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Jean Van Schaftingen

Publications and source records attributed to Jean Van Schaftingen.

At least 19 recordsLinked to original sources
arXiv2026

Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N})$ which is not a weak limit of smooth maps into $ \mathcal{N} $ due to a mechanism of analytical obstruction. For $ p = 4n - 1 $, the target manifold can be taken to be the sphere $ \mathbb{S}^{2n} $ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for $ p = 4n -1 = 3$. The results extend to higher order Sobolev spaces $ W^{s,p} $, with $ s \in \mathbb{R} $, $s \geq 1 $, $ sp \in \mathbb{N}$, and $ sp \ge 2 $.

math.FA↗
arXiv2026

Equi-integrable approximation of Sobolev mappings between manifolds

We show that limits of sequences of smooth maps between compact Riemannian manifolds with equi-integrable $W^{1, p}$-Sobolev energy can always be strongly approximated by smooth maps, giving a counterpart of Hang's density result in $W^{1, 1}$ for the Sobolev space $W^{1, p}$ with integer $p \ge 2$. Our result extends to higher-order Sobolev spaces and is straightforward in fractional Sobolev spaces. We also provide a proof based on the weak continuity of Jacobians in the cases where the cohomological criterion of Bethuel, Demengel, Colon and Hélein applies.

math.AP↗
arXiv2026

Heterotopic energy for Sobolev mappings

We study the notion of heterotopic energy defined as the limit of Sobolev energies of Sobolev mappings in a given homotopy class approximating almost everywhere a given Sobolev mapping. We show that the heterotopic energy is finite if and only if the mappings in the corresponding homotopy classes are homotopic on a codimension one skeleton of a triangulation of the domain. When this is the case, the heterotopic energy of a mapping is the sum of its Sobolev energy and its disparity energy, defined as the minimum energy of a bubble to pass between these homotopy classes. At the more technical level, we rely on a framework that works when the target and domain manifolds are not simply connected and there is no canonical isomorphism between homotopy groups with different basepoints.

math.CA↗
arXiv2025

Morrey--Sobolev inequalities with power weights on the half-space

Morrey--Sobolev inequalities are established for functions in weighted Sobolev spaces on the $n$-dimensional half-space, where the weight is a power of the distance to the boundary, as well as for Sobolev spaces on the $n$-dimensional hyperbolic space. All the estimates are optimal up to a multiplicative constant.

math.FA↗
arXiv2025

Trace conjunction inequalities

Trace conjunction integrals are introduced and studied. They appear in trace conjunction inequalities which unify the Hardy inequality on a halfspace and the classical Gagliardo trace inequality. At the endpoint they satisfy a Bourgain-Brezis-Mironescu formula for smooth maps, which raises some new open problems.

math.FA↗
arXiv2025

Extensions of traces for Sobolev mappings into manifolds at the endpoint $p=1$

We give direct proofs and constructions of the trace and extension theorems for Sobolev mappings in $W^{1, 1} (M, N)$, where $M$ is Riemannian manifold with compact boundary $\partial M$ and $N$ is a complete Riemannian manifold. The analysis is also applicable to halfspaces and strips. The extension is based on a tiling the domain of the considered applications by suitably chosen dyadic cubes to construct the desired extension. Along the way, we obtain asymptotic characterizations of the $L^1$-energy of mappings.

math.AP↗
arXiv2025

Generic topological screening and approximation of Sobolev maps

This manuscript develops a framework for the strong approximation of Sobolev maps with values in compact manifolds, emphasizing the interplay between local and global topological properties. Building on topological concepts adapted to VMO maps, such as homotopy and the degree of continuous maps, it introduces and analyzes extendability properties, focusing on the notions of $\ell$-extendability and its generalization, $(\ell, e)$-extendability. We rely on Fuglede maps, providing a robust setting for handling compositions with Sobolev maps. Several constructions -- including opening, thickening, adaptive smoothing, and shrinking -- are carefully integrated into a unified approach that combines homotopical techniques with precise quantitative estimates. Our main results establish that a Sobolev map $u \in W^{k, p}$ defined on a compact manifold of dimension $m > kp$ can be approximated by smooth maps if and only if $u$ is $(\lfloor kp \rfloor, e)$-extendable with $e = m$. When $e < m$, the approximation can still be carried out using maps that are smooth except on structured singular sets of rank $m - e - 1$.

math.FA↗
arXiv2025

Trace theory for gauge-covariant Sobolev spaces

The traces of gauge-covariant Sobolev spaces on a Riemannian vector bundle for some connection are characterised as some gauge-covariant fractional Sobolev spaces when the curvature of the connection is bounded. The constants in the trace and extension theorems only depend on this curvature. When the connection is abelian, one recovers known results for magnetic Sobolev spaces.

math.FA↗
arXiv2024

On the local character of the extension of traces for Sobolev mappings

We prove that a mapping $u \colon \mathcal{M}'\to \mathcal{N}$, where $\mathcal{M}'$ and $ \mathcal{N}$ are compact Riemannian manifolds, is the trace of a Sobolev mapping $U \colon \mathcal{M}' \times [0, 1) \to \mathcal{N}$ if and only if it is on some open covering of $\mathcal{M}'$. In the global case where $\mathcal{M}$ is a compact Riemannian manifold with boundary, this implies that the analytical obstructions to the extension of a mapping $u \colon \partial \mathcal{M}\to \mathcal{N}$ to some Sobolev mapping $U \colon \mathcal{M} \to \mathcal{N}$ are purely local.

math.AP↗
arXiv2024

Injective ellipticity, cancelling operators, and endpoint Gagliardo-Nirenberg-Sobolev inequalities for vector fields

Although Ornstein's nonestimate entails the impossibility to control in general all the $L^1$-norm of derivatives of a function by the $L^1$-norm of a constant coefficient homogeneous vector differential operator, the corresponding endpoint Sobolev inequality has been known to hold in many cases: 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). The class of differential operators for which estimates holds can be characterized by a cancelling condition. The proof of the estimates rely on a duality estimate for $L^1$-vector fields lying in the kernel of a cocancelling differential operator, combined with classical linear algebra and harmonic analysis techniques. This characterization unifies classes of known Sobolev inequalities and extends to fractional Sobolev and Hardy inequalities. A similar weaker condition introduced by Raiţă characterizes the operators for which there is an $L^\infty$-estimate on lower-order derivatives.

math.AP↗
arXiv2024

Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group

We study the limiting behavior of minimizing $p$-harmonic maps from a bounded Lipschitz domain $Ω\subset \mathbb{R}^{3}$ to a compact connected Riemannian manifold without boundary and with finite fundamental group as $p \nearrow 2$. We prove that there exists a closed set $S_{*}$ of finite length such that minimizing $p$-harmonic maps converge to a locally minimizing harmonic map in $Ω\setminus S_{*}$. We prove that locally inside $Ω$ the singular set $S_{*}$ is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains. Furthermore, we establish local and global estimates for the limiting singular harmonic map. Under additional assumptions, we prove that globally in $\overlineΩ$ the set $S_{*}$ is a finite union of straight line segments, and it minimizes the mass in the appropriate class of admissible chains, which is defined by a given boundary datum and $Ω$.

math.AP↗
arXiv2024

Singular extension of critical Sobolev mappings under an exponential weak-type estimate

Given $m \in \mathbb{N} \setminus \{0\}$ and a compact Riemannian manifold $\mathcal{N}$, we construct for every map $u$ in the critical Sobolev space $W^{m/(m + 1), m + 1} (\mathbb{S}^m, \mathcal{N})$, a map $U : \mathbb{B}^{m + 1} \to \mathcal{N}$ whose trace is $u$ and which satisfies an exponential weak-type Sobolev estimate. The result and its proof carry on to the extension to a half-space of maps on its boundary hyperplane and to the extension to the hyperbolic space of maps on its boundary sphere at infinity.

math.AP↗
arXiv2024

The extension of traces for Sobolev mappings between manifolds

The compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$ for which the trace operator from the first-order Sobolev space of mappings $\smash{\dot{W}}^{1, p} (\mathcal{M}, \mathcal{N})$ to the fractional Sobolev-Slobodecki\uı space $\smash{\smash{\dot{W}}^{1 - 1/p, p}} (\partial \mathcal{M}, \mathcal{N})$ is surjective when $1 < p < \dim \mathcal{M}$ are characterised. The traces are extended using a new construction which can be carried out assuming the absence of the known topological and analytical obstructions. When $p \ge \dim \mathcal{M}$ the same construction provides a Sobolev extension with linear estimates for maps that have a continuous extension, provided that there are no known analytical obstructions to such a control.

math.AP↗
arXiv2024

Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities

We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities \begin{align*} \lVert P\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}\le c\big(\lVert\mathscr{A}[P]\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}(\mathbb{R}^n)}+\lVert\mathbb{B}P\rVert_{L^{1}(\mathbb{R}^n)}\big) \end{align*} to hold for all $P\in C_{c}^{\infty}(\mathbb{R}^{n};V)$, where $\mathscr{A}$ is a linear map between finite dimensional vector spaces and $\mathbb{B}$ is a $k$-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the $L^{1}$-norm of the differential expression $\mathbb{B}P$ on the right-hand side, such inequalities generalise previously known estimates to the borderline case $p=1$, and thereby answer an open problem due to Müller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative.

math.AP↗
arXiv2023

Asymptotic behavior of minimizing $p$-harmonic maps when $p \nearrow 2$ in dimension 2

We study $p$--harmonic maps with Dirichlet boundary conditions from a planar domain into a general compact Riemannian manifold. We show that as $p$ approaches $2$ from below, they converge up to a subsequence to a minimizing singular renormalizable harmonic map. The singularities are imposed by topological obstructions to the existence of harmonic mappings; the location of the singularities being governed by a renormalized energy. Our analysis is based on lower bounds on growing balls and also yields some uniform weak-$L^p$ bounds (also known as Marcinkiewicz or Lorentz $L^{p,\infty}$).

math.AP↗
arXiv2023

Lifting of fractional Sobolev mappings to noncompact covering space

Given compact Riemannian manifolds $\mathcal{M}$ and $\mathcal{N}$, a Riemannian covering $π: \smash{\widetilde{\mathcal{N}}} \to \mathcal{N}$ by a noncompact covering space $\smash{\widetilde{\mathcal{N}}}$, $1 < p < \infty$ and $0 < s < 1$, the space of liftings of fractional Sobolev maps in $\smash{\dot{W}^{s, p}} (\mathcal{M}, \mathcal{N})$ is characterized when $sp > 1$ and an optimal nonlinear fractional Sobolev estimate is obtained when moreover $sp \ge \dim \mathcal{M}$. A nonlinear characterization of the sum of spaces $\smash{\dot{W}^{s, p}} (\mathcal{M}, \mathbb{R}) + \smash{\dot{W}^{1, sp}} (\mathcal{M}, \mathbb{R})$ is also provided.

math.AP↗
arXiv2023

Limiting Sobolev estimates for vector fields and cancelling differential operators

These notes present Sobolev-Gagliardo-Nirenberg endpoint estimates for classes of homogeneous vector differential operators. Away of the endpoint cases, the classical Calderón-Zygmund estimates show that the ellipticity is necessary and sufficient to control all the derivatives of the vector field. In the endpoint case, Ornstein showed that there is no nontrivial estimate on same-order derivatives. On the other hand endpoint Sobolev estimates were proved for the deformation operator (Korn-Sobolev inequality by M.J. Strauss) and for the Hodge complex (Bourgain and Brezis). The class of operators for which such Sobolev estimates holds can be characterized by a cancelling condition. The estimates rely on a duality estimate for $L^1$ vector fields satisfying some conditions on the derivatives, combined with classical algebraic and harmonic analysis techniques. This characterization unifies classes of known inequalities and extends to the case of Hardy inequalities.

math.AP↗