Papers by Jean Van Schaftingen
We study when and how the norm of a function u in the homogeneous Sobolev spacesẆ s,p
Cross field generation is often used as the basis for the construction of block-structured quadra... more Cross field generation is often used as the basis for the construction of block-structured quadrangular meshes, and the field singularities have a key impact on the structure of the resulting meshes. In this paper, we extend Ginzburg-Landau cross field generation methods with a new formulation that allows a user to impose inner singularities. The cross field is computed via the optimization of a linear objective function with localized quadratic constraints. This method consists in fixing singularities in small holes drilled in the computational domain with specific degree conditions on their boundaries, which leads to non-singular cross fields on the drilled domain. We also propose a way to calculate the Ginzburg-Landau energy of these cross fields on the perforated domain by solving a Neumann linear problem. This energy converges to the energy of the Ginzburg-Landau functional as epsilon and the radius of the holes tend to zero. To obtain insights concerning the sum of the inner singularity degrees, we give: (i) an extension of the Ginzburg-Landau energy to the piecewise smooth domain allowing to identify the positions and degrees of the boundary singularities, and (ii) an interpretation of the Poincaré-Hopf theorem focusing on internal singularities.
Rendiconti Lincei - Matematica e Applicazioni
We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on R ... more We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on R n , using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the L p norm of functions in L p (R n).
Annales Fennici Mathematici
We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified ... more We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn-Sobolev and Korn-Hardy inequalities on stratified homogeneous groups.
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
We define the notion of colocally weakly differentiable maps from a manifold M to a manifold N. I... more We define the notion of colocally weakly differentiable maps from a manifold M to a manifold N. If p ≥ 1 and M and N are endowed with a Riemannian metric, this allows us to define intrinsically the homogeneous Sobolev spaceẆ 1,p (M, N). This new definition is equivalent with the definition by embedding in the Euclidean space and with the definition of Sobolev maps into a metric space. The colocal weak derivative is an approximate derivative. The colocal weak differentiability is stable under the suitable weak convergence. The Sobolev spaces can be endowed with various intrinsinc distances that induce the same topology and for which the space is complete.
A simpler proof of a recent inequality of Bourgain, Brezis and Mironescu is given. To cite this a... more A simpler proof of a recent inequality of Bourgain, Brezis and Mironescu is given. To cite this article:
ArXiv, 2021
We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung i... more We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space Ẇ . The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodeckĭı spaces. We establish the equivalence between Fourier analytic definitions and definitions via difference operators acting on measurable functions. We prove various new results on embeddings and non-embeddings, and give applications to harmonic and caloric extensions. For suitable wavelet bases we obtain a characterization of the approximation spaces for best n-term approximation from a wavelet basis via smoothness conditions on the function; this extends a classical result by DeVore, Jawerth and Popov.
Communications in Contemporary Mathematics, 2022
We give a quantitative characterization of traces on the boundary of Sobolev maps in [Formula: se... more We give a quantitative characterization of traces on the boundary of Sobolev maps in [Formula: see text], where [Formula: see text] and [Formula: see text] are compact Riemannian manifolds, [Formula: see text]: the Borel-measurable maps [Formula: see text] that are the trace of a map [Formula: see text] are characterized as the maps for which there exists an extension energy density [Formula: see text] that controls the Sobolev energy of extensions from [Formula: see text]-dimensional subsets of [Formula: see text] to [Formula: see text]-dimensional subsets of [Formula: see text].
Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of v... more Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side data is manufactured from divergence free vector fields which are in L^1. Such a result was proved earlier by Jean Bourgain and Haim Brezis on Euclidean space.
A simpler proof of a recent inequality of Bourgain, Brezis and Mironescu is given. To cite this a... more A simpler proof of a recent inequality of Bourgain, Brezis and Mironescu is given. To cite this article: J. Van
A simple proof of an integral inequality involving L1- vector fields is provided. This gives a sh... more A simple proof of an integral inequality involving L1- vector fields is provided. This gives a short proof of estimates of Bourgain and Brezis for elliptic and div-curl systems. To cite this article: J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 338 (2004). Résumé Nous donnons une preuve simple d’une inégalite ́ faisant intervenir un champ de vecteurs dans L1. Nous en tirons une preuve courte d’estimées de Bourgain et Brezis pour des systèmes elliptiques et des systèmes div-rot. Pour citer cet article: J. Van Schaftingen, C. R. Acad. Sci. Paris, Ser. I 338 (2004). Version française abrégée Nous donnons une preuve élémentaire de l’inégalité∣∣∣∣∣∣ RN f · u dx
Received *****; accepted after revision +++++ Presented by????? Given two compact Riemannian mani... more Received *****; accepted after revision +++++ Presented by????? Given two compact Riemannian manifolds Mm, Nn without boundary and m − 1 < 2p < m, we show that maps which are smooth except on finitely many points are dense in W 2,p(M;N). If in addition pim−1(N) is trivial, then
I propose a proof of the existence of the existence of eigenvectors and eigenvalues in the spirit... more I propose a proof of the existence of the existence of eigenvectors and eigenvalues in the spirit of Argand's proof of the fundamental theorem of algebra. The proof only relies on Weierstrass's theorem, the definition of the inverse of a linear operator and algebraic identities.
We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estima... more We prove interpolation estimates between Morrey-Campanato spaces and Sobolev spaces. These estimates give in particular concentration-compactness inequalities in the translation-invariant and in the translation-and dilation-invariant case. They also give in particular interpolation estimates between Sobolev spaces and functions of bounded mean oscillation. The proofs rely on Sobolev integral representation formulae and maximal function theory. Fractional Sobolev spaces are also covered.
The existence of solutions to −ε 2 ∆u + V u = u p in R N , is proved for small ε, where N ≥ 3, V ... more The existence of solutions to −ε 2 ∆u + V u = u p in R N , is proved for small ε, where N ≥ 3, V is a nonnegative continuous potential which has a positive local minimum, and N N −2 < p < N +2 N −2. No restriction is imposed on the rate of decay of V ; this includes in particular the case where V is compactly supported. To cite this article: V.
Abstract. We study the continuity of functions u whose mixed derivative ∂1 · · · ∂Nu is a measure... more Abstract. We study the continuity of functions u whose mixed derivative ∂1 · · · ∂Nu is a measure. If u ∈W 1,1(RN), then we prove that u is continu-ous. The same conclusion holds for u ∈ Wk,p(Q), with kp> N − 1, where Q denotes a cube in RN. The key step in the proof consists in showing that the measure ∂1 · · · ∂Nu does not charge hyperplanes orthogonal to the coordinate axes. 1.
Comptes Rendus Mathematique, 2018
We have recently introduced the trimming property for a complete Riemannian manifold N n as a nec... more We have recently introduced the trimming property for a complete Riemannian manifold N n as a necessary and sufficient condition for bounded maps to be strongly dense in W 1,p (B m ; N n) when p ∈ {1,. .. , m}. We prove in this note that even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.
We obtain new characterizations of the Sobolev spaces Ẇ (R ) and the bounded variation space ̇ BV... more We obtain new characterizations of the Sobolev spaces Ẇ (R ) and the bounded variation space ̇ BV (R ). The characterizations are in terms of the functionals νγ(Eλ,γ/p[u]) where Eλ,γ/p[u] = { (x, y) ∈ R × R : x 6= y, |u(x) − u(y)| |x − y|1+γ/p > λ } and the measure νγ is given by dνγ(x, y) = |x−y| dxdy. We provide characterizations which involve the L-quasi-norms supλ>0 λ νγ(Eλ,γ/p[u]) 1/p and also exact formulas via corresponding limit functionals, with the limit for λ → ∞ when γ > 0 and the limit for λ → 0 when γ < 0. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For p > 1 the characterizations hold for all γ 6= 0. For p = 1 the upper bounds for the L quasi-norms fail in the range γ ∈ [−1, 0); moreover in this case the limit functionals represent the L norm of the gradient for C c -functions but not for generic Ẇ -functions. For this situation we provide new counterexamples which are built on self-similar...
arXiv: Differential Geometry, 2019
We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of t... more We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.
We study the incompressible two dimensional Navier–Stokes equation with initial vorticity in the ... more We study the incompressible two dimensional Navier–Stokes equation with initial vorticity in the homogeneous Sobolev space \(\dot{W}^{1,1}(\mathbb{R}^{2})\). This complements our earlier work for the case when the initial vorticity is in the inhomogeneous Sobolev space \(W^{1,1}(\mathbb{R}^{2})\).
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Papers by Jean Van Schaftingen