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Quantitative Isoperimetric Inequalities on the Real Line

Yohann de Castro (2011)

Annales mathématiques Blaise Pascal

In a recent paper A. Cianchi, N. Fusco, F. Maggi, and A. Pratelli have shown that, in the Gauss space, a set of given measure and almost minimal Gauss boundary measure is necessarily close to be a half-space.Using only geometric tools, we extend their result to all symmetric log-concave measures on the real line. We give sharp quantitative isoperimetric inequalities and prove that among sets of given measure and given asymmetry (distance to half line, i.e. distance to sets of minimal perimeter),...

Rectifiability and perimeter in step 2 Groups

Bruno Franchi, Raul Serapioni, Francesco Serra Cassano (2002)

Mathematica Bohemica

We study finite perimeter sets in step 2 Carnot groups. In this way we extend the classical De Giorgi’s theory, developed in Euclidean spaces by De Giorgi, as well as its generalization, considered by the authors, in Heisenberg groups. A structure theorem for sets of finite perimeter and consequently a divergence theorem are obtained. Full proofs of these results, comments and an exhaustive bibliography can be found in our preprint (2001).

Regular mappings between dimensions

Guy David, Stephen Semmes (2000)

Publicacions Matemàtiques

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...

Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marco Marchi (2014)

Annales de l’institut Fourier

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type ☆ ” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type ☆ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step...

Semicontinuity in L ∞ for polyconvex integrals

Emilio Acerbi, Giuseppe Buttazzo, Nicola Fusco (1982)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Viene studiata la semicontinuità rispetto alla topologia di L ∞ ⁢ ( Ω ; 𝐑 m ) per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.

Sets of finite perimeter associated with vector fields and polyhedral approximation

Francescopaolo Montefalcone (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let X = X 1 , ⋯ , X m be a family of bounded Lipschitz continuous vector fields on R n . In this paper we prove that if E is a set of finite X -perimeter then his X -perimeter is the limit of the X -perimeters of a sequence of euclidean polyhedra approximating E in L 1 -norm. This extends to Carnot-Carathéodory geometry a classical theorem of E. De Giorgi.

Shape Sensitivity Analysis of the Dirichlet Laplacian in a Half-Space

Cherif Amrouche, Šárka Nečasová, Jan Sokołowski (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

Material and shape derivatives for solutions to the Dirichlet Laplacian in a half-space are derived by an application of the speed method. The proposed method is general and can be used for shape sensitivity analysis in unbounded domains for the Neumann Laplacian as well as for the elasticity boundary value problems.

Some Liouville theorems for PDE problems in periodic media

Luis Caffarelli (2003)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Liouville problems in periodic media (i.e. the study of properties of global solutions to PDE) arise both in homogenization and dynamical systems. We discuss some recent results for minimal surfaces and free boundaries.

Some relations among volume, intrinsic perimeter and one-dimensional restrictions of B V functions in Carnot groups

Francescopaolo Montefalcone (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let 𝔾 be a k -step Carnot group. The first aim of this paper is to show an interplay between volume and 𝔾 -perimeter, using one-dimensional horizontal slicing. What we prove is a kind of Fubini theorem for 𝔾 -regular submanifolds of codimension one. We then give some applications of this result: slicing of B V 𝔾 functions, integral geometric formulae for volume and 𝔾 -perimeter and, making use of a suitable notion of convexity, called 𝔾 -convexity, we state a Cauchy type formula for 𝔾 -convex sets. Finally,...

Some Results on Maps That Factor through a Tree

Roger Züst (2015)

Analysis and Geometry in Metric Spaces

We give a necessary and sufficient condition for a map deffned on a simply-connected quasi-convex metric space to factor through a tree. In case the target is the Euclidean plane and the map is Hölder continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over winding number functions. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carathéodory metric and the Hölder exponent of the map is bigger than...

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