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Francesco Maggi

Publications and source records attributed to Francesco Maggi.

At least 37 records · Page 2Linked to original sources
arXiv2016

A bridge between Sobolev and Escobar inequalities and beyond

The classical Sobolev and Escobar inequalities are embedded into the same one-parameter family of sharp trace-Sobolev inequalities on half-spaces. Equality cases are characterized for each inequality in this family by tweaking a well-known mass transportation argument and lead to a new comparison theorem for trace Sobolev inequalities. The case $p=2$ corresponds to a family of variational problems on conformally flat metrics which was previously settled by Carlen and Loss with their method of competing symmetries. In this case minimizers interpolate between conformally flat spherical and hyperbolic geometries, passing through the Euclidean geometry defined by the fundamental solution of the Laplacian.

math.AP↗
arXiv2016

Asymptotic expansions of the contact angle in nonlocal capillarity problems

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels $|z|^{-n-s}$, with $s\in(0,1)$ and $n$ the dimension of the ambient space. The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit as $s\to 1^-$, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for $s$ close to $1$, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient $σ$ is negative, and larger if $σ$ is positive. In addition, we address the asymptotics of the fractional Young's law in the limit case $s\to 0^+$ of interaction kernels with heavy tails. Interestingly, near $s=0$, the dependence of the contact angle from the relative adhesion coefficient becomes linear.

math.AP↗
arXiv2016

Isoperimetry with upper mean curvature bounds and sharp stability estimates

It was proved by Almgren that among boundaries whose mean curvature is bounded from above, perimeter is uniquely minimized by balls. We obtain sharp stability estimates for Almgren's isoperimetric principle and, as an application, we deduce a sharp description of boundaries with almost constant mean curvature under a total perimeter bound which prevents bubbling.

math.OC↗
arXiv2016

The sharp quantitative Euclidean concentration inequality

The Euclidean concentration inequality states that, among sets with fixed volume, balls have $r$-neighborhoods of minimal volume for every $r>0$. On an arbitrary set, the deviation of this volume growth from that of a ball is shown to control the square of the volume of the symmetric difference between the set and a ball. This sharp result is strictly related to the physically significant problem of understanding near maximizers in the Riesz rearrangement inequality with a strictly decreasing radially decreasing kernel. Moreover, it implies as a particular case the sharp quantitative Euclidean isoperimetric inequality from \cite{fuscomaggipratelli}.

math.AP↗
arXiv2016

Capillarity problems with nonlocal surface tension energies

We explore the possibility of modifying the classical Gauss free energy functional used in capillarity theory by considering surface tension energies of nonlocal type. The corresponding variational principles lead to new equilibrium conditions which are compared to the mean curvature equation and Young's law found in classical capillarity theory. As a special case of this family of problems we recover a nonlocal relative isoperimetric problem of geometric interest.

math.AP↗
arXiv2015

Stability for the Brunn-Minkowski and Riesz rearrangement inequalities, with applications to Gaussian concentration and finite range non-local isoperimetry

We provide a simple, general argument to obtain improvements of concentration-type inequalities starting from improvements of their corresponding isoperimetric-type inequalities. We apply this argument to obtain robust improvements of the Brunn-Minkowski inequality (for Minkowski sums between generic sets and convex sets) and of the Gaussian concentration inequality. The former inequality is then used to obtain a robust improvement of the Riesz rearrangement inequality under certain natural conditions. These conditions are compatible with the applications to a finite-range nonlocal isoperimetric problem arising in statistical mechanics.

math.OC↗
arXiv2015

Improved convergence theorems for bubble clusters. I. The planar case

We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this construction to isoperimetric problems for planar bubble clusters. In this setting we develop an improved convergence theorem, showing that a sequence of almost-minimizing planar clusters converging in $L^1$ to a limit cluster has actually to converge in a strong $C^{1,α}$-sense. Applications of this improved convergence result to the classification of isoperimetric clusters and the qualitative description of perturbed isoperimetric clusters are also discussed. Analogous results for three-dimensional clusters are presented in part two, while further applications are discussed in some companion papers.

math.AP↗
arXiv2015

Improved convergence theorems for bubble clusters. II. The three-dimensional case

Given a sequence $\{\mathcal{E}_{k}\}_{k}$ of almost-minimizing clusters in $\mathbb{R}^3$ which converges in $L^{1}$ to a limit cluster $\mathcal{E}$ we prove the existence of $C^{1,α}$-diffeomorphisms $f_k$ between $\partial\mathcal{E}$ and $\partial\mathcal{E}_k$ which converge in $C^1$ to the identity. Each of these boundaries is divided into $C^{1,α}$-surfaces of regular points, $C^{1,α}$-curves of points of type $Y$ (where the boundary blows-up to three half-spaces meeting along a line at 120 degree) and isolated points of type $T$ (where the boundary blows up to the two-dimensional cone over a one-dimensional regular tetrahedron). The diffeomorphisms $f_k$ are compatible with this decomposition, in the sense that they bring regular points into regular points and singular points of a kind into singular points of the same kind. They are almost-normal, meaning that at fixed distance from the set of singular points each $f_k$ is a normal deformation of $\partial\mathcal{E}$, and at fixed distance from the points of type $T$, $f_k$ is a normal deformation of the set of points of type $Y$. Finally, the tangential displacements are quantitatively controlled by the normal displacements. This improved convergence theorem is then used in the study of isoperimetric clusters in $\mathbb{R}^3$.

math.AP↗
arXiv2015

Sharp stability inequalities for planar double bubbles

In this paper we address the global stability problem for double-bubbles in the plane. This is accomplished by combining the "improved convergence theorem" for planar clusters developed in arXiv:1409.6652 with an ad hoc analysis of the problem, which addresses the delicate interaction between the (possible) dislocation of singularities and the multiple-volumes constraint.

math.AP↗
arXiv2015

On the shape of compact hypersurfaces with almost constant mean curvature

The distance of an almost constant mean curvature boundary from a finite family of disjoint tangent balls with equal radii is quantitatively controlled in terms of the oscillation of the scalar mean curvature. This result allows one to quantitatively describe the geometry of volume-constrained stationary sets in capillarity problems.

math.AP↗
arXiv2015

Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case, we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controls its $C^2$-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.

math.AP↗
arXiv2014

A sharp quantitative version of Hales' isoperimetric honeycomb theorem

We prove a sharp quantitative version of Hales' isoperimetric honeycomb theorem by exploiting a quantitative isoperimetric inequality for polygons and an improved convergence theorem for planar bubble clusters. Further applications include the description of isoperimetric tilings of the torus with respect to almost unit-area constraints or with respect to almost flat Riemannian metrics.

math.AP↗
arXiv2014

A direct approach to Plateau's problem

We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David.

math.AP↗
arXiv2014

Regularity of free boundaries in anisotropic capillarity problems and the validity of Young's law

Local volume-constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers. We prove the regularity of the free boundary outside a closed negligible set, showing in particular the validity of Young's law at almost every point of the free boundary. Our regularity results are not specific to capillarity problems, and actually apply to sets of finite perimeter (and thus to codimension one integer rectifiable currents) arising as minimizers in other variational problems with free boundaries.

math.AP↗
arXiv2013

Rigidity of equality cases in Steiner's perimeter inequality

Characterization results for equality cases and for rigidity of equality cases in Steiner's perimeter inequality are presented. (By rigidity, we mean the situation when all equality cases are vertical translations of the Steiner's symmetral under consideration.) We achieve this through the introduction of a suitable measure-theoretic notion of connectedness and a fine analysis of barycenter functions for sets of finite perimeter having segments as orthogonal sections with respect to an hyperplane.

math.AP↗