A weighted Sobolev-Poincaré's inequality on infinite networks (Q2731096)
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scientific article; zbMATH DE number 1625541
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A weighted Sobolev-Poincaré's inequality on infinite networks |
scientific article; zbMATH DE number 1625541 |
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25 April 2002
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infinite graphs
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discrete Laplacian
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Poincare-Sobolev inequality
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0.9091558
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0.88799524
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0.8809382
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0.88039625
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A weighted Sobolev-Poincaré's inequality on infinite networks (English)
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The authors consider a locally finite and connected, countable graph with nodes \(X\) and arcs \(Y\). A resistance function \(r:Y\to (0,\infty)\) determines a discrete Dirichlet form \(D\) on its set of functions of finite energy. It is analyzed on the weighted space \(L^2(X,m)\) with weight function \(m:X \to (0,\infty)\). As in the classical continuous case the optimal constant \(c\) in the weighted Poincaré-Sobolev inequality \(\|u\|_{L^2(X,m)}^2 \leq c D(u)\), for \(u\) with compact support, is characterized variationally and in terms of the lowest eigenvalue of the Schrödinger operator with potential \(m\).
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