Blaschke--Santaló Diagram for Volume, Perimeter, and First Dirichlet Eigenvalue

Abstract

We are interested in the study of Blaschke-Santaló diagrams describing the possible inequalities involving the first Dirichlet eigenvalue, the perimeter and the volume, for different classes of sets. We give a complete description of the diagram for the class of open sets in R d , basically showing that the isoperimetric and Faber-Krahn inequalities form a complete system of inequalities for these three quantities. We also give some qualitative results for the Blaschke-Santaló diagram for the class of planar convex domains: we prove that in this case the diagram can be described as the set of points contained between the graphs of two continuous and increasing functions. This shows in particular that the diagram is simply connected, and even horizontally and vertically convex. We also prove that the shapes that fill the upper part of the boundary of the diagram are smooth (C 1,1 ), while those on the lower one are polygons (except for the ball). Finally, we perform some numerical simulations in order to have an idea on the shape of the diagram; we deduce both from theoretical and numerical results some new conjectures about geometrical inequalities involving the functionals under study in this paper.

A Shape Optimization Problem on Planar Sets with Prescribed Topology

“…Finally, shape functionals F( ) involving quantities other than perimeter and torsional rigidity are interesting to be studied: we point out some recent results in [4,9,11] and references therein. However, to our knowledge, the study of these shape functionals under topological constraints as the ones of classes A k is still missing.…”

A shape optimization problem on planar sets with prescribed topology

“…Finally, shape functionals F (Ω) involving quantities other than perimeter and torsional rigidity are interesting to be studied: we point out some recent results in [18], [19] and references therein. However, to our knowledge, the study of these shape functionals under topological constraints as the ones of classes A k is still missing.…”

On Blaschke-Santaló diagrams for the torsional rigidity and the first Dirichlet eigenvalue

“…Proof. Some ideas of the following proof are inspired by [16], in which the authors study the Blaschke-Santaló diagram of the pair (perimeter, λ 1 ), under volume constraint.…”

“…A complete description would amount to characterize the relations between the shape functionals X and Y , by means of (optimal) upper and lower bounds in terms of the boundary points of the diagram and the associated shapes. Since shape functionals and their bounds appear in several mathematical areas (e.g., Poincaré inequalities in functional analysis), Blaschke-Santaló diagrams are very useful tools, and the literature on the subject is quite vast (see for instance [4,29,15,2,11,6,3,23] and more recently [16,5]). As it appears from the literature, the theoretical analysis, even if very fine, is in general not enough for an accurate description, and some aspects remain unsolved.…”