Gian Paolo Leonardi

An Overview on the Cheeger Problem

This is an expository article presenting various results about the Cheeger problem and its connections with other relevant mathematical problems. In section 2 we synthetically describe three mathematical problems (eigenvalue estimates, prescribed mean curvature equation, ROF model for image segmentation) focusing on their link with the Cheeger problem. Then, a selection of classical as well as more recent results obtained in collaboration with Aldo Pratelli will be described in the remaining sections.

Topics in mathematical analysis

Complex Variables and Potential Theory: Integral Representations in Complex, Hypercomplex and Clifford Analysis (H Begehr) Nonlinear Potential Theory in Metric Spaces (O Martio) Differential Equations and Nonlinear Analysis: Geometric Evolution Problems (G Bellettini) Introduction to Bifurcation Theory (P Drabek) Nonlinear Eigenvalue Problems (P Lindqvist) Nonlinear Elliptic Equations with Critical and Supercritical Sobolev Exponents (D Passaseo) Discrete Spectrum Analysis of Elliptic Operators (G Rozenblum) Introduction to Continuous Semigroups (E Vesentini) Harmonic Analysis: Spectral Analysis of the Laplace Operator and Integral Geometry (M Agranovsky) Average Decay of the Fourier Transform and Applications to Harmonic Analysis and Geometric Measure Theory (A Losevich) Eigenfunctions of the Laplacian (C D Sogge) An Introduction to Harmonic Analysis: From the Basic Equations of the Physics to Singular and Oscillatory Integrals (F Soria) Spectral Theory of Differential and Integral Operators Related to Fractals and Metric Spaces (H Triebel).

Extremal Curves in Nilpotent Lie Groups

We classify extremal curves in free nilpotent Lie groups. The classification is obtained via an explicit integration of the adjoint equation in Pontryagin maximum principle. It turns out that abnormal extremals are precisely the horizontal curves contained in algebraic varieties of a specific type. We also extend the results to the nonfree case.

Best constants for the isoperimetric inequality in quantitative form

We prove existence and regularity of minimizers for a class of functionals defined on Borel sets in

Locality of the mean curvature of rectifiable varifolds

R^n

The prescribed mean curvature equation in weakly regular domains

. Combining these results with a refinement of the selection principle introduced by the authors in arXiv:0911.0786, we describe a method suitable for the determination of the best constants in the quantitative isoperimetric inequality with higher order terms. Then, applying Bonnesens annular symmetrization in a very elementary way, we show that, for

Isodiametric sets in the Heisenberg group

n=2

Infiltrations in immiscible fluids systems

, the above-mentioned constants can be explicitly computed through a one-parameter family of convex sets known as ovals. This proves a further extension of a conjecture posed by Hall in J. Reine Angew. Math. 428 (1992).

Quantitative isoperimetric inequalities in \mathbb {H}^n

Abstract The aim of this paper is to investigate whether, given two rectifiable k-varifolds in ℝ n with locally bounded first variations and integer-valued multiplicities, their mean curvatures coincide ℋ k -almost everywhere on the intersection of the supports of their weight measures. This so-called locality property, which is well known for classical C 2 surfaces, is far from being obvious in the context of varifolds. We prove that the locality property holds true for integral 1-varifolds, while for k-varifolds, k > 1, we are able to prove that it is verified under some additional assumptions (local inclusion of the supports and locally constant multiplicities on their intersection). We also discuss a couple of applications in elasticity and computer vision.

On minimizing partitions with infinitely many components

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 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 weak Young’s law for