arXiv subjects
Publications and source records attributed to Gisella Croce.
We study the existence of an optimizer for a quantitative isoperimetric ratio $\mathcal{Q}_*$ in $\mathbb{R}^3$ involving the Hausdorff asymmetry. We prove that $\mathcal{Q}_*$ attains its minimum over the class $\mathcal{A}$ of convex bodies of fixed volume.
In a recent paper, C. Gambicchia and A. Pratelli proved a quantitative isoperimetric inequality involving the isoperimetric deficit $\delta(K)$ and the barycentric distance $\lambda_0(K)$ for sets $K\subset \mathbb{R}^N$ with given diameter $D$ and measure. In this work we are interested in the optimal sets for this inequality in the plane, i.e. sets that minimize the ratio $\delta(K)/\lambda_0(K)^2$. We prove existence of optimal sets (at least when $D$ is large enough), regularity and express the optimality conditions. Moreover, we prove that the optimal sets have exactly two connected components and their boundary does not contain any arc of circle.
We study the shape of solutions to some variational problems in Sobolev spaces with weights that are powers of |x|. In particular, we detect situations when the extremal functions lack symmetry properties such as radial symmetry and antisymmetry. We also prove an isoperimetric inequality for the first non-zero eigenvalue of a weighted Neumann problem.
In this note we prove the existence of a set $E_0\subset\mathbb{R}^2$, different from a ball, which minimizes, among the convex sets that satisfy a suitable interior cone condition, the ratio \begin{equation} \label{eq:0} \frac{D(E)}{\lambda_\mathcal{H}^2(E)}, \end{equation} where $D$ is the isoperimetric deficit and $\lambda_\mathcal{H}$ the deviation from the spherical shape of a set $E\subset \mathbb{R}^2$.
In this paper, we study a reverse isoperimetric inequality for planar convex bodies whose radius of curvature is between two positive numbers 0 < $\alpha$ < $\beta$, called ($\alpha$, $\beta$)--convex bodies. We show that among planar ($\alpha$, $\beta$)--convex bodies of fixed perimeter, the extremal shape is a domain whose boundary is composed by two arcs of circles of radius $\alpha$ joined by two arcs of circles of radius $\beta$.
In this paper, we consider a problem in calculus of variations motivated by a quantitative isoperimetric inequality in the plane. More precisely, the aim of this article is the computation of the minimum of the variational problem $$\inf_{u\in\mathcal{W}}\frac{\displaystyle\int_{-\pi}^{\pi}[(u')^2-u^2]d\theta}{\displaystyle\left[\int_{-\pi}^{\pi}|u| d\theta\right]^2}$$where $u\in \mathcal{W}$ is a $H^1(-\pi,\pi)$ periodic function, with zero average on $(-\pi,\pi)$ and orthogonal to sine and cosine.
In this paper we study the following quantitative isoperimetric inequality in the plane: $\lambda_0^2(\Omega) \leq C \delta(\Omega)$ where $\delta$ is the isoperimetric deficit and $\lambda_0$ 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 $\delta(\Omega)/\lambda_0^2(\Omega)$ in the class of compact connected sets and in the class of convex sets.
For $\mathrm{H} \in C^2(\mathbb{R}^{N \times n})$ and $u : Ω\subseteq \mathbb{R}^n \to \mathbb{R}^N$, consider the system \[ \label{1}\mathrm{A}\_\infty u\, :=\,\Big(\mathrm{H}\_P \otimes \mathrm{H}\_P + \mathrm{H}[\mathrm{H}\_P]^\bot \mathrm{H}\_{PP}\Big)(\mathrm{D} u): \mathrm{D}^2 u\, =\,0. \tag{1}\]We construct $\mathcal{D}$-solutions to the Dirichlet problem for (1), an apt notion of generalised solutions recently proposed for fully nonlinear systems. Our $\mathcal{D}$-solutions are $W^{1,\infty}$-submersions and are obtained without any convexity hypotheses for $\mathrm{H}$, through a result of independent interest involving existence of strong solutions to the singular value problem for general dimensions $n\neq N$.
We consider the vectorial system \[ \begin{cases} Du \in \mathcal{O}(2), & \mbox{a.e. in}\;Ω, u=0, & \mbox{on} \;\partial Ω, \end{cases} \] where $Ω$ is a subset of $\R^2$, $u:Ω\to \R^2$ and $\mathcal{O}(2)$ is the orthogonal group of $\R^2$. We provide a variational method to select, among the infinitely many solutions, the ones that minimize an appropriate weighted measure of the singular set of the gradient.
In this paper we study the existence of solutions of thedegererate elliptic system.
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.
We study a degenerate elliptic equation, proving existence results of distributional solutions in some borderline cases.
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a W^{1,1}_0 solution which is distributional or entropic, according to the growth assumptions on a lower order term in divergence form.
We study a degenerate elliptic equation, proving the existence of a W^{1,1}_0 distributional solution.
We prove an isoperimetric inequality of the Rayleigh-Faber-Krahn type for a nonlinear generalization of the first twisted Dirichlet eigenvalue. More precisely, we show that the minimizer among sets of given volume is the union of two equal balls.
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a unique W^{1,1}_0 distributional solution under suitable summability assumptions on the source in Lebesgue spaces. Moreover, we prove that our problem has no solution if the source is a Radon measure concentrated on a set of zero harmonic capacity.
We study an integral non coercive functional defined on H^1_0, proving the existence of a minimum in W^{1,1}_0.
We study a Dirichlet problem for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side. We will show that the right-hand side has some regularizing effects on the solutions, even if it is singular.