arXiv subjects

V. Ferone

Publications and source records attributed to V. Ferone.

5 recordsLinked to original sources

arXiv2018

On a P\'olya functional for rhombi, isosceles triangles, and thinning convex sets

Let $\Omega$ be an open convex set in ${\mathbb R}^m$ with finite width, and let $v_{\Omega}$ be the torsion function for $\Omega$, i.e. the solution of $-\Delta v=1, v\in H_0^1(\Omega)$. An upper bound is obtained for the product of $\Vert v_{\Omega}\Vert_{L^{\infty}(\Omega)}\lambda(\Omega)$, where $\lambda(\Omega)$ is the bottom of the spectrum of the Dirichlet Laplacian acting in $L^2(\Omega)$. The upper bound is sharp in the limit of a thinning sequence of convex sets. For planar rhombi and isosceles triangles with area $1$, it is shown that $\Vert v_{\Omega}\Vert_{L^{1}(\Omega)}\lambda(\Omega)\ge \frac{\pi^2}{24}$, and that this bound is sharp.

math.AP↗

arXiv2018

A sharp estimate for the first Robin-Laplacian eigenvalue with negative boundary parameter

In this paper we prove that the ball maximizes the first eigenvalue of the Robin Laplacian operator with negative boundary parameter, among all convex sets of \mathbb{R}^n with prescribed perimeter. The key of the proof is a dearrangement procedure of the first eigenfunction of the ball on the level sets of the distance function to the boundary of the convex set, which controls the boundary and the volume energies of the Rayleigh quotient.

math.AP↗

arXiv2016

The quantitative Faber-Krahn inequality for the Robin Laplacian

We prove a quantitative Faber-Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.

math.AP↗

arXiv2016

On Polya's inequality for torsional rigidity and first Dirichlet eigenvalue

Let $Ω$ be an open set in Euclidean space with finite Lebesgue measure $|Ω|$. We obtain some properties of the set function $F:Ω\mapsto \R^+$ defined by $$ F(Ω)=\frac{T(Ω)λ_1(Ω)}{|Ω|} ,$$ where $T(Ω)$ and $λ_1(Ω)$ are the torsional rigidity and the first eigenvalue of the Dirichlet Laplacian respectively. We improve the classical Pólya bound $F(Ω)\le 1,$ and show that $$F(Ω)\le 1- ν_m T(Ω)|Ω|^{-1-\frac2m},$$ where $ν_m$ depends only on $m$. For any $m=2,3,\dots$ and $ε\in (0,1)$ we construct an open set $Ω_ε\subset \R^m$ such that $F(Ω_ε)\ge 1-ε$.

math.AP↗

arXiv2010

The longest shortest fence and sharp Poincaré-Sobolev inequalities

We prove a long standing conjecture concerning the fencing problem in the plane: among planar convex sets of given area, prove that the disc, and only the disc maximizes the length of the shortest area-bisecting curve. Although it may look intuitive, the result is by no means trivial since we also prove that among planar convex sets of given area the set which maximizes the length of the shortest bisecting chords is the so-called Auerbach triangle.

math.OC↗