Spectrum of the Laplacian on Planar Domains with Fractal Boundary
Julia Sets
Computation of the Area (Iter. 170)
Image size 5120x3840: Data Slice at Im(c)=0
Image size 640x480: Data 2D 3D
Computation of the Box-Counting Dimension (Iter. 170)
Data: Rectangle Slice at Im(c)=0 Slice at Re(c)=-0.12Graphs: Slice at Im(c)=0 (Zoom) Slice at Re(c)=-0.12 Power Law (Log-Log) 2D Full 3D Full
Eigenvalues with Dirichlet boundary conditions (Iter. 170, Res. 200)
-0.50+0.00i -0.50+0.05i -0.50+0.10i -0.50+0.15i -0.50+0.20i -0.50+0.25i -0.50+0.30i -0.50+0.35i -0.50+0.40i -0.50+0.45i-0.45+0.00i -0.45+0.05i -0.45+0.10i -0.45+0.15i -0.45+0.20i -0.45+0.25i -0.45+0.30i -0.45+0.35i -0.45+0.40i -0.45+0.45i
-0.40+0.00i -0.40+0.05i -0.40+0.10i -0.40+0.15i -0.40+0.20i -0.40+0.25i -0.40+0.30i -0.40+0.35i -0.40+0.40i -0.40+0.45i
-0.35+0.00i -0.35+0.05i -0.35+0.10i -0.35+0.15i -0.35+0.20i -0.35+0.25i -0.35+0.30i -0.35+0.35i -0.35+0.40i -0.35+0.45i
-0.30+0.00i -0.30+0.05i -0.30+0.10i -0.30+0.15i -0.30+0.20i -0.30+0.25i -0.30+0.30i -0.30+0.35i -0.30+0.40i -0.30+0.45i
-0.25+0.00i -0.25+0.05i -0.25+0.10i -0.25+0.15i -0.25+0.20i -0.25+0.25i -0.25+0.30i -0.25+0.35i -0.25+0.40i -0.25+0.45i
-0.20+0.00i -0.20+0.05i -0.20+0.10i -0.20+0.15i -0.20+0.20i -0.20+0.25i -0.20+0.30i -0.20+0.35i -0.20+0.40i -0.20+0.45i
-0.15+0.00i -0.15+0.05i -0.15+0.10i -0.15+0.15i -0.15+0.20i -0.15+0.25i -0.15+0.30i -0.15+0.35i -0.15+0.40i -0.15+0.45i
-0.10+0.00i -0.10+0.05i -0.10+0.10i -0.10+0.15i -0.10+0.20i -0.10+0.25i -0.10+0.30i -0.10+0.35i -0.10+0.40i -0.10+0.45i
0.00+0.00i 0.00+0.05i 0.00+0.10i 0.00+0.15i 0.00+0.20i 0.00+0.25i 0.00+0.30i 0.00+0.35i 0.00+0.40i 0.00+0.45i
0.05+0.00i 0.05+0.05i 0.05+0.10i 0.05+0.15i 0.05+0.20i 0.05+0.25i 0.05+0.30i 0.05+0.35i 0.05+0.40i 0.05+0.45i
0.10+0.00i 0.10+0.05i 0.10+0.10i 0.10+0.15i 0.10+0.20i 0.10+0.25i 0.10+0.30i 0.10+0.35i 0.10+0.40i 0.10+0.45i
0.15+0.00i 0.15+0.05i 0.15+0.10i 0.15+0.15i 0.15+0.20i 0.15+0.25i 0.15+0.30i 0.15+0.35i 0.15+0.40i 0.15+0.45i
0.20+0.00i 0.20+0.05i 0.20+0.10i 0.20+0.15i 0.20+0.20i 0.20+0.25i 0.20+0.30i 0.20+0.35i 0.20+0.40i 0.20+0.45i
Graph Graph Graph Graph Graph Graph Graph Graph Graph Graph
Eigenfunctions with Dirichlet boundary conditions (Iter. 170, Res. 200)
-0.50+0.00i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
-0.50+0.45i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
0.00+0.00i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
0.20+0.00i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
0.20+0.45i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
Eigenfunctions with Neumann boundary conditions (Iter. 170, Res. 200)
0.00+0.00i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
0.20+0.00i: 2D 1-25 26-50 51-75 76-100 Show No.: Go
3D 1-25 26-50 51-75 76-100 Show No.: Go
Dirichlet boundary conditions on the Basilica (c=-1.00+0.00i, Res. 30)
1000 Eigenvalues: 10 Iterations 15 Iterations 20 Iterations 25 Iterations 30 IterationsDirichlet boundary conditions on the Rabbit (c=-0.12+0.75i, Res. 30)
1000 Eigenvalues: 10 Iterations 15 Iterations 20 Iterations 25 Iterations 30 IterationsDirichlet boundary conditions on the junction point to the Basilica-bulb (c=-0.75+0.00i, Res. 30)
1000 Eigenvalues: 10 Iterations 15 Iterations 20 Iterations 25 Iterations 30 IterationsDirichlet boundary conditions on the junction point to the Rabbit-bulb (c=-0.12+0.70i, Res. 30)
1000 Eigenvalues: 10 Iterations 15 Iterations 20 Iterations 25 Iterations 30 IterationsDirichlet boundary conditions on the Basilica (c=-1.00+0.00i, Iter. 30, Res. 30)
2D: 1-25 26-50 51-75 76-1003D: 1-25 26-50 51-75 76-100
Dirichlet boundary conditions on the Rabbit (c=-0.12+0.75i, Iter. 30, Res. 30)
2D: 1-25 26-50 51-75 76-1003D: 1-25 26-50 51-75 76-100
Dirichlet boundary conditions on the Rabbit (c=-0.12+0.75i, Iter. 30, Res. 30)
2D: 1-25 26-50 51-75 76-1003D: 1-25 26-50 51-75 76-100
Dirichlet boundary conditions on the junction point to the Rabbit-bulb (c=-0.12+0.70i, Iter. 30, Res. 30)
2D: 1-25 26-50 51-75 76-1003D: 1-25 26-50 51-75 76-100
Dirichlet boundary conditions on Julia sets from the main bulb moving to the junction point to the Basilica-bulb (c=-0.75+0.00i, Iter. 30, Res. 30)
1000 Eigenvalues: -0.70+0.00i -0.71+0.00i -0.72+0.00i -0.73+0.00i -0.74+0.00iDirichlet boundary conditions on Julia sets from the main bulb moving to the junction point to the Rabbit-bulb (c=-0.12+0.70i, Iter. 30, Res. 30)
Coming soon!Dirichlet boundary conditions on quasicircles of the Basilica (c=-1.00+0.00i, Iter. 170, Res. 70): Full
1: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
2: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
3: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
4: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
Neumann boundary conditions on quasicircles of the Basilica (c=-1.00+0.00i, Iter. 170, Res. 70): Full
1: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
2: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
3: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
4: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
Dirichlet boundary conditions on quasicircles of the Rabbit (c=-0.12+0.75i, Iter. 170, Res. 70): Full
1: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
2: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
3: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
Neumann boundary conditions on quasicircles of the Rabbit (c=-0.12+0.75i, Iter. 170, Res. 70): Full
1: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
2: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
3: 1-25 26-50 51-75 76-100 Show No.: Go (Eigenvalues)
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Contact Information
Robert Strichartz, Cornell University, Department of Mathematics, str@math.cornell.edu
Samuel Wiese, University of Leipzig, Department of Mathematics, sw31hiqa@studserv.uni-leipzig.de