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Camillo De Lellis

Publications and source records attributed to Camillo De Lellis.

At least 73 records · Page 4Linked to original sources
arXiv2016

Regularity theory for $2$-dimensional almost minimal currents I: Lipschitz approximation

We construct Lipschitz $Q$-valued functions which approximate carefully integral currents when their cylindrical excess is small and they are almost minimizing in a suitable sense. This result is used in two subsequent works to prove the discreteness of the singular set for the following three classes of $2$-dimensional integral currents: area minimizing in Riemannian manifolds, semicalibrated and spherical cross sections of $3$-dimensional area minimizing cones.

math.AP↗
arXiv2016

Lossless Linear Analog Compression

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ from the noiseless linear measurements ${\boldsymbol{\mathsf{y}}}=\boldsymbol{A}{\boldsymbol{\mathsf{x}}}$ with measurement matrix $\boldsymbol{A}\in{\mathbb R}^{n\times m}$. Specifically, for a random vector ${\boldsymbol{\mathsf{x}}}\in{\mathbb R}^m$ of arbitrary distribution we show that ${\boldsymbol{\mathsf{x}}}$ can be recovered with zero error probability from $n>\inf\underline{\operatorname{dim}}_\mathrm{MB}(U)$ linear measurements, where $\underline{\operatorname{dim}}_\mathrm{MB}(\cdot)$ denotes the lower modified Minkowski dimension and the infimum is over all sets $U\subseteq{\mathbb R}^{m}$ with $\mathbb{P}[{\boldsymbol{\mathsf{x}}}\in U]=1$. This achievability statement holds for Lebesgue almost all measurement matrices $\boldsymbol{A}$. We then show that $s$-rectifiable random vectors---a stochastic generalization of $s$-sparse vectors---can be recovered with zero error probability from $n>s$ linear measurements. From classical compressed sensing theory we would expect $n\geq s$ to be necessary for successful recovery of ${\boldsymbol{\mathsf{x}}}$. Surprisingly, certain classes of $s$-rectifiable random vectors can be recovered from fewer than $s$ measurements. Imposing an additional regularity condition on the distribution of $s$-rectifiable random vectors ${\boldsymbol{\mathsf{x}}}$, we do get the expected converse result of $s$ measurements being necessary. The resulting class of random vectors appears to be new and will be referred to as $s$-analytic random vectors.

cs.IT↗
arXiv2016

Transport equation with integral terms

We prove some theorems on the existence, uniqueness, stability and compactness properties of solutions to inhomogeneous transport equations with Sobolev coefficients, where the inhomogeneous term depends upon the solution through an integral operator. Contrary to the usual DiPerna-Lions approach, the essential step is to formulate the problem in the Lagrangian setting. Some motivations to study the above problem arise from the description of polymeric flows, where such kind of equations are coupled with other Navier-Stokes type equations. Using the results for the transport equation we will provide, in a separate paper, a sequential stability theorem for the full problem of the flow of concentrated polymers.

math.AP↗
arXiv2015

Regularity of area minimizing currents II: center manifold

This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing current. Such center manifold is complemented with a Lipschitz multi-valued map on its normal bundle, which approximates the current with a highe degree of accuracy. In the third and final paper these objects are used to conclude a new proof of Almgren's celebrated dimension bound on the singular set.

math.DG↗
arXiv2015

Regularity of area minimizing currents III: blow-up

This is the last of a series of three papers in which we give a new, shorter proof of a slightly improved version of Almgren's partial regularity of area minimizing currents in Riemannian manifolds. Here we perform a blow-up analysis deducing the regularity of area minimizing currents from that of Dir-minimizing multiple valued functions.

math.DG↗
arXiv2015

Regularity theory for $2$-dimensional almost minimal currents III: blowup

We analyze the asymptotic behavior of a $2$-dimensional integral current which is almost minimizing in a suitable sense at a singular point. Our analysis is the second 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↗
arXiv2015

Uniqueness of tangent cones for $2$-dimensional almost minimizing currents

We consider $2$-dimensional integer rectifiable currents which are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated $2$-dimensional currents and spherical cross sections of $3$-dimensional area minimizing cones.

math.AP↗
arXiv2015

Endpoint regularity of $2$d Mumford-Shah minimizers

We prove an $\varepsilon$-regularity theorem at the endpoint of connected arcs for $2$-dimensional Mumford-Shah minimizers. In particular we show that, if in a given ball $B_r (x)$ the jump set of a given Mumford-Shah minimizer is sufficiently close, in the Hausdorff distance, to a radius of $B_r (x)$, then in a smaller ball the jump set is a connected arc which terminates at some interior point $y_0$ and it is $C^{1,α}$ up to $y_0$.

math.AP↗
arXiv2014

SBV regularity for Hamilton-Jacobi equations in $\mathbb R^n$

In this paper we study the regularity of viscosity solutions to the following Hamilton-Jacobi equations $$ \partial_t u + H(D_{x} u)=0 \qquad \textrm{in} Ω\subset \mathbb R\times \mathbb R^{n} . $$ In particular, under the assumption that the Hamiltonian $H\in C^2(\mathbb R^n)$ is uniformly convex, we prove that $D_{x}u$ and $\partial_t u$ belong to the class $SBV_{loc}(Ω)$.

math.AP↗
arXiv2014

Regularity of area minimizing currents I: gradient L^p estimates

In a series of papers, including the present one, we give a new, shorter proof of Almgren's partial regularity theorem for area minimizing currents in a Riemannian manifold, with a slight improvement on the regularity assumption for the latter. This note establishes a new a priori estimate on the excess measure of an area minimizing current, together with several statements concerning approximations with Lipschitz multiple valued graphs. Our new a priori estimate is an higher integrability type result, which has a counterpart in the theory of Dir-minimizing multiple valued functions and plays a key role in estimating the accuracy of the Lipschitz approximations.

math.DG↗
arXiv2014

A direct approach to Plateau's problem

We provide a compactness principle which is applicable to different formulations of Plateau's problem in codimension one and which is exclusively based on the theory of Radon measures and elementary comparison arguments. Exploiting some additional techniques in geometric measure theory, we can use this principle to give a different proof of a theorem by Harrison and Pugh and to answer a question raised by Guy David.

math.AP↗
arXiv2014

Dissipative Euler flows with Onsager-critical spatial regularity

For any $ε>0$ we show the existence of continuous periodic weak solutions $v$ of the Euler equations which do not conserve the kinetic energy and belong to the space $L^1_t (C_x^{\frac{1}{3}-ε})$, namely $x\mapsto v (x,t)$ is $(\frac{1}{3}-ε)$-Hölder continuous in space at a.e. time $t$ and the integral $\int [v(\cdot, t)]_{\frac{1}{3}-ε} dt$ is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class $L^\infty_t (C_x^{\frac{1}{3}-ε})$.

math.AP↗
arXiv2014

A note on the hausdorff dimension of the singular set for minimizers of the mumford-shah energy

We give a more elementary proof of a result by Ambrosio, Fusco and Hutchinson to estimate the Hausdorff dimension of the singular set of minimizers of the Mumford-Shah energy (see [2, Theorem 5.6]). On the one hand, we follow the strategy of the above mentioned paper; but on the other hand our analysis greatly simplifies the argument since it relies on the compactness result proved by the first two Authors in [4, Theorem 13] for sequences of local minimizers with vanishing gradient energy, and the regularity theory of minimal Caccioppoli partitions, rather than on the corresponding results for Almgren's area minimizing sets.

math.AP↗
arXiv2013

Transporting microstructure and dissipative Euler flows

Recently the second and third author developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces (arXiv:1202.1751 and arXiv:1205.3626 (2012)). The motivation comes from Onsager's conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper P. Isett (arXiv:1211.4065) has improved upon our methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent - albeit below the one conjectured by Onsager. In this paper we give a shorter proof of Isett's final result, adhering more to the original scheme and introducing some new devices. More precisely we show that for any positive $ε$ there exist periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy and belong to the Hölder class $C^{\frac{1}{5}-ε}$.

math.AP↗
arXiv2013

Higher integrability and approximation of minimal currents

We give a new, simpler proof of the main approximation theorem for area minimizing current contained in Almgren's Big regularity paper. Our proof relies on a new estimate concerning the higher integrability of the quantity called here the excess density, which in turn we link to an analogous property of harmonic multiple valued functions.

math.AP↗
arXiv2013

Multiple valued functions and integral currents

We prove several results on Almgren's multiple valued functions and their links to integral currents. In particular, we give a simple proof of the fact that a Lipschitz multiple valued map naturally defines an integer rectifiable current; we derive explicit formulae for the boundary, the mass and the first variations along certain specific vector-fields; and exploit this connection to derive a delicate reparametrization property for multiple valued functions. These results play a crucial role in our new proof of the partial regularity of area minimizing currents.

math.DG↗