arXiv subjects

Gloria Paoli

Publications and source records attributed to Gloria Paoli.

24 records · Page 2Linked to original sources

arXiv2021

The orthotropic $p$-Laplace eigenvalue problem of Steklov type as $p\to+\infty$

We study the Steklov eigenvalue problem for the $\infty-$orthotropic Laplace operator defined on convex sets of $\mathbb{R}^N$, with $N\geq2$, considering the limit for $p\to+\infty$ of the Steklov problem for the $p-$orthotropic Laplacian. We find a limit problem that is satisfied in the viscosity sense and a geometric characterization of the first non trivial eigenvalue. Moreover, we prove Brock-Weinstock and Weinstock type inequalities among convex sets, stating that the ball in a suitable norm maximizes the first non trivial eigenvalue for the Steklov $\infty-$orthotropic Laplacian, once we fix the volume or the anisotropic perimeter.

math.AP↗

arXiv2020

An estimate for the anisotropic maximum curvature in the planar case

We fix a Finsler norm $F$ and, using the anisotropic curvature flow, we prove that the anisotropic maximum curvature $k^F_{\max}$ of a smooth Jordan curve is such that $ k^F_{\max}(γ)\geq \sqrt{κ/A}$ , where $A$ is the area enclosed by $γ$ and $κ$ the area of the unitary Wulff shape associated to the anisotropy $F$.

math.AP↗

arXiv2020

Sharp estimates for the first $p$-Laplacian eigenvalue and for the $p$-torsional rigidity on convex sets with holes

We study, in dimension $n\geq2$, 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.

math.AP↗

arXiv2019

A quantitative Weinstock inequality

The paper is devoted to the study of a quantitative Weinstock inequality in higher dimension for the first non trivial Steklov eigenvalue of Laplace operator for convex sets. The key rule is played by a quantitative isoperimetric inequality which involves the boundary momentum, the volume and the perimeter of a convex open set of $\mathbb R^n$, $n \ge 2$.

math.AP↗

arXiv2019

Anisotropic Isoperimetric Inequalities involving Boundary Momentum, Perimeter and Volume

We consider a scale invariant functional involving the anisotropic $p-$momentum, the anisotropic perimeter and the volume. We show that the Wulff shape, associated with the Finsler norm $F$ considered and centered at the origin, is the unique minimizer of the anisotropic functional taken into consideration among all bounded convex sets.

math.AP↗

arXiv2018

Two estimates for the first Robin eigenvalue of the Finsler Laplacian with negative boundary parameter

We prove two bounds for the first Robin eigenvalue of the Finsler Laplacian with negative boundary parameter in the planar case. In the constant area problem, we show that the Wulff shape is the maximizer only for values which are close to 0 of the boundary parameter and, in the fixed perimeter case, that the Wulff shape maximizes the first eigenvalue for all values of the parameter.

math.AP↗