scholarly journals On relations between principal eigenvalue and torsional rigidity

2020 ◽  
pp. 2050093
Author(s):  
Michiel van den Berg ◽  
Giuseppe Buttazzo ◽  
Aldo Pratelli
Keyword(s):  
Optimization Problem ◽  
Torsional Rigidity ◽  
Principal Eigenvalue ◽  
Higher Dimensions ◽  
First Eigenvalue ◽  
Thin Domains ◽  
Convex Domains ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Dimension One

We consider the problem of minimizing or maximizing the quantity [Formula: see text] on the class of open sets of prescribed Lebesgue measure. Here [Formula: see text] is fixed, [Formula: see text] denotes the first eigenvalue of the Dirichlet Laplacian on [Formula: see text], while [Formula: see text] is the torsional rigidity of [Formula: see text]. The optimization problem above is considered in the class of all domains [Formula: see text], in the class of convex domains [Formula: see text], and in the class of thin domains. The full Blaschke–Santaló diagram for [Formula: see text] and [Formula: see text] is obtained in dimension one, while for higher dimensions we provide some bounds.

An eigenvalue optimization problem for the p-Laplacian

2015 ◽  
Vol 145 (6) ◽  
pp. 1145-1151 ◽  
Author(s):  
Anisa M. H. Chorwadwala ◽  
Rajesh Mahadevan
Keyword(s):  
Eigenvalue Problem ◽  
Optimization Problem ◽  
First Eigenvalue ◽  
Eigenvalue Optimization ◽  
Nonlinear Case ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Monotonicity Result ◽  
Open Question

It has been shown by Kesavan (Proc. R. Soc. Edinb. A (133) (2003), 617–624) that the first eigenvalue for the Dirichlet Laplacian in a punctured ball, with the puncture having the shape of a ball, is maximum if and only if the balls are concentric. Recently, Emamizadeh and Zivari-Rezapour (Proc. Am. Math. Soc.136 (2007), 1325–1331) have tried to generalize this result to the case of the p-Laplacian but could succeed only in proving a domain monotonicity result for a weighted eigenvalue problem in which the weights need to satisfy some artificial conditions. In this paper we generalize the result of Kesavan to the case of the p-Laplacian (1 < p < ∞) without any artificial restrictions, and in the process we simplify greatly the proof, even in the case of the Laplacian. The uniqueness of the maximizing domain in the nonlinear case is still an open question.

scholarly journals A spectral shape optimization problem with a nonlocal competing term

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Dario Mazzoleni ◽  
Berardo Ruffini
Keyword(s):  
Shape Optimization ◽  
Optimization Problem ◽  
Critical Value ◽  
Relative Strength ◽  
Spectral Shape ◽  
First Eigenvalue ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Existence Of Minimizers ◽  
Type Interaction

AbstractWe study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.

scholarly journals Convex duality for principal frequencies

2021 ◽  
Vol 4 (4) ◽  
pp. 1-28
Author(s):  
Lorenzo Brasco ◽  
Keyword(s):  
Lower Bounds ◽  
Variational Formulation ◽  
Torsional Rigidity ◽  
First Eigenvalue ◽  
Convex Minimization Problem ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Divergence Type ◽  
Poincaré Constant ◽  
Type Constraint

<abstract><p>We consider the sharp Sobolev-Poincaré constant for the embedding of $ W^{1, 2}_0(\Omega) $ into $ L^q(\Omega) $. We show that such a constant exhibits an unexpected dual variational formulation, in the range $ 1 &lt; q &lt; 2 $. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $ q = 1 $) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $ q = 2 $).</p></abstract>

scholarly journals A variation on the Donsker–Varadhan inequality for the principal eigenvalue

2017 ◽  
Vol 473 (2204) ◽  
pp. 20160877
Author(s):  
Jianfeng Lu ◽  
Stefan Steinerberger
Keyword(s):  
Lower Bound ◽  
Exit Time ◽  
Principal Eigenvalue ◽  
Short Paper ◽  
First Eigenvalue ◽  
Drift Diffusion ◽  
First Exit Time ◽  
The First Eigenvalue ◽  
Order Elliptic Operator ◽  
The Mean

The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .

scholarly journals An optimization problem for the first eigenvalue of the p-fractional Laplacian

2018 ◽  
Vol 291 (4) ◽  
pp. 632-651 ◽  
Author(s):  
Leandro Del Pezzo ◽  
Julián Fernández Bonder ◽  
Luis López Ríos
Keyword(s):  
Fractional Laplacian ◽  
Optimization Problem ◽  
First Eigenvalue ◽  
The First Eigenvalue

scholarly journals Some Inequalities Involving Perimeter and Torsional Rigidity

2020 ◽  
Author(s):  
Luca Briani ◽  
Giuseppe Buttazzo ◽  
Francesca Prinari
Keyword(s):  
Lebesgue Measure ◽  
Torsional Rigidity ◽  
Thin Domains ◽  
Convex Domains

Abstract We consider shape functionals of the form $$F_q(\Omega )=P(\Omega )T^q(\Omega )$$ F q ( Ω ) = P ( Ω ) T q ( Ω ) on the class of open sets of prescribed Lebesgue measure. Here $$q>0$$ q > 0 is fixed, $$P(\Omega )$$ P ( Ω ) denotes the perimeter of $$\Omega $$ Ω and $$T(\Omega )$$ T ( Ω ) is the torsional rigidity of $$\Omega $$ Ω . The minimization and maximization of $$F_q(\Omega )$$ F q ( Ω ) is considered on various classes of admissible domains $$\Omega $$ Ω : in the class $$\mathcal {A}_{all}$$ A all of all domains, in the class $$\mathcal {A}_{convex}$$ A convex of convex domains, and in the class $$\mathcal {A}_{thin}$$ A thin of thin domains.

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.

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