Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Freitas, Pedro; Siudeja, Bartłomiej

doi:10.1051/cocv/2009018

Cited by 18 publications

(21 citation statements)

References 14 publications

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“…Analogous conjectures were formulated also for the torsional rigidity and for the logarithmic capacity. For N = 3 and N = 4 these conjectures were proved by Pólya and Szegő themselves [27, p. 158], via the classical tool of Steiner symmetrization (recently, improved bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals have been obtained in [16]). For N ≥ 5, as explained in [20,Sect.…”

Section: Theorem 1 Among All Simple Polygons With a Given Area And Atmentioning

confidence: 99%

A Faber–Krahn Inequality for the Cheeger Constant of $$N$$ N -gons

Bucur

Fragalà

2014

J Geom Anal

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Section: Theorem 1 Among All Simple Polygons With a Given Area And Atmentioning

confidence: 99%

A Faber–Krahn Inequality for the Cheeger Constant of $$N$$ N -gons

Bucur

Fragalà

2014

J Geom Anal

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“…Freitas [13] showed for arbitrary triangles that λ 1 A 2 S 2 ≤ π 2 3 , which is slightly weaker than (10.1); quadrilaterals have been studied too [16]. Conjectures involving λ 1 and geometric functionals have been raised by Antunes and Freitas [1].…”

Section: Survey Of Dirichlet Eigenvalue Estimatesmentioning

confidence: 99%

Maximizing Neumann fundamental tones of triangles

Laugesen

Siudeja

2009

Journal of Mathematical Physics

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“…In the process we found that working with the explictly defined Stieltjes function φ 2 is overwhelmingly neater than the elementary, but detailed, calculations that arise when working with µ (2) , the inverse of φ. Many other inequalities on λ 1 , and related functionals, for domains more general than boxes, have been proved and others have been conjectured: see, for example, [13,28] and the long arXiv article [22]. In [41] it is asked if, amongst all n-gons of given area, that which has the least λ 1 is the regular n-gon.…”

Section: The Eigenvalue Problemmentioning

confidence: 99%

Inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional rectangular parallelepipeds

Keady¹,

Wiwatanapataphee²

2018

Mathematical Inequalities &Amp; Applications

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This document consists of two papers, both submitted, and supplementary material. The submitted papers are here given as Parts I and II. Their abstracts are given at pages 2 and 37 respectively. The supplementary material is given in appendices.The supplementary material includes material directed at answering questions stated in, or implicit in, the submitted papers. The maple code referenced in the supplements is available via https://sites.google.com/site/keadyperthunis/home/papers One of the questions is as follows. Let φ 1 (z) = arctan(1/z)/z. (φ 1 is, itself, completely monotone.) Is the inverse of φ 1 , denoted µ in the following, also completely monotone? Various approaches have been considered. Direct calculation of the higher derivatives of µ -with maple to handle the lengthy expressions -and an inductive argument is considered in work presented at page 25. Some complex variable approaches are instigated at page 33.Abstract Any function f from (0, ∞) onto (0, ∞) which is decreasing and convex has an inverse g which is positive and decreasing -and convex. When f has some form of generalized convexity we determine additional convexity properties inherited by g. When f is positive, decreasing and (p, q)-convex, its inverse g is (q, p)-convex. Related properties which pertain when f is a Stieltjes function are developed. The results are illustrated with the Stieltjes function f (x) = arctan(1/ √ x)/ √ x, a function which arises, via a transcendental equation, in an application presented in Part II.

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Bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals

Abstract: Abstract. We

Cited by 18 publications

References 14 publications

A Faber–Krahn Inequality for the Cheeger Constant of $$N$$ N -gons

A Faber–Krahn Inequality for the Cheeger Constant of $$N$$ N -gons

Maximizing Neumann fundamental tones of triangles

Inequalities for the fundamental Robin eigenvalue for the Laplacian on N-dimensional rectangular parallelepipeds

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