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Harmonic Functions and the Spectrum of the Laplacian

• n the Sierpinski Carpet

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz October 3, 2010

Construction of SC

The Sierpinski Carpet, SC, is constructed by eight contraction mappings. The maps contract the unit square by a factor of 1/3 and translate to one

• f the eight points along the boundary. SC is the unique nonempty

compact set satisfying the self-similar identity SC =

7

• i=0

Fi(SC). F0 F1 F2 F3 F4 F5 F6 F7

• Matthew Begu´

e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 2 / 23

Constructing the Laplacian

The Laplacian ∆ on SC was independently constructed by Barlow & Bass (1989) and Kusuoka & Zhou (1992). In 2009, Barlow, Bass, Kumagai, & Teplyaev showed that both methods construct the same unique Laplacian on SC. We will be following Kusuoka & Zhou’s approach in which we consider average values of a function on any level m-cell. We approximate the Laplacian on the carpet by calculating the graph Laplacian on the approximation graphs where verticies of the graph are cells of level m:

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 3 / 23

Constructing the Laplacian

∆mu(x) =

• y∼x

m

(u(y) − u(x)).

• ⑦

⑦

a b c d

• ⑦

⑦

x ˜ x ˜ x′ y z For example the graph Laplacian of interior cell a is ∆mu(a) = −3u(a) + u(b) + u(c) + u(d). For boundary cells, we include its neighboring virtual cells. eg: ∆mu(x) = −4u(x) + u(y) + u(z) + u(˜ x) + u( ˜ x′).

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 4 / 23

Construction of the Laplacian

The Laplacian on the whole carpet is the limit of the approximating graph Laplacians ∆ = lim

m→∞ r−m∆m

where r is the renormalization constant r = (8ρ)−1. So far, ρ has only been determined experimentally. ρ ≈ 1.251 and therefore 1/r ≈ 10.011.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 5 / 23

Harmonic Functions

A harmonic function, h, minimizes the graph energy given a function defined along the boundary as well as satisfying ∆h(x) = 0 for all interior cells x. The boundary of SC is defined to be the unit square containing all of SC. Example: Set three edges of the boundary of SC to 0 and assign sin πx along the remaining edge and extend harmonically. sin πx

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 6 / 23

More Harmonic Functions

sin 2πx sin 3πx

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 7 / 23

Boundary Value Problems

We also wish to solve the eigenvalue problem on the Sierpinski Carpet: −∆u = λu We have two types of boundary value problems: Neumann ∂nu |∂SC= 0 Corresponds to even reflections about boundary. ie: ˜ x = x Dirichlet u|∂SC = 0 Corresponds to odd reflections about the boundary. ie: ˜ x = −x

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 8 / 23

∆mu(x) =

• y∼x

m

(u(y) − u(x))

• ⑦

⑦

a b c d

• ⑦

⑦

x ˜ x ˜ x′ y z Therefore the Laplacian operator is determined by 8m linear equations. This can be represented in an 8m square matrix. The matrix is created in MATLAB and the eigenvalues and eigenfunctions are calculated using the built-in eigs function.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 9 / 23

Some eigenfunctions

Neumann: ∂nu|∂SC = 0 Dirichlet: u|∂SC = 0

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 10 / 23

Refinement

On level m + 1 we expect to see all 8m eigenfunctions from level m but refined. The eigenvalue is renormalized by r = 10.011.

Figure: φ(4)

5

and φ(5)

5

with respective eigenvalues λ(4)

5

= 0.00328 and λ(4)

5

= 0.000328.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 11 / 23

Miniaturization

Any level m eigenfunction and eigenvalue miniaturizes on the level m + 1

• carpet. It will consist of 8 copies of φ(4) or −φ(4).

Figure: φ(4)

4

and φ(5)

20 with respective eigenvalues λ(4) 4

= λ(5)

20 = 0.00177.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 12 / 23

Describing the eigenvalue data

Eigenvalue counting function: N(t) = #{λ : λ ≤ t} N(t) is the number of eigenvalues less than or equal to t. Describes the spectrum of eigenvalues. We expect the N(t) to asymptotically grow like tα as t → ∞ where α = log 8/ log 10.011 ≈ 0.9026.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 13 / 23

N(t) = #{λ : λ ≤ t}

Weyl Ratio: W (t) = N(t)

tα

α ≈ 0.9

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 15 / 23

Neumann vs. Dirichlet eigenvalues

By the min-max property, we can say that λ(N)

j

≤ λ(D)

j

for each j. Therefore, N(D)(t) ≤ N(N)(t). What is the growth rate of N(N)(t) − N(D)(t). We suspect there is some power β such that N(N)(t) − N(D)(t) ∼ tβ. β ≈

log 3 log 10.011 = 0.4769

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 16 / 23

N(N)(t) − N(D)(t)

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 17 / 23

N(N)(t)−N(D)(t) tβ

A stronger periodicity is apparent here.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 18 / 23

Fractafolds

We can eliminate the boundary of SC by gluing its boundary in specific

• rientations. We examined three types of SC fractafolds:

❅ ❅

• ❅

❅

• ❅

❅ ❅ ❅

• ❅

❅

• ❅

❅

Torus Klein Bottle Projective Space

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 19 / 23

Some Eigenfunctions for the Fractafolds

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 20 / 23

How to define the normal derivative on the boundary of SC

We wish to define ∂nu on ∂SC so that the Gauss-Green formula holds: E(u, v) = −

• SC

(∆u)v dµ +

• ∂SC

(∂nu)v dµ′. We know that Em(u, v) = 1 ρm

• x∼y

(u(x) − u(y))(v(x) − v(y)) and −∆mu(x) = 8m ρm

• x∼y

(u(x) − u(y)).

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 21 / 23

Sketch of how ∂nu is defined

Let x be a point on ∂SC and xm be the m-cell containing x. We can use the equations from the previous slide to find ∂mu remembering to give special treatment to cells on the border of SC because we must incorporate their virtual cells. After much rearrangement we obtain

• ∂SC

v∂nu dx = 2 · 3m ρm

• xm∼∂SC

v(xm)(f (x) − u(x)) 1 3m which lets us define the normal derivative as: ∂nu(x) = lim

m→∞

2 · 3m ρm (u(x) − u(xm)). The normal derivative most likely only exists as a measure.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 22 / 23

Website

More data is available (and more to come) on www.math.cornell.edu/∼reu/sierpinski-carpet including: Full list of eigenvalues and pictures of eigenfunctions for both Dirichlet and Neumann boundary value problems. Eigenvalue counting function data & Weyl ratios. Eigenvalue data on covering spaces of SC. All MATLAB scripts used. Trace of the Heat Kernel data. Dirichlet and Poisson kernel data.

Matthew Begu´ e, Tristan Kalloniatis, & Robert Strichartz () Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet October 3, 2010 23 / 23