A convex surface with fractal curvature. (English) Zbl 1441.28006
Summary: We construct a surface that is obtained from the octahedron by pushing out four of the faces so that the curvature is supported in a copy of the Sierpinski gasket (SG) in each of them, and is essentially the self similar measure on SG. We then compute the bottom of the spectrum of the associated Laplacian using the finite element method on polyhedral approximations of our surface, and speculate on the behavior of the entire spectrum.
MSC:
| 28A80 | Fractals |
Software:
TriangleReferences:
| [1] | Strichartz, R. S., Differential Equations on Fractals: A Tutorial (Princeton University Press, Princeton, New Jersey, 2006). · Zbl 1190.35001 |
| [2] | Shewchuk, J. R., Delaunay refinement algorithms for triangular mesh generation, Comp. Geom. Theor. Appl.22 (2002) 21-74. · Zbl 1016.68139 |
| [3] | Bobenko, A. I. and Izmestiev, I., Alexandrov’s theorem, weighted delaunay triangulations, and mixed volumes, Ann. Inst. Fourier58 (2008) 447-505. · Zbl 1154.52005 |
| [4] | R. S. Strichartz and S. C. Wiese, A convex surface with fractal curvature (2020), http://pi.math.cornell.edu/\( \sim\) sw972/fractal_curvature.html. |
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