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Camillo De Lellis

Publications and source records attributed to Camillo De Lellis.

At least 55 records · Page 3Linked to original sources
arXiv2020

Positive solutions of transport equations and classical nonuniqueness of characteristic curves

The seminal work of DiPerna and Lions [Invent. Math., 98, 1989] guarantees the existence and uniqueness of regular Lagrangian flows for Sobolev vector fields. The latter is a suitable selection of trajectories of the related ODE satisfying additional compressibility/semigroup properties. A long-standing open question is whether the uniqueness of the regular Lagrangian flow is a corollary of the uniqueness of the trajectory of the ODE for a.e. initial datum. Using Ambrosio's superposition principle we relate the latter to the uniqueness of positive solutions of the continuity equation and we then provide a negative answer using tools introduced by Modena and Székelyhidi in the recent groundbreaking work [Ann. PDE, 4, 2018]. On the opposite side, we introduce a new class of asymmetric Lusin-Lipschitz inequalities and use them to prove the uniqueness of positive solutions of the continuity equation in an integrability range which goes beyond the DiPerna-Lions theory.

math.AP↗
arXiv2019

Geometric measure theory and differential inclusions

In this paper we consider Lipschitz graphs of functions which are stationary points of strictly polyconvex energies. Such graphs can be thought as integral currents, resp. varifolds, which are stationary for some elliptic integrands. The regularity theory for the latter is a widely open problem, in particular no counterpart of the classical Allard's theorem is known. We address the issue from the point of view of differential inclusions and we show that the relevant ones do not contain the class of laminates which are used in [22] and [25] to construct nonregular solutions. Our result is thus an indication that an Allard's type result might be valid for general elliptic integrands. We conclude the paper by listing a series of open questions concerning the regularity of stationary points for elliptic integrands.

math.AP↗
arXiv2019

Dirichlet energy-minimizers with analytic boundary

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and provides a first step to) a conjecture by B. White \cite{White97} that area minimizing $2$-dimensional currents with real analytic boundaries have a finite number of singularities. We also show that, in any dimension, Dirichlet energy-minimizers with a $C^1$ boundary interface are Hölder continuous at the interface.

math.AP↗
arXiv2019

Nonclassical minimizing surfaces with smooth boundary

We construct a Riemannian metric $g$ on $\mathbb{R}^4$ (arbitrarily close to the euclidean one) and a smooth simple closed curve $Γ\subset \mathbb R^4$ such that the unique area minimizing surface spanned by $Γ$ has infinite topology. Furthermore the metric is almost Kähler and the area minimizing surface is calibrated.

math.DG↗
arXiv2019

$C^{1,α}$ Isometric Embeddings of Polar Caps

We study isometric embeddings of $C^2$ Riemannian manifolds in the Euclidean space and we establish that the Hölder space $C^{1,\frac{1}{2}}$ is critical in a suitable sense: in particular we prove that for $α> \frac{1}{2}$ the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any $α< \frac{1}{2}$ we construct $C^{1,α}$ isometric embeddings of portions of the standard $2$-dimensional sphere for which such property fails.

math.AP↗
arXiv2018

Almgren's center manifold in a simple setting

We aim at explaining the most basic ideas underlying two fundamental results in the regularity theory of area minimizing oriented surfaces: De Giorgi's celebrated $\varepsilon$-regularity theorem and Almgren's center manifold. Both theorems will be proved in a very simplified situation, which however allows to illustrate some of the most important PDE estimates.

math.AP↗
arXiv2018

Boundary regularity of mass-minimizing integral currents and a question of Almgren

This short note is the announcement of a forthcoming work in which we prove a first general boundary regularity result for area-minimizing currents in higher codimension, without any geometric assumption on the boundary, except that it is an embedded submanifold of a Riemannian manifold, with a mild amount of smoothness ($C^{3, a_0}$ for a positive $a_0$ suffices). Our theorem allows to answer a question posed by Almgren at the end of his Big Regularity Paper. In this note we discuss the ideas of the proof and we also announce a theorem which shows that the boundary regularity is in general weaker that the interior regularity. Moreover we remark an interesting elementary byproduct on boundary monotonicity formulae.

math.AP↗
arXiv2018

Min-max embedded geodesic lines in asymptotically conical surfaces

We employ min-max methods to construct uncountably many, geometrically distinct, properly embedded geodesic lines in any asymptotically conical surface of non-negative scalar curvature, a setting where minimization schemes are doomed to fail. Our construction provides control of the Morse index of the geodesic lines we produce, which will be always less or equal than one (with equality under suitable curvature or genericity assumptions), as well as of their precise asymptotic behaviour. In fact, we can prove that in any such surface for every couple of opposite half-lines there exists an embedded geodesic line whose two ends are asymptotic, in a suitable sense, to those half-lines.

math.DG↗
arXiv2017

Regularity theory for $2$-dimensional almost minimal currents II: branched center manifold

We construct a branched center manifold in a neighborhood of a singular point of a $2$-dimensional integral current which is almost minimizing in a suitable sense. Our construction is the first half of an argument which shows the discreteness of the singular set for the following three classes of $2$-dimensional currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.

math.AP↗
arXiv2017

Ill-posedness of Leray solutions for the ipodissipative Navier-Stokes equations

We prove the ill-posedness of Leray solutions to the Cauchy problem for the ipodissipative Navier--Stokes equations, when the dissipative term is a fractional Laplacian $(-Δ)^α$ with exponent $α< \frac{1}{5}$. The proof follows the ''convex integration methods'' introduced by the second author and László Székelyhidi Jr. for the incomprresible Euler equations. The methods yield indeed some conclusions even for exponents in the range $[\frac{1}{5}, \frac{1}{2}[$.

math.AP↗
arXiv2017

Min-max theory for minimal hypersurfaces with boundary

In this note we propose a min-max theory for embedded hypersurfaces with a fixed boundary and apply it to prove several theorems about the existence of embedded minimal hypersurfaces with a given boundary. A simpler variant of these theorems holds also for the case of the free boundary minimal surfaces.

math.AP↗
arXiv2017

A direct approach to the anisotropic Plateau problem

We prove a compactness principle for the anisotropic formulation of the Plateau problem in codimension one, along the same lines of previous works of the authors [DGM14, DPDRG15]. In particular, we perform a new strategy for proving the rectifiability of the minimal set, avoiding the Preiss' Rectifiability Theorem [Pre87].

math.AP↗
arXiv2017

Fractional Sobolev Regularity for the Brouwer Degree

We prove that if $Ω\subset \mathbb R^n$ is a bounded open set and $nα> {\rm dim}_b (\partial Ω) = d$, then the Brouwer degree deg$(v,Ω,\cdot)$ of any Hölder function $v\in C^{0,α}\left (Ω, \mathbb R^{n}\right)$ belongs to the Sobolev space $W^{β, p} (\mathbb R^n)$ for every $0\leq β< \frac{n}{p} - \frac{d}α$. This extends a summability result of Olbermann and in fact we get, as a byproduct, a more elementary proof of it. Moreover we show the optimality of the range of exponents in the following sense: for every $β\geq 0$ and $p\geq 1$ with $β> \frac{n}{p} - \frac{n-1}α$ there is a vector field $v\in C^{0, α} (B_1, \mathbb R^n)$ with $\mbox{deg}\, (v, Ω, \cdot)\notin W^{β, p}$, where $B_1 \subset \mathbb R^n$ is the unit ball.

math.CA↗
arXiv2017

Onsager's conjecture for admissible weak solutions

We prove that given any $β<1/3$, a time interval $[0,T]$, and given any smooth energy profile $e \colon [0,T] \to (0,\infty)$, there exists a weak solution $v$ of the three-dimensional Euler equations such that $v \in C^β([0,T]\times \mathbb{T}^3)$, with $e(t) = \int_{\mathbb{T}^3} |v(x,t)|^2 dx$ for all $t\in [0,T]$. Moreover, we show that a suitable $h$-principle holds in the regularity class $C^β_{t,x}$, for any $β<1/3$. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.

math.AP↗
arXiv2016

A Nash-Kuiper theorem for $C^{1,\frac{1}{5}-δ}$ immersions of surfaces in $3$ dimensions

We prove that, given a $C^2$ Riemannian metric $g$ on the $2$-dimensional disk $D_2$, any short $C^1$ immersion of $(D_2,g)$ into $\mathbb R^3$ can be uniformly approximated with $C^{1,α}$ isometric immersions for any $α< \frac{1}{5}$. This statement improves previous results by Yu.F. Borisov and of a joint paper of the first and third author with S. Conti.

math.DG↗
arXiv2016

High dimensionality and h-principle in PDE

In this note we would like to present "an analysts' point of view" on the Nash-Kuiper theorem and in particular highlight the very close connection to some aspects of turbulence -- a paradigm example of a high-dimensional phenomenon.

math.AP↗