Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet

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1 Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet Matthew Begué, Tristan Kalloniatis, & Robert Strichartz October 3, 2010

2 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 of the eight points along the boundary. SC is the unique nonempty compact set satisfying the self-similar identity SC = 7 F i (SC). i=0 F 0 F 1 F 2 F 7 F 3 F 6 F 5 F 4 Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 2 / 23

3 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 Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 3 / 23

4 Constructing the Laplacian m u(x) = y x m b 7 a d c (u(y) u(x)). x x x z y 7 For example the graph Laplacian of interior cell a is m u(a) = 3u(a) + u(b) + u(c) + u(d). For boundary cells, we include its neighboring virtual cells. eg: m u(x) = 4u(x) + u(y) + u(z) + u( x) + u( x ). Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 4 / 23

5 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 and therefore 1/r Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 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.

6 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 Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 6 / 23

More Harmonic Functions sin 2πx 0 0 0 sin 3πx 0 0 0 Matthew Begué, Tristan Kalloniatis, & RobertHarmonic

7 More Harmonic Functions sin 2πx sin 3πx Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 7 / 23

8 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 n u SC = 0 Dirichlet u SC = 0 Corresponds to even reflections about boundary. ie: x = x Corresponds to odd reflections about the boundary. ie: x = x atthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 8 / 23

9 m u(x) = y x m (u(y) u(x)) b 7 a c d x x x z y 7 Therefore the Laplacian operator is determined by 8 m linear equations. This can be represented in an 8 m square matrix. The matrix is created in MATLAB and the eigenvalues and eigenfunctions are calculated using the built-in eigs function. Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on the October Sierpinski 3, 2010 Carpet 9 / 23

10 Some eigenfunctions Neumann: n u SC = 0 Dirichlet: u SC = 0 atthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 10 / 23

11 Refinement On level m + 1 we expect to see all 8 m eigenfunctions from level m but refined. The eigenvalue is renormalized by r = Figure: φ (4) 5 and φ (5) 5 with respective eigenvalues λ (4) 5 = and λ (4) 5 = Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 11 / 23

12 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 = Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 12 / 23

13 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 t α as t where α = log 8/ log Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 13 / 23

14 N(t) = #{λ : λ t}

15 Weyl Ratio: W (t) = N(t) t α α 0.9 Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 15 / 23

16 Neumann vs. Dirichlet eigenvalues By the min-max property, we can say that λ (N) j Therefore, N (D) (t) N (N) (t). λ (D) j for each j. 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) t β. β log 3 log = Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 16 / 23

17 N (N) (t) N (D) (t) Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 17 / 23

18 N (N) (t) N (D) (t) t β A stronger periodicity is apparent here. Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 18 / 23

19 Fractafolds We can eliminate the boundary of SC by gluing its boundary in specific orientations. We examined three types of SC fractafolds: Torus Klein Bottle Projective Space Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 19 / 23

20 Some Eigenfunctions for the Fractafolds Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 20 / 23

21 How to define the normal derivative on the boundary of SC We wish to define n u on SC so that the Gauss-Green formula holds: E(u, v) = ( u)v dµ + ( n u)v dµ. We know that SC SC E m (u, v) = 1 ρ m (u(x) u(y))(v(x) v(y)) x y and m u(x) = 8m ρ m (u(x) u(y)). x y Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 21 / 23

22 Sketch of how n u is defined Let x be a point on SC and x m be the m-cell containing x. We can use the equations from the previous slide to find m u 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 n u dx = 2 3m ρ m x m SC which lets us define the normal derivative as: n u(x) = lim m v(x m )(f (x) u(x)) 1 3 m 2 3 m ρ m (u(x) u(x m)). The normal derivative most likely only exists as a measure. Matthew Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 22 / 23

23 Website More data is available (and more to come) on 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 Begué, Tristan Kalloniatis, & RobertHarmonic Strichartz Functions () and the Spectrum of the Laplacian on October the Sierpinski 3, 2010 Carpet 23 / 23

Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet

Harmonic Functions and the Spectrum of the Laplacian on the Sierpinski Carpet Matthew Begué, Tristan Kalloniatis, & Robert Strichartz October 4, 2011 Norbert Wiener Center Seminar Construction of the Sierpinski