arXiv subjects
Publications and source records attributed to Luigi De Masi.
We show that the set of points where the blow-up, in the sense of Preiss, of a signed Radon measure on $\mathbb{R}^n$ is unique and its invariant subspace has dimension $k$ is $k$-rectifiable. As simple applications, we obtain a rectifiability criterion for signed Radon measures and the extension of a result, due to Mattila, on measures having unique blow-up almost everywhere.
Let $μ$ be a finite Radon measure on an open set $Ω\subset\mathbb{R}^d$, singular with respect to the Lebesgue measure. We prove Lusin-type solvability results for the prescribed divergence equation and the prescribed Jacobian equation with Lipschitz solutions. More precisely, for every $\varepsilon>0$ and every Borel datum $f \colon Ω\to \mathbb{R}$ there exists a vector field $V\in C^1_c(Ω;\mathbb{R}^d)$ such that $\operatorname{div} V=f$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(V)\le (1+\varepsilon)\|f\|_{L^\infty(Ω,μ)}$. Similarly, for every Borel datum $g\colon Ω\to \mathbb{R}$ there exists a map $Φ$ with $Φ-\operatorname{Id}\in C^1_c(Ω;\mathbb{R}^d)$ such that $\det DΦ=g$ on a compact set $K\subsetΩ$ with $μ(Ω\setminus K)<\varepsilon$, and $\operatorname{Lip}(Φ-\operatorname{Id})\le (1+\varepsilon)\|g-1\|_{L^\infty(Ω,μ)}$. The maps $V$ and $Φ-\operatorname{Id}$ can be chosen arbitrarily small in supremum norm.
This paper introduces two variational formulations for a model of robust optimal transport, that is, the problem of designing optimal transport networks that are resilient to potential damages, balancing construction costs against the benefit of maintaining partial functionality when parts of the network are damaged. We propose a Eulerian formulation, where the network is modeled by a rectifiable measure and recovery plans are represented by 1-dimensional normal currents. This framework allows for changes in the direction of the transportation in response to damages but restricts damages to be characteristic functions of closed sets. We also propose a Lagrangian formulation, where the network is a traffic plan (that is, a measure on the space of Lipschitz curves) and recovery plans are sub-traffic plans. This approach prescribes the network's orientation but allows for a wider class of damages. We prove existence of minimizers in both settings. The two models are compared through examples that illustrate their main differences: the Eulerian formulation's necessity for an unoriented network to achieve existence, the Lagrangian formulation's ability to handle general damages and its requirement for a positive distance between the supports of the source and target measures.
We generalize a result by Alberti, showing that, if a first-order linear differential operator $\mathcal{A}$ belongs to a certain class, then any $L^1$ function is the absolutely continuous part of a measure $μ$ satisfying $\mathcal{A}μ=0$. When $\mathcal{A}$ is scalar valued, we provide a necessary and sufficient condition for the above property to hold true and we prove dimensional estimates on the singular part of $μ$. Finally, we show that operators in the above class satisfy a Lusin-type property.
We prove a refined version of the celebrated Lusin type theorem for gradients by Alberti, stating that any Borel vector field $f$ coincides with the gradient of a $C^1$ function $g$, outside a set $E$ of arbitrarily small Lebesgue measure. We replace the Lebesgue measure with any Radon measure $μ$, and we obtain that the estimate on the $L^p$ norm of $Dg$ does not depend on $μ(E)$, if the value of $f$ is $μ$-a.e. orthogonal to the decomposability bundle of $μ$. We observe that our result implies the 1-dimensional version of the flat chain conjecture by Ambrosio and Kirchheim on the equivalence between metric currents and flat chains with finite mass in $\mathbb{R}^n$ and we state a suitable generalization for $k$-forms, which would imply the validity of the conjecture in full generality.
We prove $\varepsilon$-regularity theorems for varifolds with capillary boundary condition in a Riemannian manifold. These varifolds were first introduced by Kagaya-Tonegawa \cite{KaTo}. We establish a uniform first variation control for all such varifolds (and free-boundary varifolds generally) satisfying a sharp density bound and prove that if a capillary varifold has bounded mean curvature and is close to a capillary half-plane with angle not equal to $\tfracπ{2}$, then it coincides with a $C^{1,α}$ properly embedded hypersurface. We apply our theorem to deduce regularity at a generic point along the boundary in the region where the density is strictly less than $1$.
We prove the existence of minimal surfaces in a bounded convex subset of $\mathbb R^3$, $\mathcal M$, intersecting the boundary of $\mathcal M$ with a fixed contact angle. The proof is based on a min-max construction in the spirit of Almgren-Pitts for the capillarity functional.
We establish a partial rectifiability result for the free boundary of a $k$-varifold $V$. Namely, we first refine a theorem of Grüter and Jost by showing that the first variation of a general varifold with free boundary is a Radon measure. Next we show that if the mean curvature $H$ of $V$ is in $L^p$ for some $p \in [1,k]$, then the set of points where the $k$-density of $V$ does not exist or is infinite has Hausdorff dimension at most $k-p$. We use this result to prove, under suitable assumptions, that the part of the first variation of $V$ with positive and finite $(k-1)$-density is $(k-1)$-rectifiable.