scholarly journals A Sharp estimate for the first Robin–Laplacian eigenvalue with negative boundary parameter

2019 ◽  
Vol 30 (4) ◽  
pp. 665-676 ◽  
Author(s):  
Dorin Bucur ◽  
Vincenzo Ferone ◽  
Carlo Nitsch ◽  
Cristina Trombetti
Keyword(s):  
Sharp Estimate ◽  
Laplacian Eigenvalue ◽  
Boundary Parameter ◽  
Robin Laplacian

scholarly journals A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter

2020 ◽  
Author(s):  
Simone Cito ◽  
Domenico Angelo La Manna
Keyword(s):  
Convex Sets ◽  
Type Inequality ◽  
First Eigenvalue ◽  
Type Problem ◽  
Laplacian Eigenvalue ◽  
Quantitative Form ◽  
Boundary Parameter ◽  
Robin Laplacian ◽  
Krahn Inequality ◽  
Optimal Set

The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalue λ β with negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer for λ β and the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.

scholarly journals Extremal Domains and Pólya-type Inequalities for the Robin Laplacian on Rectangles and Unions of Rectangles

2019 ◽  
Author(s):  
Pedro Freitas ◽  
James B Kennedy
Keyword(s):  
Boundary Conditions ◽  
Robin Boundary Conditions ◽  
Bounded Domains ◽  
Fixed Domain ◽  
Boundary Parameter ◽  
Eigenvalues Of The Laplacian ◽  
Robin Laplacian ◽  
Robin Boundary ◽  
Original Approach

Abstract We investigate the question of whether the eigenvalues of the Laplacian with Robin boundary conditions can satisfy inequalities of the same type as those in Pólya’s conjecture for the Dirichlet and Neumann Laplacians and, if so, what form these inequalities should take. Motivated in part by Pólya’s original approach and in part by recent analogous works treating the Dirichlet and Neumann Laplacians, we consider rectangles and unions of rectangles and show that for these two families of domains, for any fixed positive value $\alpha$ of the boundary parameter, Pólya-type inequalities do indeed hold, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in both cases, showing that they are different in the two situations. Our approach to proving these results includes a characterization of the corresponding extremal domains for the $k^{\textrm{}}$th eigenvalue in regions of the $(k,\alpha )$-plane, which in turn supports recent conjectures on the nature of the extrema among all bounded domains.

Geometric Control of the Robin Laplacian Eigenvalues: The Case of Negative Boundary Parameter

2019 ◽  
Vol 30 (4) ◽  
pp. 4356-4385 ◽  
Author(s):  
Dorin Bucur ◽  
Simone Cito
Keyword(s):  
Geometric Control ◽  
Laplacian Eigenvalues ◽  
Boundary Parameter ◽  
Robin Laplacian

scholarly journals On the Boundary Dieudonné–Pick Lemma

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1108
Author(s):  
Olga Kudryavtseva ◽  
Aleksei Solodov
Keyword(s):  
Fixed Point ◽  
Interior Point ◽  
Sharp Estimate ◽  
General Boundary ◽  
Extremal Functions ◽  
Angular Derivative ◽  
Additional Information ◽  
Carathéodory Theorem ◽  
Class Of Functions ◽  
Boundary Fixed Point

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

scholarly journals On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 512
Author(s):  
Maryam Baghipur ◽  
Modjtaba Ghorbani ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Graph Parameters ◽  
Simple Connected Graph ◽  
Second Largest Eigenvalue ◽  
Reciprocal Distance Matrix

The signless Laplacian reciprocal distance matrix for a simple connected graph G is defined as RQ(G)=diag(RH(G))+RD(G). Here, RD(G) is the Harary matrix (also called reciprocal distance matrix) while diag(RH(G)) represents the diagonal matrix of the total reciprocal distance vertices. In the present work, some upper and lower bounds for the second-largest eigenvalue of the signless Laplacian reciprocal distance matrix of graphs in terms of various graph parameters are investigated. Besides, all graphs attaining these new bounds are characterized. Additionally, it is inferred that among all connected graphs with n vertices, the complete graph Kn and the graph Kn−e obtained from Kn by deleting an edge e have the maximum second-largest signless Laplacian reciprocal distance eigenvalue.

Heat generation/absorption on MHD flow of a micropolar fluid over a heated stretching surface in the presence of the boundary parameter

Heat Transfer ◽  
2021 ◽  
Author(s):  
Shankar Goud Bejawada ◽  
Zafar Hayat Khan ◽  
Muhammad Hamid
Keyword(s):  
Heat Generation ◽  
Micropolar Fluid ◽  
Mhd Flow ◽  
Stretching Surface ◽  
Boundary Parameter

scholarly journals Domination number and Laplacian eigenvalue distribution

2016 ◽  
Vol 53 ◽  
pp. 66-71 ◽  
Author(s):  
Stephen T. Hedetniemi ◽  
David P. Jacobs ◽  
Vilmar Trevisan
Keyword(s):  
Domination Number ◽  
Eigenvalue Distribution ◽  
Laplacian Eigenvalue

A Sharp Lower Bound on the Least Signless Laplacian Eigenvalue of a Graph

2016 ◽  
Vol 41 (4) ◽  
pp. 2011-2018 ◽  
Author(s):  
Xiaodan Chen ◽  
Yaoping Hou
Keyword(s):  
Lower Bound ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sharp Lower Bound

On signed graphs whose second largest Laplacian eigenvalue does not exceed 3

2015 ◽  
Vol 64 (9) ◽  
pp. 1785-1799 ◽  
Author(s):  
Francesco Belardo ◽  
Paweł Petecki ◽  
Jianfeng Wang
Keyword(s):  
Signed Graphs ◽  
Laplacian Eigenvalue
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