Isoperimetric upper bound for the first eigenvalue of discrete Steklov\n problems

2020/02/20 by Hélène Perrin, Perrin, Hélène · 5 citations
Computer Science · Mathematics · #58J50 39A12 15A42 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2002.08751

openalex publication_date 2020/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study upper bounds for the first non-zero eigenvalue of the Steklov\nproblem defined on finite graphs with boundary. For finite graphs with boundary\nincluded in a Cayley graph associated to a group of polynomial growth, we give\nan upper bound for the first non-zero Steklov eigenvalue depending on the\nnumber of vertices of the graph and of its boundary. As a corollary, if the\ngraph with boundary also satisfies a discrete isoperimetric inequality, we show\nthat the first non-zero Steklov eigenvalue tends to zero as the number of\nvertices of the graph tends to infinity. This extends recent results of Han and\nHua, who obtained a similar result in the case of \ℤⁿ. We obtain the\nresult using metric properties of Cayley graphs associated to groups of\npolynomial growth.\n

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