arXiv subjects

Giorgio Saracco

Publications and source records attributed to Giorgio Saracco.

24 records · Page 2Linked to original sources

arXiv2018

On the isoperimetric problem with double density

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.

math.AP↗

arXiv2018

Two examples of minimal Cheeger sets in the plane

We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of Cheeger sets, as its perimeter does not coincide with the $1$-dimensional Hausdorff measure of its topological boundary. The second one is a kind of porous set, whose boundary is not locally a graph at many of its points, yet it is a weakly regular open set admitting a unique (up to vertical translations) non--parametric solution to the prescribed mean curvature equation, in the extremal case corresponding to the capillarity for perfectly wetting fluids in zero gravity.

math.AP↗

arXiv2018

The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a \textit{generalized Gauss-Green theorem} based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a \textit{weak Young's law} for $(Λ,r_{0})$-minimizers of the perimeter.

math.AP↗

arXiv2018

Weighted Cheeger sets are domains of isoperimetry

We consider a generalization of the Cheeger problem in a bounded, open set $Ω$ by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer $A$ of this weighted Cheeger problem such that $H^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric inequality. If $Ω$ itself is a connected minimizer such that $H^{n-1}(Ω^{(1)} \cap \partial Ω)=0$, then it allows the classical Sobolev and $BV$ embeddings and the classical $BV$ trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and $Ω$ is such that $|\partial Ω|=0$ and $H^{n-1}(Ω^{(1)} \cap \partial Ω)=0$.

math.FA↗

arXiv2018

The Cheeger constant of a Jordan domain without necks

We show that the maximal Cheeger set of a Jordan domain $Ω$ without necks is the union of all balls of radius $r = h(Ω)^{-1}$ contained in $Ω$. Here, $h(Ω)$ denotes the Cheeger constant of $Ω$, that is, the infimum of the ratio of perimeter over area among subsets of $Ω$, and a Cheeger set is a set attaining the infimum. The radius $r$ is shown to be the unique number such that the area of the inner parallel set $Ω^r$ is equal to $πr^2$. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.

math.AP↗

arXiv2018

On the generalized Cheeger problem and an application to 2d strips

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.

math.MG↗