Abstract
Abstract We prove the “hot spots” conjecture on the Vicsek set. Specifically, we will show that every eigenfunction of the second smallest eigenvalue of the Neumann Laplacian on the Vicsek set attains its maximum and minimum on the boundary.
Highlights
The “hot spots” conjecture studies whether a flat piece of metal that is given an initial heat distribution will achieve its highest temperatures on its boundary given enough time
We prove the “hot spots” conjecture on the Vicsek set
The “hot spots” conjecture has been shown to hold on the Sierpinski gasket and higher dimensional variants [14,15,16] but fail on the hexagasket fractal [17]
Summary
The “hot spots” conjecture studies whether a flat piece of metal that is given an initial heat distribution will achieve its highest temperatures on its boundary given enough time. We are going to use the theory developed by Kigami [8], see [9] that applies to the class of post critically finite (p.c.f.) fractals For many such fractals, eigenvalues and eigenfunctions of the Laplacian can be computed explicitly via a method called spectral decimation [10,11,12,13]. The boundary of the Sierpinski gasket and its higher dimensional variants mentioned above consists of all of the fixed points of the iterated function system that determines the gasket. We study in this paper the Vicsek set VS2 that is generated by five contractions; its analytic boundary consists of only four of the five fixed points of the iterated function systems. Our main theorem states that, unlike the hexagasket, the “hot spots” conjecture on the Vicsek set is true. We placed the statement and proof of some formulas used throughout the paper in the Appendix in order to help with the readability of Section 4
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