arXiv subjects
Publications and source records attributed to Giacomo Del Nin.
We present a new proof that an integer rectifiable current with finite mass, and whose boundary has also finite mass, is integral. We deduce the result from De Giorgi's structure theorem for integer-valued $BV$ functions and a cylindrical projection argument. As a consequence, we also give a new proof of the compactness of integral currents that is ultimately based on the $BV$ theory.
For any smooth bounded domain $Ω\subset \mathbb{R}^3$, we construct a divergence-free velocity field $u \in L_t^1 W^{1,p}(Ω)$ for all $p < \infty$, and magnetic fields $B^ε\in L_t^p C^{m}(Ω)$ for all $p < \infty$ and $m\in \mathbb{N}$, that solve the kinematic dynamo equation and exhibit arbitrarily fast growth of any magnetic energy mode, uniformly in the vanishing-diffusivity limit $ε\to 0$. The construction is based on the convex integration scheme of Modena-Székelyhidi and Cheskidov-Luo. The main novelty lies in the introduction of explicit potentials, which allow the solutions to be localized and avoid the need to work with the anti-curl operator. In addition, we present a unified scheme for the geometric transport equation (GTE), which encompasses both the transport and Maxwell equations.
We show that if a non-trivial measure in the plane admits, at almost every point, positive and finite $α$-dimensional density with respect to some norm, then $α$ must be an integer.
We prove existence and uniqueness for the transport equation for currents (Geometric Transport Equation) when the driving vector field is time-dependent, Lipschitz in space and merely integrable in time. This extends previous work where well-posedness was shown in the case of a time-independent, Lipschitz vector field. The proof relies on the decomposability bundle and requires to extend some of its properties to the class of functions that in one direction are only absolutely continuous, rather than Lipschitz.
We consider a variant of the sticky disk energy where distances between particles are evaluated through the sup norm $\lVert\cdot\rVert_\infty$ in the plane. We first prove crystallization of minimizers in the square lattice, for any fixed number $N$ of particles. Then we consider the limit as $N\to\infty$: in contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the $Γ$-limit to be an anisotropic perimeter with octagonal Wulff shape. The results are based on an energy decomposition for graphs that generalizes the one proved by De Luca-Friesecke [J. Nonlinear Sci. 28 (2018), 69-90] in the triangular case.
Given any divergence-free vector field of Sobolev class $W^{m,p}_0(Ω)$ in a bounded open subset $Ω\subset \mathbb{R}^2$, we are interested in approximating it in the $W^{m,p}$ norm with divergence-free smooth vector fields compactly supported in $Ω$. We show that this approximation property holds in the following cases: For $p>2$, this holds given that $\partial Ω$ has zero Lebesgue measure (a weaker but more technical condition is sufficient); For $p \leq 2$, this holds if $Ω^c$ can be decomposed into finitely many disjoint closed sets, each of which is connected or $d$-Ahlfors regular for some $d\in[0,2)$. This has links to the uniqueness of weak solutions to the Stokes equation in $Ω$. For Hölder spaces, we prove this approximation property in general bounded domains.
We introduce the concept of local Poincaré constant of a $BV$ function as a tool to understand the relation between its mean oscillation and its total variation at small scales. This enables us to study a variant of the BMO-type seminorms on $\varepsilon$-size cubes introduced by Ambrosio, Bourgain, Brezis, and Figalli. More precisely, we relax the size constraint by considering a family of functionals that allow cubes of sidelength smaller than or equal to $\varepsilon$. These new functionals converge, as $\varepsilon$ tends to zero, to a local functional defined on $BV$, which can be represented by integration in terms of the local Poincaré constant and the total variation. This contrasts with the original functionals, whose limit is defined on $SBV$ and may not exist for functions with a non-trivial Cantor part. Moreover, we characterize the local Poincaré constant of a function with a cell-formula given by the maximum mean oscillation of its $BV$ blow-ups. As a corollary of this characterization, we show that the new limit functional extends the original one to all $BV$ functions. Finally, we discuss rigidity properties and other challenging questions relating the local Poincaré constant of a function to its fine properties.
We prove finite crystallization for particles in the plane interacting through a soft disc potential, as originally shown by C. Radin \cite{Radin_soft}. We give an alternative proof that relies on the geometric decomposition of the energy proved in \cite{DLF1}, and that is based on showing that any minimizer has at least as many boundary points as the canonical ``spiral'' configuration.
We prove that given an $n$-dimensional integral current space and a $1$-Lipschitz map, from this space onto the $n$-dimensional Euclidean ball, that preserves the mass of the current and is injective on the boundary, then the map has to be an isometry. We deduce as a consequence a stability result with respect to the intrinsic flat distance, which implies the stability of the positive mass theorem for graphical manifolds as originally formulated by Huang--Lee--Sormani.
This work establishes the existence and uniqueness of solutions to the initial-value problem for the geometric transport equation $$ \frac{\mathrm{d}}{\mathrm{d} t}T_t+\mathcal{L}_b T_t=0 $$ in the class of $k$-dimensional integral or normal currents $T_t$ ($t$ being the time variable) under the natural assumption of Lipschitz regularity of the driving vector field $b$. Our argument relies crucially on the notion of decomposability bundle introduced recently by Alberti and Marchese. In the particular case of $0$-currents, this also yields a new proof of the uniqueness for the continuity equation in the class of signed measures.
The transport of many kinds of singular structures in a medium, such as vortex points/lines/sheets in fluids, dislocation loops in crystalline plastic solids, or topological singularities in magnetism, can be expressed in terms of the geometric (Lie) transport equation \[ \frac{\mathrm{d}}{\mathrm{d} t} T_t + \mathcal{L}_{b_t} T_t = 0 \] for a time-indexed family of integral or normal $k$-currents $t \mapsto T_t$ in $\mathbb{R}^d$. Here, $b_t$ is the driving vector field and $\mathcal{L}_{b_t} T_t$ is the Lie derivative of $T_t$ with respect to $b_t$. Written in coordinates for different values of $k$, this PDE encompasses the classical transport equation ($k = d$), the continuity equation ($k = 0$), as well as the equations for the transport of dislocation lines in crystals ($k = 1$) and membranes in liquids ($k =d-1$). The top-dimensional and bottom-dimensional cases have received a great deal of attention in connection with the DiPerna--Lions and Ambrosio theories of Regular Lagrangian Flows. On the other hand, very little is rigorously known at present in the intermediate-dimensional cases. This work develops the theory of the geometric transport equation for arbitrary $k$ and in the case of boundaryless currents $T_t$, covering in particular existence and uniqueness of solutions, structure theorems, rectifiability, and a number of Rademacher-type differentiability results. The latter yield, given an absolutely continuous (in time) path $t \mapsto T_t$, the existence almost everywhere of a ''geometric derivative'', namely a driving vector field $b_t$. This subtle question turns out to be intimately related to the critical set of the evolution, a new notion introduced in this work, which is closely related to Sard's theorem and concerns singularities that are ''smeared out in time''. Our differentiability results are sharp, which we demonstrate through an explicit example.
We address a question raised by Ambrosio, Bourgain, Brezis, and Figalli, proving that the $Γ$-limit, with respect to the $L^1_{\rm loc}$ topology, of a family of $BMO$-type seminorms is given by $\tfrac14$ times the total variation seminorm. Our method also yields an alternative proof of previously known lower bounds for the pointwise limit and conveys a compactness result in $L^1_{\rm loc}$ in terms of the boundedness of the $BMO$-type seminorms.
We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.
We show that if a measure of dimension $s$ on $\mathbb{R}^d$ admits $(p,q)$ Fourier restriction for some endpoint exponents allowed by its dimension, namely $q=\tfrac{s}{d}p'$ for some $p>1$, then it is either absolutely continuous or $1$-purely unrectifiable.
In the setting of complete metric spaces, we prove that integral currents can be decomposed as a sum of indecomposable components. In the special case of one-dimensional integral currents, we also show that the indecomposable ones are exactly those associated with injective Lipschitz curves or injective Lipschitz loops, therefore extending Federer's characterisation to metric spaces. Moreover, some applications of our main results will be discussed.
In this note we show that for every measurable function on $\mathbb{R}^n$ the set of points where the blowup exists and is not constant is $(n-1)$-rectifiable. In particular, for every $u\in L^1_{loc}(\mathbb{R}^n)$ the jump set $J_u$ is $(n-1)$-rectifiable.
We minimize a linear combination of the Willmore and the length functional among networks in $\mathbb{R}^d$ belonging to a given class determined by the number of curves, the order of the junctions and the angles between curves at the junctions. Since this class lacks compactness, we characterize the set of limits of sequences of networks bounded in energy, providing an explicit representation of the relaxed problem. This is expressed in terms of the new notion of degenerate elastic networks that, rather surprisingly, involves only the properties of the given class, without reference to the curvature. In the case of $d=2$ we also give an equivalent description of degenerate elastic networks by means of a combinatorial definition easy to validate by a finite algorithm. Moreover we provide examples, counterexamples, and additional results that motivate our study and show the sharpness of our characterization.
We prove criteria for $\mathcal{H}^k$-rectifiability of subsets of $\mathbb{R}^n$ with $C^{1,α}$ maps, $0<α\leq 1$, in terms of suitable approximate tangent paraboloids. We also provide a version for the case when there is not an a priori tangent plane, measuring on dyadic scales how close the set is to lying in a $k$-plane. We then discuss the relation with similar criteria involving Peter Jones' $β$ numbers, in particular proving that a sufficient condition is the boundedness for small $r$ of $r^{-α}β_p(x,r)$ for $\mathcal{H}^k$-a.e. $x$ and for any $1\leq p\leq \infty$.