Searcharxiv⌕ Search

arXiv subjects

Alba Lia Masiello

Publications and source records attributed to Alba Lia Masiello.

At least 19 recordsLinked to original sources
arXiv2026

On some functionals for which the ball is a saddle shape

In the present paper, we study the maximization problem of the $k$-Torsional rigidity under quermassintegral constraint, with particular emphasis on the role of the ball. Our main result shows a phenomenon which, as far as we know, has not previously been observed: the ball, rather than being an extremal shape, is a saddle point. More precisely, for suitable quermassintegral constraints, we show that the $k$-torsional rigidity admits both increasing and decreasing perturbations around the ball. The proof is based on a second-order analysis of the corresponding variational problem. To the best of our knowledge, this is the first use of a second-order shape derivative approach in the study of variational problems for $k$-Hessian operators. We also establish an analogous saddle-point behavior for a related curvature functional under a volume constraint, highlighting a structural distinction between the intermediate Hessian regime and the classical Laplace and Monge-Ampère cases.

math.AP↗
arXiv2026

Asymptotic behavior of the first Robin eigenvalue of nonlinear operators

Let $Ω$ be a bounded Lipschitz domain of $\mathbb R^N$, $N\geq 2$. In this paper, we study the asymptotic behavior of the first Robin eigenvalue of the $p$-Laplace operator as $β$ goes to $0$ and as $β$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $β\to+\infty$ is obtained under the additional assumption that $\partialΩ$ is of class $C^{1,1}$.

math.AP↗
arXiv2026

Geometrical bounds for the torsion and the first eigenvalue of the Laplacian with Robin boundary condition

In this paper, we deal with functionals involving the torsion and the first eigenvalue of the Laplacian with Robin boundary conditions (to which we refer as Robin Torsion and Robin Eigenvalue), with other geometrical quantities, in the class of convex sets. Firstly, we prove an upper bound for the Robin Torsion in terms of the $L^1$ and $L^2$ norms of the distance function from the boundary, which allows us to prove a generalization of the Makai inequality involving the Robin Torsion, the Lebeasgue measure, and the inradius of a convex set. Subsequently, we prove quantitative estimates for the Robin Makai functional and for the Robin Pólya functionals, which link the Lebesgue measure and the perimeter with the Robin Torsion and the Robin Eigenvalue respectively. In particular, we prove that the optimal values of all these shape functionals are achieved by slab domains.

math.AP↗
arXiv2025

A quantitative Talenti-type comparison result with Robin boundary conditions

The purpose of this paper is to establish a quantitative version of the Talenti comparison principle for solutions to the Poisson equation with Robin boundary conditions. This quantitative enhancement is proved in terms of the asymmetry of domain. The key role is played by a careful analysis of the propagation of asymmetry for the level sets of the solutions of a PDE. As a byproduct, we obtain an alternative proof of the quantitative Saint-Venant inequality for the Robin torsion and, in the planar case, of the quantitative Faber-Krahn inequality for the first Robin eigenvalue. In addition, we complete the framework of the rigidity result of the Talenti inequalities with Robin boundary conditions.

math.AP↗
arXiv2025

On the second eigenvalue of the infinity Laplacian with Robin boundary conditions

We study the behaviour, as $p \to +\infty$, of the second eigenvalues of the $p$-Laplacian with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that, up to some regularity of the set, the limit of the second eigenvalues is actually the second eigenvalue of the so-called $\infty$-Laplacian.

math.AP↗
arXiv2025

Some shape functionals for the $k$-Hessian equation

For a non-empty, bounded, open, and convex set of class $C^2$, we consider the Torsional Rigidity associated to the $k$-Hessian operator. We first prove Pólya type lower bound for the $k$-Torsional Rigidity in any dimension; then, in order to investigate optimal sets in the Pólya type inequality, we provide two quantitative estimates.

math.AP↗
arXiv2025

Hessian operators, overdetermined problems, and higher order mean curvatures: symmetry and stability results

It is well known that there is a deep connection between Serrin's symmetry result -- dealing with overdetermined problems involving the Laplacian -- and the celebrated Alexandrov's Soap Bubble Theorem (SBT) -- stating that, if the mean curvature $H$ of the boundary of a smooth bounded connected open set $\Om$ is constant, then $\Om$ must be a ball. One of the main aims of the paper is to extend the study of such a connection to the broader case of overdetermined problems for Hessian operators and constant higher order mean curvature boundaries. Our analysis will not only provide new proofs of the higher order SBT (originally established by Alexandrov) and of the symmetry for overdetermined Serrin-type problems for Hessian equations (originally established by Brandolini, Nitsch, Salani, and Trombetti), but also bring several benefits, including new interesting symmetry results and quantitative stability estimates. In fact, leveraging the analysis performed in the classical case (i.e., with classical mean curvature and classical Laplacian) by Magnanini and Poggesi in a series of papers, we will extend their approach to the higher order setting (i.e., with $k$-order mean curvature and $k$-Hessian operator, for $k \ge 1$) achieving various quantitative estimates of closeness to the symmetric configuration. Finally, leveraging the quantitative analysis in presence of bubbling phenomena performed in arXiv:2405.06376, we also provide a quantitative stability result of closeness of almost constant $k$-mean curvature boundaries to a set given by the union of a finite number of disjoint balls of equal radii. In passing, we will also provide two alternative proofs of the result established by Brandolini, Nitsch, Salani, and Trombetti, one of which provides the extension to Hessian operators of the approach famously pioneered by Weinberger for the classical Laplacian.

math.AP↗
arXiv2025

A quantitative result for the $k$-Hessian equation

In this paper, we study a symmetrization that preserves the mixed volume of the sublevel sets of a convex function, under which, a Pólya-Szeg\H o type inequality holds. We refine this symmetrization to obtain a quantitative improvement of the Pólya-Szeg\H o inequality for the $k$-Hessian integral, and, with similar arguments, we show a quantitative inequality for the comparison proved by Tso \cite{tso} for solutions to the $k$-Hessian equation. As an application of the first result, we prove a quantitative version of the Faber-Krahn and Saint-Venant inequalities for these equations.

math.AP↗
arXiv2024

Sharp inequalities involving the Cheeger constant of planar convex sets

We are interested in finding sharp bounds for the Cheeger constant $h$ via different geometrical quantities, namely the area $|\cdot|$, the perimeter $P$, the inradius $r$, the circumradius $R$, the minimal width $ω$ and the diameter $d$. We provide new sharp inequalities between these quantities for planar convex bodies and enounce new conjectures based on numerical simulations. In particular, we completely solve the Blaschke-Santaló diagrams describing all the possible inequalities involving the triplets $(P,h,r)$, $(d,h,r)$ and $(R,h,r)$ and describe some parts of the boundaries of the diagrams of the triplets $(ω,h,d)$, $(ω,h,R)$, $(ω,h,P)$, $(ω,h,|\cdot|)$, $(R,h,d)$ and $(ω,h,r)$.

math.AP↗
arXiv2024

Finite element approximation of the Hardy constant

We consider finite element approximations to the optimal constant for the Hardy inequality with exponent $p=2$ in bounded domains of dimension $n=1$ or $n \geq 3$. For finite element spaces of piecewise linear and continuous functions on a mesh of size $h$, we prove that the approximate Hardy constant converges to the optimal Hardy constant at a rate proportional to $1/| \log h |^2$. This result holds in dimension $n=1$, in any dimension $n \geq 3$ if the domain is the unit ball and the finite element discretization exploits the rotational symmetry of the problem, and in dimension $n=3$ for general finite element discretizations of the unit ball. In the first two cases, our estimates show excellent quantitative agreement with values of the discrete Hardy constant obtained computationally.

math.NA↗
arXiv2024

The Talenti comparison result in a quantitative form

In this paper, we obtain a quantitative version of the classical comparison result of Talenti for elliptic problems with Dirichlet boundary conditions. The key role is played by quantitative versions of the Pólya-Szego inequality and of the Hardy-Littlewood inequality.

math.AP↗
arXiv2023

A remark on solutions to semilinear equations with Robin boundary conditions

Symmetry properties of solutions to elliptic quasilinear equations have been widely studied in the context of Dirichlet boundary conditions. We show that, in the context of Robin boundary conditions, the symmetry property á la Gidas, Ni and Nirenberg does not hold in dimension $n\geq 2$, even for superharmonic functions, and we provide an explicit example.

math.AP↗
arXiv2023

A Rigidity Result for the Robin Torsion Problem

Let $Ω\subset \mathbb{R}^2$ be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to $Ω$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in arXiv:1909.11950. We prove that the equality is achieved only if $Ω$ is a disk and the torsion function $u$ is radial.

math.AP↗
arXiv2023

Rigidity results for the $p$-Laplacian Poisson problem with Robin boundary conditions

Let $Ω\subset \mathbb{R}^n$ be an open, bounded and Lipschitz set. We consider the Poisson problem for the $p-$Laplace operator associated to $Ω$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison stated in \cite{AGM}. We prove that the equality is achieved only if $Ω$ is a ball and both the function $u$ and the right hand side $f$ of the Poisson equation are radial.

math.AP↗
arXiv2022

Sharp and quantitative estimates for the $p-$Torsion of convex sets

Let $Ω\subset\mathbb{R}^n$, $n\geq 2$, be a bounded, open and convex set and let $f$ be a positive and non-increasing function depending only on the distance from the boundary of $Ω$. We consider the $p-$torsional rigidity associated to $Ω$ for the Poisson problem with Dirichlet boundary conditions, denoted by $T_{f,p}(Ω)$. Firstly, we prove a Pólya type lower bound for $T_{f,p}(Ω)$ in any dimension; then, we consider the planar case and we provide two quantitative estimates in the case $f\equiv 1 $.

math.AP↗
arXiv2022

Estimates for Robin $p$-Laplacian eigenvalues of convex sets with prescribed perimeter

In this paper, we prove an upper bound for the first Robin eigenvalue of the $p$-Laplacian with a positive boundary parameter and a quantitative version of the reverse Faber-Krahn type inequality for the first Robin eigenvalue of the $p$-Laplacian with negative boundary parameter, among convex sets with prescribed perimeter. The proofs are based on a comparison argument obtained by means of inner sets, introduced by Payne, Weimberger and Polya.

math.AP↗