arXiv subjects
Publications and source records attributed to Aldo Pratelli.
In this paper we show the stability of the ball as maximizer of the Riesz potential among sets of given volume. The stability is proved with sharp exponent $1/2$, and is valid for any dimension $N\geq 2$ and any power $1<α<N$.
In this paper we consider the isoperimetric problem with double density in an Euclidean space, that is, we study the minimisation of the perimeter among subsets of $\mathbb{R}^n$ with fixed volume, where volume and perimeter are relative to two different densities. The case of a single density, or equivalently, when the two densities coincide, has been well studied in the last years; nonetheless, the problem with two different densities is an important generalisation, also in view of applications. We will prove the existence of isoperimetric sets in this context, extending the known results for the case of single density.
In this paper we consider the generalization of the Cheeger problem which comes by considering the ratio between the perimeter and a certain power of the volume. This generalization has been already sometimes treated, but some of the main properties were still not studied, and our main aim is to fill this gap. We will show that most of the first important properties of the classical Cheeger problem are still valid, but others fail; more precisely, long and thin rectangles will give a counterexample to the property of Cheeger sets of being the union of all the balls of a certain radius, as well as to the uniqueness. The shape of Cheeger set for rectangles and strips is then studied as well as their Cheeger constant.
We characterize the (sequentially) weak and strong closure of planar diffeomorphisms in the Sobolev topology and we show that they always coincide. We also provide some sufficient condition for a planar map to be approximable by diffeomorphisms in terms of the connectedness of its counter-images, in the spirit of Young's characterisation of monotone functions. We finally show that the closure of diffeomorphisms in the Sobolev topology is strictly contained in the class INV introduced by Muller and Spector.
In this paper we investigate the behavior of the eigenvalues of the Dirichlet Laplacian on sets in $\mathbb{R}^N$ whose first eigenvalue is close to the one of the ball with the same volume. In particular in our main Theorem we prove that, for all $k\in\mathbb{N}$, there is a positive constant $C=C(k,N)$ such that for every open set $Ω\subseteq \mathbb{R}^N$ with unit measure and with $λ_1(Ω)$ not excessively large one has \begin{align*} |λ_k(Ω)-λ_k(B)|\leq C (λ_1(Ω)-λ_1(B))^β\,, && λ_k(B)-λ_k(Ω)\leq Cd(Ω)^{β'}\,, \end{align*} where $d(Ω)$ is the Fraenkel asymmetry of $Ω$, and where $β$ and $β'$ are explicit exponents, not depending on $k$ nor on $N$; for the special case $N=2$, a better estimate holds.
In this paper we consider the mass transport problem in the case of a relativistic cost; we can establish the continuity of the total cost, together with a general estimate about the directions in which the mass can actually move, under mild assumptions. These results generalize those recently obtained by J.Bertrand, A.Pratelli and M.Puel, also positively answering some of the open questions there.
The first goal of this paper is to give a short description of the planar bi-Sobolev homeomorphisms, providing simple and self-contained proofs for some already known properties. In particular, for any such homeomorphism $u:Ω\to Δ$, one has $Du(x)=0$ for almost every point $x$ for which $J_u(x)=0$. As a consequence, one can prove that \begin{equation} \int_Ω|Du| = \int_Δ|Du^{-1}|\,. \end{equation} Notice that this estimate holds trivially if one is allowed to use the change of variables formula, but this is not always the case for a bi-Sobolev homeomorphism.\par As a corollary of our construction, we will show that any $W^{1,1}$ homeomorphism $u$ with $W^{1,1}$ inverse can be approximated with smooth diffeomorphisms (or piecewise affine homeomorphisms) $u_n$ in such a way that $u_n$ converges to $u$ in $W^{1,1}$ and, at the same time, $u_n^{-1}$ converges to $u^{-1}$ in $W^{1,1}$. This positively answers an open conjecture (see for instance~\cite[Question~4]{arXiv:1009.0286}) for the case $p=1$.
In this paper we deal with the task of uniformly approximating an $L$-biLipschitz curve by means of piecewise linear ones. This is rather simple if one is satisfied to have approximating functions which are $L'$-biLipschitz, for instance this was already done with $L'= 4L$ in [Daneri-Pratelli, Lemma 5.5]. The main result of this paper is to do the same with $L'=L+ \varepsilon$ (which is of course the best possible result); in the end, we generalize the result to the case of closed curves.
We consider the isoperimetric problem in $\mathbb R^n$ with density for the planar case $n=2$. We show that, if the density is ${\rm C}^{0,α}$, then the boundary of any isoperimetric is of class ${\rm C}^{1,\frac α{3-2α}}$. This improves the previously known regularity.
Let $Ω\subseteq\mathbb R^2$ be a domain and let $f\in W^{1,1}(Ω,\mathbb R^2)$ be a homeomorphism (between $Ω$ and $f(Ω)$). Then there exists a sequence of smooth diffeomorphisms $f_k$ converging to $f$ in $W^{1,1}(Ω,\mathbb R^2)$ and uniformly.
In this paper, we consider the isoperimetric problem in the space $\mathbb{R}^N$ with density. Our result states that, if the density f is l.s.c. and converges to a positive limit at infinity, being smaller than this limit far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities positively answers a conjecture made in [10].
In this paper we consider the Cheeger problem for non-convex domains, with a particular interest in the case of planar strips, which have been extensively studied in recent years. Our main results are an estimate on the Cheeger constant of strips, which is stronger than the previous one known from a recent result by D. Krejcirik and the second-named author, and the proof that strips share with convex domains a number of crucial properties with respect to the Cheeger problem. Moreover, we present several counterexamples showing that the same properties are not valid for generic non-convex domains.
We study the optimal sets $Ω^\ast\subset\mathbb{R}^d$ for spectral functionals $F\big(λ_1(Ω),\dots,λ_p(Ω)\big)$, which are bi-Lipschitz with respect to each of the eigenvalues $λ_1(Ω),\dots,λ_p(Ω)$ of the Dirichlet Laplacian on $Ω$, a prototype being the problem $$ \min{\big\{λ_1(Ω)+\dots+ λ_p(Ω)\;:\;Ω\subset\mathbb{R}^d,\ |Ω|=1\big\}}. $$ We prove the Lipschitz regularity of the eigenfunctions $u_1,\dots,u_p$ of the Dirichlet Laplacian on the optimal set $Ω^*$ and, as a corollary, we deduce that $Ω^*$ is open. For functionals depending only on a generic subset of the spectrum, as for example $λ_k(Ω)$ or $λ_{k_1}(Ω)+\dots+λ_{k_p}(Ω)$, our result proves only the existence of a Lipschitz continuous eigenfunction in correspondence to each of the eigenvalues involved.
We show that every isoperimetric set in R^N with density is bounded if the density is continuous and bounded by above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical "ε-ε" property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an "ε-ε^β" version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the sole continuity is enough. We conclude by deriving some consequences of our result about the existence and regularity of isoperimetric sets.
We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighbourhoods of curves in the plane. If the reference curve is complete and finite (a "curved annulus"), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about non-complete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.
Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(λ_1(Ω),λ_2(Ω))$ for all the open sets $Ω\subseteq\R^N$ with unit measure, and by $Θ\subseteq\R^N$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that $\partial\E$ has horizontal tangent at its lowest point $(λ_1(Θ),λ_2(Θ))$.
In this paper we show that any increasing functional of the first k eigenvalues of the Dirichlet Laplacian admits a (quasi-)open minimizer among the subsets of R^N of unit measure. In particular, there exists such a minimizer which is bounded, where the bound depends on k and N, but not on the functional. In the meantime, we show that the ratio λ_k(Ω)/λ_1(Ω) is uniformly bounded for sets Ω\in R^N.
We prove the existence of isoperimetric regions in $\R^n$ with density under various hypotheses on the growth of the density. Along the way we prove results on the boundedness of isoperimetric regions.