Ilias Ftouhi, Friedrich-Alexander Universität
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PDL C-038
A fundamental question that frequently arises in mathematics is to
understand inequalities between different quantities and, ideally,
identifying the sharpest ones. Blaschke--Santal\'o diagrams provide an
elegant and effective way to visualize the best possible inequalities
relating various quantities.
As an example, in this talk, we focus on the spectrum of the Laplace
operator with Dirichlet boundary conditions on $\partial \Omega$, where
$\Omega \subset \mathbb{R}^d$, specifically its first eigenvalue, also
known as the fundamental frequency $\lambda_1(\Omega)$. Unfortunately,
an explicit formula for $\lambda_1(\Omega)$ is generally unavailable.
This motivates the search for estimates using other functionals that are
easier to manipulate, such as the perimeter $P(\Omega)$ and the volume
$|\Omega|$.
To better understand the inequalities linking the first eigenvalue, the
perimeter and the volume, we introduce the Blaschke--Santal\'o diagram
of the triplet $(P, \lambda_1, |\cdot|)$, defined as follows:
where $F_{ad}$ is a given class of subsets of $\mathbb{R}^d$. It is
important to note that characterizing this diagram is equivalent to
identifying all possible inequalities involving the considered
quantities.
We fully describe the diagram for open sets, demonstrating that no
inequalities exist beyond the classical Faber--Krahn and the
isoperimetric inequalities. This motivates the exploration of other
classes of sets, such as convex ones, for which we provide an advanced
description of the corresponding diagram. Finally, we discuss the use of
numerical tools and shape optimization methods to obtain an optimal
characterization of such diagrams, illustrating our results with various
examples. This talk is based on joint works with Jimmy Lamboley
(Sorbonne University, France).
understand inequalities between different quantities and, ideally,
identifying the sharpest ones. Blaschke--Santal\'o diagrams provide an
elegant and effective way to visualize the best possible inequalities
relating various quantities.
As an example, in this talk, we focus on the spectrum of the Laplace
operator with Dirichlet boundary conditions on $\partial \Omega$, where
$\Omega \subset \mathbb{R}^d$, specifically its first eigenvalue, also
known as the fundamental frequency $\lambda_1(\Omega)$. Unfortunately,
an explicit formula for $\lambda_1(\Omega)$ is generally unavailable.
This motivates the search for estimates using other functionals that are
easier to manipulate, such as the perimeter $P(\Omega)$ and the volume
$|\Omega|$.
To better understand the inequalities linking the first eigenvalue, the
perimeter and the volume, we introduce the Blaschke--Santal\'o diagram
of the triplet $(P, \lambda_1, |\cdot|)$, defined as follows:
where $F_{ad}$ is a given class of subsets of $\mathbb{R}^d$. It is
important to note that characterizing this diagram is equivalent to
identifying all possible inequalities involving the considered
quantities.
We fully describe the diagram for open sets, demonstrating that no
inequalities exist beyond the classical Faber--Krahn and the
isoperimetric inequalities. This motivates the exploration of other
classes of sets, such as convex ones, for which we provide an advanced
description of the corresponding diagram. Finally, we discuss the use of
numerical tools and shape optimization methods to obtain an optimal
characterization of such diagrams, illustrating our results with various
examples. This talk is based on joint works with Jimmy Lamboley
(Sorbonne University, France).