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Jean Van Schaftingen

Publications and source records attributed to Jean Van Schaftingen.

At least 37 records · Page 2Linked to original sources
arXiv2023

Endpoint Sobolev inequalities for vector fields and cancelling operators

The injectively elliptic vector differential operators $A (\mathrm{D})$ from $V$ to $E$ on $\mathbb{R}^n$ such that the estimate \[ \Vert D^\ell u\Vert_{L^{n/(n - \ell)} (\mathbb{R}^n)} \le \Vert A (\mathrm{D}) u\Vert_{L^1 (\mathbb{R}^n)} \] holds can be characterized as the operators satisfying a cancellation condition \[ \bigcap_{ξ\in \mathbb{R}^n \setminus \{0\}} A (ξ)[V] = \{0\}\;. \] These estimates unify existing endpoint Sobolev inequalities for the gradient of scalar functions (Gagliardo and Nirenberg), the deformation operator (Korn-Sobolev inequality by M.J. Strauss) and the Hodge complex (Bourgain and Brezis). Their proof is based on the fact that $A (\mathrm{D}) u$ lies in the kernel of a cocancelling differential operator.

math.AP↗
arXiv2022

Spaces of Besov-Sobolev type and a problem on nonlinear approximation

We study fractional variants of the quasi-norms introduced by Brezis, Van Schaftingen, and Yung in the study of the Sobolev space $\dot W^{1,p}$. The resulting spaces are identified as a special class of real interpolation spaces of Sobolev-Slobodecki\uı 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.

math.FA↗
arXiv2022

Boundary ellipticity and limiting $L^1$-estimates on halfspaces

We identify necessary and sufficient conditions on $k$th order differential operators $\mathbb{A}$ in terms of a fixed halfspace $H^+\subset\mathbb{R}^n$ such that the Gagliardo--Nirenberg--Sobolev inequality $$ \|D^{k-1}u\|_{\mathrm{L}^{\frac{n}{n-1}}(H^+)}\leq c\|\mathbb{A} u\|_{\mathrm{L}^1(H^+)}\quad\text{for }u\in\mathrm{C}^\infty_c (\mathbb{R}^{n},V) $$ holds. This comes as a consequence of sharp trace theorems on $H=\partial H^+$.

math.AP↗
arXiv2022

Limiting Sobolev and Hardy inequalities on stratified homogeneous groups

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.

math.CA↗
arXiv2022

Sobolev spaces revisited

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, 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 $L^p(\mathbb{R}^n)$.

math.CA↗
arXiv2021

Quantitative characterization of traces of Sobolev maps

We give a quantitative characterization of traces on the boundary of Sobolev maps in $\dot{W}^{1,p}(\mathcal M, \mathcal N)$, where $\mathcal{M}$ and $\mathcal{N}$ are compact Riemannian manifolds, $\partial \mathcal{M} \neq \emptyset$: the Borel-measurable maps $u\colon \partial \mathcal M \to \mathcal{N}$ that are the trace of a map $U\in \dot{W}^{1,p}(\mathcal M, \mathcal{N})$ are characterized as the maps for which there exists an extension energy density $w \colon \partial \mathcal{M} \to [0,\infty]$ that controls the Sobolev energy of extensions from $\lfloor p - 1 \rfloor$-dimensional subsets of $\partial \mathcal{M}$ to $\lfloor p\rfloor$-dimensional subsets of $\mathcal{M}$.

math.AP↗
arXiv2021

Families of functionals representing Sobolev norms

We obtain new characterizations of the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^N)$ and the bounded variation space $\dot{BV}(\mathbb{R}^N)$. The characterizations are in terms of the functionals $ν_γ (E_{λ,γ/p}[u])$ where \[ E_{λ,γ/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+γ/p}}>λ\Big\} \] and the measure $ν_γ$ is given by $\mathrm{d} ν_γ(x,y)=|x-y|^{γ-N} \mathrm{d} x \mathrm{d} y$. We provide characterizations which involve the $L^{p,\infty}$-quasi-norms $\sup_{λ>0} λ\, ν_γ (E_{λ,γ/p}[u]) ^{1/p}$ and also exact formulas via corresponding limit functionals, with the limit for $λ\to\infty$ when $γ>0$ and the limit for $λ\to 0^+$ 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 $γ\neq 0$. For $p=1$ the upper bounds for the $L^{1,\infty}$ quasi-norms fail in the range $γ\in [-1,0) $; moreover in this case the limit functionals represent the $L^1$ norm of the gradient for $C^\infty_c$-functions but not for generic $\dot W^{1,1}$-functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension $γ+1$. For $γ=0$ the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions $ν_0(E_{λ,0}[u])$.

math.FA↗
arXiv2021

Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold

We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued harmonic map, which is smooth outside a finite number of points around which the energy concentrates and whose singularities' location minimises a renormalised energy, generalising known results by Bethuel, Brezis and Hélein for the circle $\mathbb{S}^1$. We also obtain $Γ$-convergence results and uniform Marcinkiewicz weak $L^2$ or Lorentz $L^2$ estimates on the derivatives. We prove that solutions to the corresponding Euler-Lagrange equation converge uniformly to the constraint and converge to harmonic maps away from singularities.

math.AP↗
arXiv2021

Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel, Brezis and Hélein for the circle (Ginzburg-Landau vortices, 1994). In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress-energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases.

math.AP↗
arXiv2021

A surprising formula for Sobolev norms

We establish the equivalence between the Sobolev semi-norm $\|\nabla u\|_{L^p}$ and a quantity obtained when replacing the strong $L^p$ by a weak $L^p$ norm in the Gagliardo semi-norm $|u|_{W^{s,p}}$ computed at $s = 1$. As corollaries we derive alternative estimates in some exceptional cases (involving $W^{1,1}$) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail.

math.FA↗
arXiv2020

Reverse superposition estimates in Sobolev spaces

We study when and how the norm of a function $u$ in the homogeneous Sobolev spaces $\dot{W}^{s, p} (\mathbb{R}^n, \mathbb{R}^m)$, with $p \ge 1$ and either $s = 1$ or $s > 1/p$, is controlled by the norm of composite function $f \circ u$ in the same space.

math.AP↗
arXiv2020

Ginzburg-Landau energy and placement of singularities in generated cross fields

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.

math.NA↗
arXiv2020

Vortex motion for the lake equations

The lake equations $$\left\{\begin{aligned} \nabla \cdot \big( b \, \mathbf{u}\big) &= 0 & & \text{on}\ \mathbb{R}\times D,\\ \partial_t\mathbf{u} + (\mathbf{u}\cdot \nabla)\mathbf{u} &= -\nabla h & & \text{on}\ \mathbb{R}\times D ,\\ \mathbf{u} \cdot \boldsymbolν &= 0 & & \text{on}\ \mathbb{R}\times\partial D . \end{aligned}\right.$$ model the vertically averaged horizontal velocity in an inviscid incompressible flow of a fluid in a basin whose variable depth $b : D \to [0, + \infty)$ is small in comparison with the size of its two-dimensional projection $D \subset \mathbb{R}^2$. When the depth $b$ is positive everywhere in $D$ and constant on the boundary, we prove that the vorticity of solutions of the lake equations whose initial vorticity concentrates at an interior point is asympotically a multiple of a Dirac mass whose motion is governed by the depth function $b$.

math.AP↗
arXiv2020

Trace theory for Sobolev mappings into a manifold

We review the current state of the art concerning the characterization of traces of the spaces $W^{1, p} (\mathbb{B}^{m-1}\times (0,1), \mathcal{N})$ of Sobolev mappings with values into a compact manifold $\mathcal{N}$. In particular, we exhibit a new analytical obstruction to the extension, which occurs when $p < m$ is an integer and the homotopy group $π_p (\mathcal{N})$ is non trivial. On the positive side, we prove the surjectivity of the trace operator when the fundamental group $π_1 (\mathcal{N})$ is finite and $π_2 (\mathcal{N}) \simeq \dotsb \simeq π_{\lfloor p - 1 \rfloor} (\mathcal{N}) \simeq \{0\}$. We present several open problems connected to the extension problem.

math.AP↗
arXiv2019

Lifting in compact covering spaces for fractional Sobolev mappings

Let $π: \widetilde{\mathcal{N}} \to \mathcal{N}$ be a Riemannian covering, with $\mathcal{N}$, $\widetilde{\mathcal{N}}$ smooth compact connected Riemannian manifolds. If $\mathcal{M}$ is an $m$-dimensional compact simply-connected Riemannian manifold, $0 1$, in dimension $1$. Our argument also leads to the existence of a lifting when $0<s<1$ and $1<sp<2\le m$, provided there is no topological obstruction on $u$, i.e., $u = π\, \circ \, \widetilde{u}$ holds in this range provided $u$ is in the strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$. However, when $0<s<1$, $sp = 1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}(\mathcal{M},\mathcal{N})$ in the strong closure of $C^\infty({\mathcal{M}}, \mathcal{N})$, such that $u = π\, \circ \, \widetilde{u}$ does not hold for any $\widetilde{u} \in W^{s, p} ({\mathcal{M}}, \widetilde{\mathcal{N}} )$.

math.AP↗