Sg < Sub > 3 < / Sub > "hot Spots" On The Suspect

"Hot spots" conjecture was posed by J.Rauch in1974. Many researchers have worked on this conjecture for kinds of domains in Euclidean space. It is already known that the conjecture holds in many case. However, there are famous examples constructed in some papers showing that the conjecture does not hold in certain domain. While all above works are based on the Euclidean space, we know that there are many works on analysis on fractals during this period. Thus it is naturel to ask whether the "hot spots" holds if the underlying space is a fractal. Although it is difficult to say something about the conjecture in all fractals, we can start the work with some special fractals that have some good property. H.J.Ruan have proven that the conjecture hold in Sierpinski gasket, SG for short, in a paper this year. In this paper, we will use spectral decimation and the method in the paper of H.J.Ruan to prove that "hot spot" conjecture also holds in SG3, i.e. every eigenfunction of the second-smallest eigenvalue of Neumann Laplacian(introduced by J.Kigarni) attains its maximum and minimum on the boundary.