Searcharxiv⌕ Search

arXiv subjects

Chiara Bianchini

Publications and source records attributed to Chiara Bianchini.

12 recordsLinked to original sources
arXiv2021

On the quantitative isoperimetric inequality in the plane with the barycentric distance

In this paper we study the following quantitative isoperimetric inequality in the plane: $λ_0^2(Ω) \leq C δ(Ω)$ where $δ$ is the isoperimetric deficit and $λ_0$ is the barycentric asymmetry. Our aim is to generalize some results obtained by B. Fuglede in \cite{Fu93Geometriae}. For that purpose, we consider the shape optimization problem: minimize the ratio $δ(Ω)/λ_0^2(Ω)$ in the class of compact connected sets and in the class of convex sets.

math.OC↗
arXiv2018

Some overdetermined problems related to the anisotropic capacity

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler $p$-capacity of a convex set $Ω\subset \mathbb{R}^N$, with $1<p<N$. In particular we show that if the Finsler $p$-capacitary potential $u$ associated to $Ω$ has two homothetic level sets then $Ω$ is Wulff shape. Moreover, we show that the concavity exponent of $u$ is $q=-(p-1)/(N-p)$ if and only if $Ω$ is Wulff shape.

math.AP↗
arXiv2017

Wulff shape characterizations in overdetermined anisotropic elliptic problems

We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems.

math.AP↗
arXiv2016

An overdetermined problem for the anisotropic capacity

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in $\mathbb{R}^N$, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of W. Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm $H$. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm $H_0$).

math.AP↗
arXiv2015

On The Quantitative Isoperimetric Inequality In The Plane

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $Ω$, different from a ball, which minimizes the ratio $δ(Ω)/λ^2(Ω)$, where $δ$ is the isoperimetric deficit and $λ$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

math.MG↗
arXiv2014

Elastic energy of a convex body

In this paper a Blaschke-Santaló diagram involving the area, the perimeter and the elastic energy of planar convex bodies is considered. More precisely we give a description of set $$\mathcal{E}:=\left\{(x,y)\in \R^2, x=\frac{4πA(Ω)}{P(Ω)^2},y=\frac{E(Ω)P(Ω)}{2π^2},\,Ω\mbox{convex} \right\},$$ where $A$ is the area, $P$ is the perimeter and $E$ is the elastic energy, that is a Willmore type energy in the plane. In order to do this, we investigate the following shape optimization problem: $$\min_{Ω\in\mathcal{C}}\{E(Ω)+μA(Ω)\},$$ where $\mathcal{C}$ is the class of convex bodies with fixed perimeter and $μ\ge 0$ is a parameter. Existence, regularity and geometric properties of solutions to this minimum problem are shown.

math.OC↗
arXiv2013

An overdetermined problem with non constant boundary condition

We investigate an overdetermined Torsion problem, with a non-constant positively homogeneous boundary constraint on the gradient. We interpret this problem as the Euler equation of a shape optimization problems, we prove existence and regularity of a solution. Moreover several geometric properties of the solution are shown.

math.AP↗
arXiv2011

Isoperimetry and Stability of Hyperplanes for Product Probability Measures

We investigate stationarity and stability of half-spaces as isoperimetric sets for product probability measures, considering the cases of coordinate and non-coordinate half-spaces. Moreover, we present several examples to which our results can be applied, with a particular emphasis on the logistic measure.

math.FA↗
arXiv2010

Optimal sets for a class of minimization problems with convex constraints

We look for the minimizers of the functional $\jla{\la}(\oo)=\la|\oo|-P(\oo)$ among planar convex domains constrained to lie into a given ring. We prove that, according to the values of the parameter $\la$, the solutions are either a disc or a polygon. In this last case, we describe completely the polygonal solutions by reducing the problem to a finite dimensional optimization problem. We recover classical inequalities for convex sets involving area, perimeter and inradius or circumradius and find a new one.

math.AP↗
arXiv2010

A Bernoulli problem with non constant gradient boundary constraint

We present in this paper a result about existence and convexity of solutions to a free boundary problem of Bernoulli type, with non constant gradient boundary constraint depending on the outer unit normal. In particular we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.

math.AP↗