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 Sharp estimates for the first p-Laplacian eigenvalue and for the p-torsional rigidity on convex sets with holes

2020 ◽  
Vol 26 ◽  
pp. 111 ◽  
Author(s):  
Gloria Paoli ◽  
Gianpaolo Piscitelli ◽  
Leonardo Trani
Keyword(s):  
Boundary Conditions ◽  
Convex Sets ◽  
Torsional Rigidity ◽  
Neumann Boundary ◽  
Robin Boundary Conditions ◽  
First Eigenvalue ◽  
Laplacian Eigenvalue ◽  
Sharp Estimates ◽  
The First Eigenvalue ◽  
Robin Boundary

We study, in dimension n ≥ 2, the eigenvalue problem and the torsional rigidity for the p-Laplacian on convex sets with holes, with external Robin boundary conditions and internal Neumann boundary conditions. We prove that the annulus maximizes the first eigenvalue and minimizes the torsional rigidity when the measure and the external perimeter are fixed.

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 Twin Positive Solutions for Schrödinger-Kirchhoff-Type Problem with Singularity and Critical Exponents

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Wei Han ◽  
Yangyang Zhao
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Perturbation Methods ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
The First Eigenvalue ◽  
Kirchhoff Type Problem

We study in this paper the following singular Schrödinger-Kirchhoff-type problem with critical exponent -a+b∫Ω∇u2dxΔu+u=Q(x)u5+μxα-2u+f(x)(λ/uγ) in Ω,u=0 on ∂Ω, where a,b>0 are constants, Ω⊂R3 is a smooth bounded domain, 0<α<1, λ>0 is a real parameter, γ∈(0,1) is a constant, and 0<μ<aμ1 (μ1 is the first eigenvalue of -Δu=μxα-2u, under Dirichlet boundary condition). Under appropriate assumptions on Q and f, we obtain two positive solutions via the variational and perturbation methods.

From Steklov to Neumann and Beyond, via Robin: The Szegő Way

2019 ◽  
Vol 72 (4) ◽  
pp. 1024-1043 ◽  
Author(s):  
Pedro Freitas ◽  
Richard S. Laugesen
Keyword(s):  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Steklov Eigenvalue ◽  
Robin Laplacian ◽  
Fixed Area ◽  
Neumann Laplacian ◽  
Simply Connected ◽  
Second Eigenvalue ◽  
Simply Connected Domains

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

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.

scholarly journals Ul’yanov-type inequality for bounded convex sets in Rd

2008 ◽  
Vol 151 (1) ◽  
pp. 60-85 ◽  
Author(s):  
Z. Ditzian ◽  
A. Prymak
Keyword(s):  
Convex Sets ◽  
Type Inequality ◽  
Bounded Convex
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