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Maria Rosaria Posteraro

Publications and source records attributed to Maria Rosaria Posteraro.

5 recordsLinked to original sources
arXiv2023

Isoperimetric sets for weighted twisted eigenvalues

In tis paper we prove an isoperimetric inequality for the first twisted eigenvalue $λ_{1,γ}^T(Ω)$ of a weighted operator, defined as the minimum of the usual Rayleigh quotient when the trial functions belong to the weighted Sobolev space $H_0^1(Ω,dγ)$ and have weighted mean value equal to zero in $Ω$. We are interested in positive measures $dγ=γ(x) dx$ for which we are able to identify the isoperimetric sets, namely, the sets that minimize $λ_{1,γ}^T(Ω)$ among sets of given weighted measure. In the cases under consideration, the optimal sets are given by two identical and disjoint copies of the isoperimetric sets (for the weighted perimeter with respect to the weighted measure).

math.AP↗
arXiv2019

Some isoperimetric inequalities with respect to monomial weights

We solve a class of isoperimetric problems on $\mathbb{R}^2_+ :=\left\{ (x,y)\in \mathbb{R} ^2 : y>0 \right\}$ with respect to monomial weights. Let $α$ and $β$ be real numbers such that $0\le α<β+1$, $β\le 2 α$. We show that, among all smooth sets $Ω$ in $\mathbb{R} ^2_+$ with fixed weighted measure $\iint_{Ω} y^β dxdy$, the weighted perimeter $\int_{\partial Ω} y^α\, ds$ achieves its minimum for a smooth set which is symmetric w.r.t. to the $y$--axis, and is explicitly given. Our results also imply an estimate of a weighted Cheeger constant and a lower bound for the first eigenvalue of a class of nonlinear problems.

math.AP↗
arXiv2018

The isoperimetric problem for a class of non-radial weights and applications

We study a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the densities of the weighted volume and weighted perimeter are given by two different non-radial functions of the type $|x|^k x_N^α$. Our results imply some sharp functional inequalities, like for instance, Caffarelli-Kohn-Nirenberg type inequalities.

math.AP↗
arXiv2018

On weighted isoperimetric inequalities with non-radial densities

We consider a class of isoperimetric problems on $\mathbb{R}^{N}_{+} $ where the volume and the area element carry two different weights of the type $|x|^lx_N^α$. We solve them in a special case while a more detailed study is contained in \cite{ABCMP2}. Our results imply a weighted Polya-Szëgo principle and a priori estimates for weak solutions to a class of boundary value problems for degenerate elliptic equations

math.AP↗