arXiv subjects

Giorgio Stefani

Publications and source records attributed to Giorgio Stefani.

40 records · Page 3Linked to original sources

arXiv2018

A maximal function characterization of absolutely continuous measures and Sobolev functions

In this note, we give a new characterisation of Sobolev $W^{1,1}$ functions among $BV$ functions via Hardy-Littlewood maximal function. Exploiting some ideas coming from the proof of this result, we are also able to give a new characterisation of absolutely continuous measures via a weakened version of Hardy-Littlewood maximal function. Finally, we show that the approach adopted in [Crippa and De Lellis, J. Reine Angew. Math. (2008)] and [Jabin, J. Math. Pures Appl. (2010)] to establish existence and uniqueness of regular Lagrangian flows associated to Sobolev vector fields cannot be further extended to the case of $BV$ vector fields.

math.CA↗

arXiv2016

On the monotonicity of perimeter of convex bodies

Let $n\ge2$ and let $Φ\colon\mathbb{R}^n\to[0,\infty)$ be a positively $1$-homogeneous and convex function. Given two convex bodies $A\subset B$ in $\mathbb{R}^n$, the monotonicity of anisotropic $Φ$-perimeters holds, i.e. $P_Φ(A)\le P_Φ(B)$. In this note, we prove a quantitative lower bound on the difference of the $Φ$-perimeters of $A$ and $B$ in terms of their Hausdorff distance.

math.MG↗

arXiv2016

Improved Lipschitz approximation of $H$-perimeter minimizing boundaries

We prove two new approximation results of $H$-perimeter minimizing boundaries by means of intrinsic Lipschitz functions in the setting of the Heisenberg group $\mathbb{H}^n$ with $n\ge2$. The first one is an improvement of a result of Monti and is the natural reformulation in $\mathbb{H}^n$ of the classical Lipschitz approximation in $\mathbb{R}^n$. The second one is an adaptation of the approximation via maximal function developed by De Lellis and Spadaro.

math.MG↗

arXiv2016

Sobolev subspaces of nowhere bounded functions

We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elements are nowhere bounded functions. We also prove the existence of a closed infinite dimensional linear subspace whose non zero elements are nowhere $L^q$ functions for suitable values of $q$ larger than the Sobolev exponent.

math.FA↗