The “hot spots” conjecture for the Sierpinski gasket
Introduction
The “hot spots” conjecture was posed by Rauch at a conference in 1974. There are several versions of this conjecture. In this paper, we will use the following version: every eigenfunction of the second-smallest eigenvalue of Neumann Laplacian attains its maximum and minimum on the boundary of the domain in Euclidean space. For details, please see [1].
The “hot spots” conjecture holds in many cases, especially for certain convex planar domains and lip domains. For examples, please see [1], [2], [3], [4]. In [5], Burdzy and Werner constructed an interesting counterexample to the “hot spots” conjecture, where the planar domain has two holes. In [6], Burdzy constructed another planar domain with one hole such that the “hot spots” conjecture fails.
The underlying spaces in above works are domains in Euclidean space. Since there are many works on the analysis on fractals during the past two decades (see [7], [8] and references therein) and the Laplacian always plays an essential role in the analysis, it is natural to ask whether the conjecture holds if the underlying space is a fractal. Basically, there are two methods for defining the Laplacian on fractals: the probability method and the analytic method. In this paper, we will choose the latter, which was introduced by Kigami [9], [10]. Meanwhile, we will choose the Sierpinski gasket, for short, to be the underlying space because it is the most typical set when we do analysis on fractals.
In the present paper, we will prove that the “hot spots” conjecture holds on by using spectral decimation, which was first observed for the Sierpinski pre-gasket by Rammal and Toulouse [11], and was discussed rigorously by Shima [12], [13] and Fukushima and Shima [14].
Spectral decimation is one basic tool for describing the properties of eigenfunctions and eigenvalues of the Dirichlet Laplacian and Neumann Laplacian on certain p.c.f. self-similar sets; see e.g. [15], [16], [17], [18], [19], [20], [21]. In particular, by using spectral decimation and inductive arguments, Dalrymple et al. [15] discussed the minimum and maximum of the Dirichlet eigenfunction. However, there has been no work done on the maximum and minimum of the Neumann eigenfunction until now, as far as we know. One possible reason is that Neumann eigenfunctions can take arbitrary values, while essentially, the Dirichlet eigenfunctions in the discussion in [15] are nonnegative. In the present paper, we introduce a new method (see Lemma 4.1) and arguments that are more careful than those in [15] in order to prove our main result.
The rest of the paper is organized as follows. Basic concepts and spectral decimation are recalled in Section 2. In Section 3, we characterize the second-smallest eigenvalues and their corresponding eigenfunctions for the Neumann Laplacian on . The main results are proved in Section 4.
Section snippets
Basic definitions
In this subsection, we recall some basic notation from [7], [8].
Let , , be non-collinear points in . Define for each . We name as the Sierpinski gasket, for short, the attractor of the iterated function system .
Let and for any positive integer . Define . For each , we say that is a word with length . Denote by the empty word and define . We also say that . For any with
Eigenfunctions of the second-smallest eigenvalue of the Neumann Laplacian
Define
The following result is well known.
Proposition 3.1 , , and for ,Furthermore, the multiplicities of 0 and 6 in are equal to 1 and 2 respectively, while any discrete N-eigenfunction corresponding to satisfies .[8], [12]
Notice that for any . Hence for any . We remark that while for any .
The following lemma shows how to obtain the second-smallest
The main result and its proof
In this section, we always assume that , and for any . Clearly,
The following theorem is the main result of the paper.
Theorem 4.1 Every eigenfunction of the second-smallest eigenvalue of Neumann Laplacian on Sierpinski gasket attains its maximum and minimum on the boundary .
Let with , , , , and and . Define functions by
Acknowledgments
This research is supported by Zhejiang Provincial Natural Science Foundation of China (No. Y6110128). The author wishes to thank Dr. Yuan Liu for helpful discussion. He also wishes to thank the referees for their helpful suggestions.
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2013, Nonlinear Analysis, Theory, Methods and ApplicationsCitation Excerpt :The underlying spaces in the above works are domains in Euclidean space. In [6], the first author proved that the hot spots conjecture holds on the Sierpinski gasket (SG for short); the Neumann Laplacian on the SG was introduced by Kigami [7,8]. The main tool used for proving the hot spots conjecture is spectral decimation, which was studied by Rammal and Toulouse [9], Shima [10,11] and Fukushima and Shima [12].
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