arXiv subjects
Publications and source records attributed to Giovanni Pisante.
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.
Given a bounded convex open set $Ω\subseteq \mathbb R^N$, we prove that the Poincaré-Sobolev constants $λ_{p,q}(Ω)$ can be bounded from below by the $p$-power of the ratio between the perimeter of $Ω$ and a suitable power of its volume, with an optimal constant which is explicitly given. This generalises an old result for torsional rigidity due to Makai when $N=2$. The proof relies on new geometric optimal bounds for the Lebesgue norms of the distance function from the boundary which are of independent interest. These results allow us to give a complete picture of the sharp inequalities for $λ_{p,q}(Ω)$ in terms of suitable powers of perimeter, inradius and volume of $Ω$.
We establish a Leray- Trudinger Type inequality in the anisotropic setting induced by a strongly convex Finsler norm F. The result generalizes classical exponential integrability inequalities for Sobolev functions to the framework of anisotropic Sobolev spaces $W^{1,n}_0(Ω)$, where the standard Euclidean norm is replaced by F and associated polar norm $F^o$. Moreover, in the class of anisotropically radial functions, we obtain the optimal constant in the spirit of Moser's sharp inequality.
We establish an Alhfors-regularity result for minimizers of a multiphase optimal design problem. It is a variant of the classical variational problem which involves a finite number of chambers $\mathcal{E}(i)$ of prescribed volume that partition a given domain $Ω\subset\mathbb{R}^n$. The cost functional associated with a configuration $\left(\{\mathcal{E}(i)\}_i,u\right)$ is made up of the perimeter of the partition interfaces and a Dirichlet energy term, which is discontinuous across the interfaces. We prove that the union of the optimal interfaces is $(n-1)$-Alhfors-regular via a penalization method and decay estimates of the energy.
We establish regularity results for almost-minimizers of a class of variational problems involving both bulk and interface energies. The bulk energy is of Dirichlet type. The surface energy exhibits anisotropic behaviour and is defined by means of an ellipsoidal density that is Hölder continuous with respect to the position variable.
We study a weighted eigenvalue problem with anisotropic diffusion in bounded Lipschitz domains $Ω\subset \mathbb{R}^{N} $, $N\ge1$, under Robin boundary conditions, proving the existence of two positive eigenvalues $λ^{\pm}$ respectively associated with a positive and a negative eigenfunction. Next, we analyze the minimization of $λ^{\pm}$ with respect to the sign-changing weight, showing that the optimal eigenvalues $Λ^{\pm}$ are equal and the optimal weights are of bang-bang type, namely piece-wise constant functions, each one taking only two values. As a consequence, the problem is equivalent to the minimization with respect to the subsets of $Ω$ satisfying a volume constraint. Then, we completely solve the optimization problem in one dimension, in the case of homogeneous Dirichlet or Neumann conditions, showing new phenomena induced by the presence of the anisotropic diffusion. The optmization problem for $λ^{+}$ naturally arises in the study of the optimal spatial arrangement of resources for a species to survive in a heterogeneous habitat.
We fill the gap left open in \cite{MT}, regarding the minimum exponent on the logarithmic correction weight so that the Leray-Trudinger inequality (see \cite{PsSp}) holds. Instead of the representation formula used in \cite{PsSp} and \cite{MT}, our proof uses expansion in spherical harmonics as in \cite{VzZ}.
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)}{λ_\mathcal{H}^2(E)}, \end{equation} where $D$ is the isoperimetric deficit and $λ_\mathcal{H}$ 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 < $α$ < $β$, called ($α$, $β$)--convex bodies. We show that among planar ($α$, $β$)--convex bodies of fixed perimeter, the extremal shape is a domain whose boundary is composed by two arcs of circles of radius $α$ joined by two arcs of circles of radius $β$.
In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of inequalities we prove some refined Hardy-type inequalities.
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.
We prove higher differentiability of bounded local minimizers to some widely degenerate functionals, verifying superquadratic anisotropic growth conditions. In the two dimensional case, we prove that local minimizers to a model functional are locally Lipschitz continuous functions, without any restriction on the anisotropy.
By a perturbative argument, we construct solutions for a plasma-type problem with two opposite-signed sharp peaks at levels $1$ and $-γ$, respectively, where $0<γ<1$. We establish some physically relevant qualitative properties for such solutions, including the connectedness of the level sets and the asymptotic location of the peaks as $γ\to0^+$.
We present results for finite time blow-up for filtration problems with nonlinear reaction under appropriate assumptions on the nonlinearities and the initial data. In particular, we prove first finite time blow up of solutions subject to sufficiently large initial data provided that the reaction term "overpowers" the nonlinear diffusion in a certain sense. Secondly, under related assumptions on the nonlinearities, we show that initial data above positive stationary state solutions will always lead to finite time blow up.
We consider a non local isoperimetric problem arising as the sharp interface limit of the Ohta-Kawasaki free energy introduced to model microphase separation of diblock copolymers. We perform a second order variational analysis that allows us to provide a quantitative second order minimality condition. We show that critical configurations with positive second variation are indeed strict local minimizers of the nonlocal perimeter. Moreover we provide, via a suitable quantitative inequality of isoperimetric type, an estimate of the deviation from minimality for configurations close to the minimum in the $L^1$ topology .
We consider a functional which models an elastic body with a cavity. We show that if a critical point has positive second variation then it is a strict local minimizer. We also provide a quantitative estimate.
Given a mean-convex domain $Ω\subset \R^n$ with boundary of class $C^{2,1}$, we provide a representation formula for a boundary integral of the type \[ \int_{\partial Ω} f(k(x)) \, d\mathcal{H}^{n-1} \] where $k\geq 0$ is the mean curvature of $\partial Ω$ and $f$ is non-increasing and sufficiently regular, in terms of volume integrals and defect measure on the ridge set.