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Rosa Barbato

Publications and source records attributed to Rosa Barbato.

8 recordsLinked to original sources
arXiv2026

Asymptotic behavior of the first Robin eigenvalue of nonlinear operators

Let $\Omega$ 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 $\beta$ goes to $0$ and as $\beta$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $\beta\to+\infty$ is obtained under the additional assumption that $\partial\Omega$ 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 P\'olya 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.

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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↗
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.

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arXiv2023

On the first Robin eigenvalue of the Finsler $p$-Laplace operator as $p\to 1$

Let $\Omega$ be a bounded, connected, sufficiently smooth open set, $p>1$ and $\beta\in\mathbb R$. In this paper, we study the $\Gamma$-convergence, as $p\rightarrow 1^+$, of the functional \[ J_p(\varphi)=\frac{\int_\Omega F^p(\nabla \varphi)dx+\beta\int_{\partial \Omega} |\varphi|^pF(\nu)d\mathcal{H}^{N-1}}{\int_\Omega |\varphi|^pdx} \] where $\varphi\in W^{1,p}(\Omega)\setminus\{0\}$ and $F$ is a sufficientely smooth norm on $\mathbb R^n$. We study the limit of the first eigenvalue $\lambda_1(\Omega,p,\beta)=\inf_{\substack{\varphi\in W^{1,p}(\Omega)\\ \varphi \ne 0}}J_p(\varphi)$, as $p\to 1^+$, that is: \begin{equation*} \Lambda(\Omega,\beta)=\inf_{\substack{\varphi \in BV(\Omega)\\ \varphi\not\equiv 0}}\dfrac{|Du|_F(\Omega)+\min\{\beta,1\}\displaystyle \int_{\partial \Omega}|\varphi|F(\nu)d\mathcal H^{N-1}}{\displaystyle s\int_\Omega |\varphi|dx}. \end{equation*} Furthermore, for $\beta>-1$, we obtain an isoperimetric inequality for $\Lambda(\Omega,\beta)$ depending on $\beta$. The proof uses an interior approximation result for $BV(\Omega)$ functions by $C^\infty(\Omega)$ functions in the sense of strict convergence on $\mathbb R^n$ and a trace inequality in $BV$ with respect to the anisotropic total variation.

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arXiv2022

Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper we consider a minimization problem of the type $$ I_{β,p}(D;Ω)=\inf\biggl\{\int_Ω\lvert{Dϕ}\rvert^pdx+β\int_{\partial^* Ω}\lvertϕ\rvert^pd\mathcal{H}^{n-1},\; ϕ\in W^{1,p}(Ω),\;ϕ\geq 1 \;\textrm{in}\;D\biggl\}, $$ where $Ω$ is a bounded connected open set in $\mathbb{R}^n$, $D\subset \barΩ$ is a compact set and $β$ is a positive constant. We let the set $D$ vary under prescribed geometrical constraints and $Ω\setminus D$ of fixed thickness, in order to look for the best (or worst) geometry in terms of minimization (or maximization) of $I_{β,p}$. In the planar case, we show that under perimeter constraint the disk maximize $I_{β,p}$. In the $n$-dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.

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