Harmonic Functions and the Spectrum of the Laplacian on 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 of the eight points along the boundary. SC is the unique nonempty compact set satisfying the self-similar identity 7 � SC = F i ( SC ) . i =0 F 0 F 1 F 2 � � � � � � � � � � F 7 F 3 � � � � � � � � � � F 6 F 5 F 4 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 � ∆ m u ( x ) = ( u ( y ) − u ( x )) . y ∼ x m b ˜ x ⑦ �� ⑦ ˜ a d x y x ′ �� ⑦ �� ⑦ c z �� For example the graph Laplacian of interior cell a is ∆ m u ( a ) = − 3 u ( a ) + u ( b ) + u ( c ) + u ( d ). For boundary cells, we include its neighboring virtual cells. x ) + u ( ˜ eg: ∆ m u ( x ) = − 4 u ( x ) + u ( y ) + u ( z ) + 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 m →∞ r − m ∆ m ∆ = lim where r is the renormalization constant r = (8 ρ ) − 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 π x along the remaining edge and extend harmonically. sin π x 0 0 0 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 2 π x 0 0 0 sin 3 π x 0 0 0 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 = λ u We have two types of boundary value problems: Neumann Dirichlet ∂ n u | ∂ SC = 0 u | ∂ SC = 0 Corresponds to odd reflections about Corresponds to even reflections about boundary. the boundary. ie: ˜ x = x 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
� ∆ m u ( x ) = ( u ( y ) − u ( x )) y ∼ x m b ˜ x �� ⑦ ⑦ ˜ a d x y x ′ �� ⑦ �� ⑦ c z �� 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 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: ∂ n u | ∂ 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 8 m eigenfunctions from level m but refined. The eigenvalue is renormalized by r = 10 . 011. Figure: φ (4) and φ (5) with respective eigenvalues λ (4) = 0 . 00328 and 5 5 5 λ (4) = 0 . 000328. 5 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) and φ (5) 20 with respective eigenvalues λ (4) = λ (5) 20 = 0 . 00177. 4 4 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 t α 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 ) α ≈ 0 . 9 t α 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 ) ≤ λ ( D ) for each j . j 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 ) ∼ t β . log 3 β ≈ 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 ) t β 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 orientations. 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 ∂ n u on ∂ SC so that the Gauss-Green formula holds: � � E ( u , v ) = − (∆ u ) v d µ + ( ∂ n u ) v d µ ′ . SC ∂ SC We know that E m ( u , v ) = 1 � ( u ( x ) − u ( y ))( v ( x ) − v ( y )) ρ m x ∼ y and − ∆ m u ( x ) = 8 m � ( u ( x ) − u ( y )) . ρ m x ∼ 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 ∂ 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 v ∂ n u dx = 2 · 3 m v ( x m )( f ( x ) − u ( x )) 1 � � ρ m 3 m ∂ SC x m ∼ ∂ SC which lets us define the normal derivative as: 2 · 3 m ∂ n u ( x ) = lim ρ m ( u ( x ) − u ( x m )) . m →∞ 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
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