The "Hot Spots" Conjecture on the Vicsek Set

TL;DR

This paper proves the Hot Spot conjecture for the Vicsek set, demonstrating that the second eigenfunction of the Neumann Laplacian reaches its extrema on the boundary, confirming a key property of eigenfunctions on fractals.

Contribution

The paper establishes the Hot Spot conjecture for the Vicsek set, a significant advance in understanding eigenfunctions on fractal structures.

Findings

Eigenfunctions of the second smallest eigenvalue attain maxima and minima on the boundary

Confirmation of the Hot Spot conjecture for the Vicsek set

Advances understanding of spectral properties on fractals

Abstract

We prove the Hot Spot conjecture on the Vicsek set. Specifically, we 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.