The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

Grieser, Daniel

doi:10.1007/s00013-005-1623-4

Cited by 23 publications

(20 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Grieser [16] shows how max flow-min cut duality leads to an elegant proof of Cheeger's inequality, giving the lower bound in (18) on the first eigenvalue of the Laplacian on Ω. The eigenfunction has u = 0 on ∂Ω, so ∂Ω N is empty:…”

Section: New Questions and Applicationsmentioning

confidence: 99%

“…Historically, the key inequality given by Cheeger [7] was a lower bound on the first eigenvalue λ 1 of the Laplacian on the domain Ω. Grieser [16] observed how neatly and directly this bound follows from Green's formula, when F = 1 and |v| ≤ 1. We expect to see the Schwarz inequality in the step from problems in L 1 and L ∞ to the eigenvalue problem in L 2 :…”

Section: Duality Coarea and Cheeger Constantsmentioning

confidence: 99%

See 1 more Smart Citation

Maximum Flows and Minimum Cuts in the Plane

Strang¹

2009

Advances in Mechanics and Mathematics

View full text Add to dashboard Cite

A continuous maximum flow problem finds the largest t such that div v = t F (x, y) is possible with a capacity constraint (v1, v2) ≤ c(x, y). The dual problem finds a minimum cut ∂S which is filled to capacity by the flow through it. This model problem has found increasing application in medical imaging, and the theory continues to develop (along with new algorithms). Remaining difficulties include explicit streamlines for the maximum flow, and constraints that are analogous to a directed graph.

show abstract

Section: New Questions and Applicationsmentioning

confidence: 99%

Section: Duality Coarea and Cheeger Constantsmentioning

confidence: 99%

Maximum Flows and Minimum Cuts in the Plane

Strang¹

2009

Advances in Mechanics and Mathematics

View full text Add to dashboard Cite

show abstract

“…One possibility is given by the following result concerning "test vector fields". Theorem 1.3 [Grieser 2006]. Let V : → ‫ޒ‬ 2 be a smooth vector field on , h ∈ ‫,ޒ‬ and assume that the pointwise inequalities |V | ≤ 1 and divV ≥ h hold in .…”

Section: Introductionmentioning

confidence: 99%

The Cheeger constant of curved strips

Krejčiřı́k¹,

Pratelli

2011

Pacific J. Math.

View full text Add to dashboard Cite

We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighborhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about noncomplete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.

show abstract

“…Mathematical study of these objects was based on limit behavior of the spectrum for shrinking wires. In particular, in [80], [46], [47] and [24], spectral properties of the Schrödinger operator on a compact network were studied, see also some recent material in [44], [45], [35] and [16].…”

Section: Introduction: Physical Environment Motivation and Main Resultsmentioning

confidence: 99%

Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

Mikhailova

Павлов

Prokhorov

2007

Mathematische Nachrichten

View full text Add to dashboard Cite

We consider the spectral problem for the Schrödinger operator L on a quantum network, represented as a union of a finite number of quantum wells and finite or semi-infinite straight quantum wires connecting them to each other and to infinity. The absolutely continuous spectrum of the Schrödinger operator, with real piecewisecontinuous potential on the wells and constant potential on the wires, has a band structure defined by the geometry and the potentials on the semi-infinite wires. We split the Schrödinger operator, depending on the Fermi level Λ, into an orthogonal sum of two operators L → lΛ ⊕ LΛ, which have the complementary branches of the absolutely continuous spectra coincident, counting multiplicity, respectively with the unionsof absolutely continuous branches σ = [λ σ , ∞) of open and closed channels in the semi-infinite wires, with thresholds λσ < Λ, λσ > Λ, respectively. This enables us to reduce the problem of evaluation of the scattering matrix on the network to the solution of a finite linear system, involving the Dirichlet-to-Neumann map of the intermediate Hamiltonian LΛ. We calculate the intermediate Dirichlet-to-Neumann map approximately, for thin networks, based on the classical DN-map of the relevant Schrödinger operator on the compact part of the network. To this end we develop a modified analytic perturbation procedure for meromorphic operatorfunctions with small one-dimensional residues.

show abstract

The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

Cited by 23 publications

References 25 publications

Maximum Flows and Minimum Cuts in the Plane

Maximum Flows and Minimum Cuts in the Plane

The Cheeger constant of curved strips

Intermediate Hamiltonian via Glazman's splitting and analytic perturbation for meromorphic matrix‐functions

Contact Info

Product

Resources

About