scholarly journals Erratum to “Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains”

2021 ◽  
Author(s):  
Simon Larson
Keyword(s):  
Shape Optimization ◽  
Convex Domains ◽  
Dirichlet Laplacian ◽  
Riesz Means ◽  
Asymptotic Shape

Riesz means associated with convex domains in the plane

2001 ◽  
Vol 236 (4) ◽  
pp. 643-676 ◽  
Author(s):  
Andreas Seeger ◽  
Sarah Ziesler
Keyword(s):  
Convex Domains ◽  
Riesz Means

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.

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 A free boundary approach to shape optimization problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140273 ◽  
Author(s):  
D. Bucur ◽  
B. Velichkov
Keyword(s):  
Free Boundary ◽  
Shape Optimization ◽  
Optimization Problem ◽  
Free Boundary Problems ◽  
Optimization Problems ◽  
General Shape ◽  
Dirichlet Laplacian ◽  
Boundary Problems ◽  
Survey Article ◽  
The Laplace Operator

The analysis of shape optimization problems involving the spectrum of the Laplace operator, such as isoperimetric inequalities, has known in recent years a series of interesting developments essentially as a consequence of the infusion of free boundary techniques. The main focus of this paper is to show how the analysis of a general shape optimization problem of spectral type can be reduced to the analysis of particular free boundary problems. In this survey article, we give an overview of some very recent technical tools, the so-called shape sub- and supersolutions, and show how to use them for the minimization of spectral functionals involving the eigenvalues of the Dirichlet Laplacian, under a volume constraint.

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