arXiv subjects
Publications and source records attributed to Pietro Aldrigo.
We give a constructive proof for existence of absolutely continuous curves connecting each pair $\mu,\nu \in \mathrm{PL}_\infty^p(\mathbb{R}^n)$, for every $1\leq p\leq \infty$, where $(\mathrm{PL}_\infty^p(\mathbb{R}^n),\mathfrak{d}_\infty^p)$ is the complete metric space of absolutely continuous measures with density in $L^p(\mathbb{R}^n)$ and bounded support introduced in [1].
We define a complete metric structure on the family $\text{PL}_q^p(\mathbb{R}^n)$ of probability measures with densities in $L^p(\mathbb{R}^n)$ and finite $q$-moments. We establish the existence of generalized minimizing movements for the isoperimetric ratio and characterize absolutely continuous curves in this space through weak solutions of the continuity equation with velocity fields satisfying a first-order integral condition. We also characterize absolutely continuous curves in the $\infty$-Wasserstein space and prove a Benamou--Brenier formula for $W_\infty$.
Let $F$ be a lower semicontinuous, 1-homogeneous positive function defined on $\mathbf{R}^n$. We provide a characterization of absolutely continuous paths that minimize the anisotropic $F$-length between two points. The characterization is achieved by establishing a connection between the minimizing paths and the geometry of the anisotropic $F$-isoperimetric set.
We establish P\'olya-Szeg\H{o}-type inequalities (PSIs) for Sobolev-functions defined on a regular $n$-dimensional submanifold $\Sigma$ (possibly with boundary) of a $(n+m)$-dimensional Euclidean space, under explicit upper bounds of the total mean curvature. The $p$-Sobolev and Gagliardo-Nirenberg inequalities, as well as the spectral gap in $W^{1,p}_0(\Sigma)$ are derived as corollaries. Using these PSIs, we prove a sharp $p$-Log-Sobolev inequality for minimal submanifolds in codimension one and two. The asymptotic sharpness of both the multiplicative constant appearing in PSIs and the assumption on the total mean curvature bound as $n\to \infty$ is provided. A second equivalent version of our PSIs is presented in the appendix of this paper, introducing the notion of model space $(\mathbb{R}^+,\mathfrak{m}_{n,K})$ of dimension $n$ and total mean curvature bounded by $K$.