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Camillo De Lellis

Publications and source records attributed to Camillo De Lellis.

At least 19 recordsLinked to original sources
arXiv2026

Scalar Dissipation Criticality in Compressible Magnetized Turbulence

We extend the Obukhov-Corrsin theory of scalar turbulence to compressible flows with spatially variable, anisotropic diffusivity. Combining density and diffusivity into a single positive matrix field, a transport landscape, permits an exact scale-by-scale balance in which the sign-indefinite commutator between filtering and diffusion is eliminated rather than estimated. If the density-weighted third-order velocity and scalar increments scale as $\ell^α$ and $\ell^β$, respectively, we prove that anomalous scalar dissipation is impossible when $α+2β>1$. Remarkably, this threshold is independent of the anisotropy and spatial regularity of the diffusivity, provided it remains uniformly elliptic. Codimension-one shocks in both velocity and scalar have $α=β=1/3$ and therefore lie exactly at the critical threshold. For statistically stationary turbulence we further obtain an exact density-weighted constant-flux relation, providing a compressible analogue of the relation underlying Yaglom's law. The results apply directly to passive-scalar transport in both gases and magnetized plasmas and provide testable diagnostics for simulations of compressible magnetohydrodynamic turbulence. Because the same operator governs anisotropic heat conduction, the results carry over to the electron temperature of a magnetized plasma, and they imply that gradient statistics in such flows must be contracted with the transport landscape rather than with the density.

physics.plasm-ph↗
arXiv2026

Stationary varifolds with singularities II

For every dimension $m\geq 3$, we show the existence of an open set $U\subset \mathbb R^{m+1}$, a smooth Riemannian metric $g$ on it, and an integral $m$-dimensional stationary varifold $V$ in $(U,g)$ with the following properties. $V$ has a flat tangent plane of multiplicity $2$ at an interior point $p$, it is a smooth immersed minimal surface in $U\setminus \{p\}$, and it has infinite topology in any neighborhood of $p$. The construction starts from an idea of three of the authors, who in \cite{DHS} tried to build a similar example with $C^{m-1, α}$ regularity for $g$. That previous attempt contained an important error, which has been overcome using a crucial suggestion of gpt-astra-6.

math.DG↗
arXiv2026

Regularity thresholds for anomalous dissipation and related phenomena in passive scalars

We prove the absence of anomalous dissipation for passive scalars driven by some random autonomous divergence-free vector fields in $\mathbb T^d$. In dimension $d=2$ we just need continuity almost surely and a mild nondegeneracy condition on the randomness. In dimension $d\geq 3$ we assume a special geometric structure and almost sure Hölder regularity with a Hölder exponent bigger than $\frac{1}{8}$. No regularity is assumed on the passive scalar except for boundedness in the initial data. The proof relies on dimension-theoretic arguments, as opposed to commutator estimates. A consequence of these results is that the same assumptions prevent (almost surely) many other expected properties of turbulent flows, such as anomalous regularization, the Yaglom-Obukhov-Corrsin law, and Richardson diffusion.

math.AP↗
arXiv2026

Level sets of fractional Sobolev functions

We prove a coarea-type result for scalar functions $f$ in fractional Sobolev spaces $W^{s, p} (Ω)$ with $Ω\subset \mathbb R^n$, $0<s<1$, and $1\leq p < \infty$. Our theorem shows that a.e. level set has zero Hausdorff $\mathcal{H}^{n-s}$ measure, where the level set $f^{-1} (y)$ is defined as the set all points at which $y$ is between the $\liminf$ and the $\limsup$ (as $r\downarrow 0$) of the averages of $f$ over the balls $B_r (y)$. A quite general construction of random series of wavelets shows also that with probability $1$ (many) level sets have indeed dimension $n-s$.

math.AP↗
arXiv2026

Stationary and stable varifolds with singularities

We construct minimal $m$-dimensional immersions in $\R^{m+1}$, equipped with a $C^{1, α}$ metric, $α\in [0,1)$, with a sequence of \emph{catenoidal necks} or \emph{floating disks} converging to an isolated, multiplicity $2$, singular flat point.

math.DG↗
arXiv2025

Area minimizing hypersurfaces modulo $p$: a geometric free-boundary problem

We consider area minimizing $m$-dimensional currents $\mathrm{mod}(p)$ in complete $C^2$ Riemannian manifolds $Σ$ of dimension $m+1$. For odd moduli we prove that, away from a closed rectifiable set of codimension $2$, the current in question is, locally, the union of finitely many smooth minimal hypersurfaces coming together at a common $C^{1,α}$ boundary of dimension $m-1$, and the result is optimal. For even $p$ such structure holds in a neighborhood of any point where at least one tangent cone has $(m-1)$-dimensional spine. These structural results are indeed the byproduct of a theorem that proves (for any modulus) uniqueness and decay towards such tangent cones. The underlying strategy of the proof is inspired by the techniques developed by Leon Simon in "Cylindrical tangent cones and the singular set of minimal submanifolds" (J. Diff. Geom. 1993) in a class of multiplicity one stationary varifolds. The major difficulty in our setting is produced by the fact that the cones and surfaces under investigation have arbitrary multiplicities ranging from $1$ to $\lfloor \frac{p}{2}\rfloor$.

math.AP↗
arXiv2025

The fine structure of the singular set of area-minimizing integral currents I: the singularity degree of flat singular points

We consider an area-minimizing integral current of dimension $m$ and codimension at least $2$ and fix an arbitrary interior singular point $q$ where at least one tangent cone is flat. For any vanishing sequence of scales around $q$ along which the rescaled currents converge to a flat cone, we define a suitable singularity degree of the rescalings, which is a real number bigger than or equal to $1$. We show that this number is independent of the chosen sequence and we prove several interesting properties linked to its value. Our study prepares the ground for two companion works, where we show that the singular set is $(m-2)$-rectifiable and the tangent cone is unique at $\mathcal{H}^{m-2}$-a.e. point.

math.AP↗
arXiv2025

$C^\infty$ rectifiability of stationary varifolds

In this paper we prove that, for every integers $m\geq 2$ and $n\geq 1$, the support of any stationary $m$-dimensional integer rectifiable varifold $V$ in an open set $U\subset \mathbb R^{m+n}$ is $C^\infty$ rectifiable, namely it can be covered, up to an $H^m$-null set, with countably many $C^\infty$ $m$-dimensional graphs.

math.AP↗
arXiv2025

Remarks and Conjectures on Stationary Varifolds

In this paper, we revisit some known results about stationary varifolds using simpler arguments. In particular, we obtain the height bound and the Lipschitz approximation along with its estimates, and as a consequence, the excess decay

math.AP↗
arXiv2024

Optimal smooth approximation of integral cycles

In this article we prove that each integral cycle $T$ in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by an integral cycle in the same homology class which is a smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 5. Moreover, if the homology class $τ$ is representable by a smooth submanifold, then $Σ$ can be chosen free of singularities.

math.DG↗
arXiv2024

Lossless Analog Compression

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where K(x) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x. We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution -- with respect to the s-dimensional Hausdorff measure -- supported on countable unions of s-dimensional differentiable submanifolds of the m-dimensional real coordinate space. Countable unions of differentiable submanifolds include essentially all signal models used in the compressed sensing literature. Specifically, we prove that, for almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n>s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n<s linear measurements. This leads us to the introduction of the new class of s-analytic random vectors, which admit a strong converse in the sense of n greater than or equal to s being necessary for recovery with probability of error smaller than one. The central conceptual tools in the development of our theory are geometric measure theory and the theory of real analytic functions.

math.FA↗
arXiv2024

The fine structure of the singular set of area-minimizing integral currents II: rectifiability of flat singular points with singularity degree larger than $1$

We consider an area-minimizing integral current $T$ of codimension higher than $1$ in a smooth Riemannian manifold $Σ$. In a previous paper we have subdivided the set of interior singular points with at least one flat tangent cone according to a real parameter, which we refer to as ``singularity degree''. This parameter determines the infinitesimal order of contact at the point in question between the ``singular part'' of $T$ and its ``best regular approximation''. In this paper we show that the set of points for which the singularity degree is strictly larger than $1$, is $(m-2)$-rectifiable. In a subsequent work we prove that the remaining flat singular points form an $(m-2)$-null set, thus concluding that the singular set of $T$ is $m-2$-rectifiable.

math.AP↗
arXiv2024

Caffarelli's work on elliptic free boundary problems

The aim of the note is to illustrate some of the ideas introduced by Luis Caffarelli in his groundbreaking works on the regularity theory for elliptic free boundary problems, in a way which can be understood by non-experts.

math.AP↗
arXiv2024

Fine Structure of Singularities in Area-Minimizing Currents Mod$(q)$

We study fine structural properties related to the interior regularity of $m$-dimensional area minimizing currents mod$(q)$ in arbitrary codimension. We show: (i) the set of points where at least one tangent cone is translation invariant along $m-1$ directions is locally a connected $C^{1,β}$ submanifold, and moreover such points have unique tangent cones; (ii) the remaining part of the singular set is countably $(m-2)$-rectifiable, with a unique flat tangent cone (with multiplicity) at $\mathcal{H}^{m-2}$-a.e. flat singular point. These results are consequences of fine excess decay theorems as well as almost monotonicity of a universal frequency function.

math.AP↗
arXiv2024

Besicovitch's 1/2 problem and linear programming

We consider the following classical conjecture of Besicovitch: a $1$-dimensional Borel set in the plane with finite Hausdorff $1$-dimensional measure $\mathcal{H}^1$ which has lower density strictly larger than $\frac{1}{2}$ almost everywhere must be countably rectifiable. We improve the best known bound, due to Preiss and Tišer, showing that the statement is indeed true if $\frac{1}{2}$ is replaced by $\frac{7}{10}$ (in fact we improve the Preiss-Tišer bound even for the corresponding statement in general metric spaces). More importantly, we propose a family of variational problems to produce the latter and many other similar bounds and we study several properties of them, paving the way for further improvements.

math.CA↗
arXiv2024

The geometry of $C^{1,α}$ flat isometric immersions

We show that any isometric immersion of a flat plane domain into $\mathbb R^3$ is developable provided it enjoys the little Hölder regulairty $c^{1,2/3}$. In particular, isometric immersions of local $C^{1,α}$ regularity with $α> 2/3$ belong to this class. The proof is based on the existence of a weak notion of second fundamental form for such immersions, the analysis of the Gauss-Codazzi-Mainardi equations in this weak setting, and a parallel result on the very weak solutions to the degenerate Monge-Ampère equation analyzed by Lewicka and the second author.

math.AP↗