the points q 0 , q 1 and q 2 are the vertices of an equilateral triangle. This means that S G = ∪ i F i (S G), where the sets F i (S G) are usually referred to as 1-cells. For a length m word w = w 1 . . . w m with letters w j ∈ {0, 1, 2} we define F w = F w 1 • · · · • F w m and call F w (S G) an m-cell. The points q i , i = 0, 1, 2 are the boundary of S G; the set of boundary points is V 0 , and we use V m to denote points of the form F w (q i ) where w is a word of length m. These V m are vertices of the usual graph approximation of S G at scale m, in which vertices x and y are joined by an edge (written x ∼ m y) if they belong to a common m-cell. Clearly V * = ∪ m V m is dense in S G. Date The Laplacian ∆ on S G is a renormalized limit of graph Laplacians ∆ m on the m-scale graphs: and we say u ∈ dom(∆) if the right side of (1.1) converges uniformly on V * \ V 0 to a continuous function. The function is extended to S G by continuity and density of V * . At a boundary point q i ∈ V 0 there is an associated normal derivative defined (with q i+3 = q i ) by A harmonic function h is one for which ∆h = 0, and for any assignment of values on V 0 there is a unique harmonic function with these boundary values. For this reason we identify the harmonic functions with the space of functions on V 0 . The harmonic functions are also graph harmonic, so it is elementary to compute the values of h on V 1 from those on V 0 , and recursively to obtain the values on V m for any m. It will be useful to formalize this by defining the harmonic extension matrices A i , which map the values of h on V 0 to those on F i (V 0 ), by We usually write this in the compact form The matrices are The structure of eigenfunctions of the Laplacian is similar to that of the harmonic functions. Specifically, it is true on S G that if m is sufficiently large then the restriction of a function u satisfying −∆u = λu from S G to V m gives an eigenfunction of the graph Laplacian −∆ m u = λ m u, with , but the positive root is only permitted to occur for finitely many values of m in order that the limit in (1.4) exists. This spectral decimation property was first recognized by Fukashima and Shima [1]. It is not true on all fractals, but on those where it is valid, it gives both a method for computing the spectrum and a recursion for the eigenfunctions [3]. Let us define provided λ 2, 5. The essence of the spectral decimation method on S G may then be summarized in the following theorem, which we have taken from Sections 3.2 and 3.3 of [5].
and we call this the spectral decimation relation. If λ is not a Dirichlet eigenvalue then we may assume m 0 = 0, at which point the condition (∆ m 0 + λ m 0 )u V m 0 = 0 is taken to be vacuous. The corresponding eigenspace is 3-dimensional and parametrized by the values of u on V 0 .
If λ is Dirichlet several possibilities occur. We indicate the initial configurations, all of which may then be continued by the spectral decimation formula. A spanning set for the configurations when m 0 = 1 are shown in Figure 1. The one on the left has λ 1 = 2 while those on the right have λ 1 = 5. If m 0 ≥ 2 then λ m 0 = 5 or λ m 0 = 6, and in the latter case λ m 0 +1 = 3. Those with λ m 0 = 5 are formed from scaled and rotated copies of the functions on the right in Figure 1, arranged so that their normal derivatives cancel. A basis of chains for m 0 = 2 is shown in Figure 2; those for general m are naturally indexed by the loops in V m , plus two strands connecting points of V 0 . In the case λ m 0 = 6 the eigenfunctions are indexed by points in V m 0 −1 \ V 0 ; a basis is obtained by scaling and rotating two copies of the function on the left in Figure 3 and gluing them at the chosen point, as shown on the right in Figure 3 for the case m 0 = 2 and a point in V 1 . T  T   T  T    T  T    T  T   T  T    T  T    T  T   T  T    T T  T   T  T    T  T    T  T   T  T    T  T    T  T   T  T    T T  T   T  T    T  T    T  T   T  T    T  T    T  T   T  T    T T   T  T    T  T    T  T   T  T    T  T    T  T   T  T    T  T   2 Define the harmonic tangent T w u of u at w to be lim m→∞ H [w] m u, if the limit exists. It should be remarked that there can be two words w and w ′ such that F w (S G) = F w ′ (S G). We will nonetheless treat the tangents T w and T w ′ separately, as they are rarely equal. The harmonic gradient is defined in the same way, but using the space of harmonic functions with average zero and the projection of the action of the matrices A i to this subspace. It is evident that if u is continuous then the gradient exists whenever the tangent exists, and conversely. See [4] for details.
If we consider the interval [0, 1] rather than the Sierpinski Gasket then it is clear that the harmonic tangent at x 0 is the vector L(0), L(1) T characterizing the unique linear function L(x) having the properties L(x 0 ) = f (x 0 ) and L ′ (x 0 ) = f ′ (x 0 ). For an eigenvalue λ ∈ R which is not equal to π 2 k 2 for any k ∈ N, it may readily be verified that the harmonic tangent is given by We cannot obtain as explicit a description of the harmonic tangent on SG; however, we produce formulae that permit its computation at any point of V * . The key observation is that when u is an eigenfunction, the computation of the gradient has a particularly elegant structure. Recall from Theorem 1.1 that the values of u on F [w] m (V 0 ) may be computed using the spectral decimation method, meaning that starting from a scale m 0 they can be obtained as in (1.8). Combining this with (2.1) we see that in which we know the limit exists by Theorem 3 of [6]. A special case occurs when F w (S G) is a point in V * , because in this case, all but finitely many letters in the word w are equal to a single letter i. By taking m 0 to be sufficiently large we see that it is useful to understand the limit and it is evident from the symmetry of the matrices A i and A i (λ) that it suffices to deal with the case i = 0.
where we used the previously computed limit for the middle factor, and the fact that For γ m the situation is a little different. Observe that however we can perform the following simplification from (1.3): Inserting this into the previous computation shows that A 0 (λ m+1 )γ m = γ m+1 provided λ m 2, 5, and therefore A 0 (λ m 0 +k ) · · · A 0 (λ m 0 +1 )γ m 0 = γ m 0 +k . To proceed we must apply A −k 0 to γ m 0 +k , which is most easily done by writing it in terms of eigenvectors as γ m 0 +k = (4, 4, 4) T − λ m 0 +k α. The result is and Proof. From (2.5) we have because w j = 0 for j ≥ k and spectral decimation applies for j ≥ m 0 . The result is therefore equivalent to which follows from Theorem 2.2.
Remark 2.4. The function τ k (λ) may appear to depend on the sequence {λ j }, but in fact this sequence is uniquely determined by λ. Indeed, there is an entire analytic function Ψ(z) with the property that λ j = Ψ(5 − j λ). To see this, let ψ(z) = z(5−z) and ψ m (z) = ψ •m ( 2 3 5 −m z). The sequence ψ m (z) consists of entire functions with ψ m (0) = 0 and ψ ′ m (0) = 2 3 , so is normal with limit Ψ(z), a power series for which may be computed recursively. It is then clear that In the same way that there are special functions associated to differential equations in Euclidean analysis, we suggest that the functions Ψ(λ) and Υ(λ) should be considered to be special functions in analysis on the Sierpinski Gasket. In terms of these functions, (2.7) has the form As a particular consequence we may compute the normal derivatives of the eigenfunctions at points of V 0 , because they are the same as the normal derivatives of the tangent functions. We expect this observation to have applications in the construction of a resolvent for the Laplacian. Corollary 2.5. If (∆ + λ)u = 0 and the spectral decimation formula holds with λ m 2, 5, 6 for m > 0, then the normal derivative of u at q 0 is ∂ n u(q 0 ) = (4 − λ 0 )u(q 0 ) − 2u(q 1 ) − 2u(q 2 ) 2λΥ(λ) 3λ 0 .
Excluding the values 2,5 and 6 from the sequence {λ m } in Theorems 2.2 and 2.3 is necessary because they occur precisely in the Dirichlet case, where the boundary data vanishes and cannot be used to determine the tangent. Nonetheless, Theorem 2.3 may be applied to find tangents to Dirichlet eigenfunctions in a simple fashion. The reason is that the description of the Dirichlet eigenfunctions given in Theorem 1.1 ensures that we need only compute the harmonic tangents of the functions in Figure 1 and the left of Figure 3. All harmonic tangents to Dirichlet eigenfunctions are then obtained from these by scaling and taking suitable linear combinations.
For the basic element used to construct the 6-series (Figure 3) we can directly apply Theorem 2.3 with λ 1 = 6 and λ 2 = 3. For example, if the top vertex in Figure 3 is q 0 and w = 0 · · · we have To calculate the harmonic tangents of the basic element of the 2-series (on the left in Figure 1) we apply Theorem 2.3 to the function shown at left in Figure 4, starting the spectral decimation at each of the values λ 1 = 5±