arXiv subjects

Antoine Detaille

Publications and source records attributed to Antoine Detaille.

7 recordsLinked to original sources

arXiv2025

A surprising threshold for the validity of the method of singular projection

Given a compact manifold $ \mathcal{N} $ embedded into $ \mathbb{R}^{\nu} $ and a projection $ P $ that retracts $ \mathbb{R}^{\nu} $ except a singular set of codimension $ \ell $ onto $ \mathcal{N} $, we investigate the maximal range of parameters $ s $ and $ p $ such that the projection $ P $ can be used to turn an $ \mathbb{R}^{\nu} $-valued $ W^{s,p} $ map into an $ \mathcal{N} $-valued $ W^{s,p} $ map. Devised by Hardt and Lin with roots in the work of Federer and Fleming, the method of projection is known to apply in $ W^{1,p} $ if and only if $ p < \ell $, and has been extended in some special cases to more general values of the regularity parameter $ s $. As a first result, we prove in full generality that, when $ s \geq 1 $, the method of projection can be applied in the whole expected range $ sp < \ell $. When $ 0 < s < 1 $, the method of projection was only known to be applicable when $ p < \ell $, a more stringent condition than $ sp < \ell $. As a second result, we show that, somehow surprisingly, the condition $ p < \ell $ is optimal, by constructing, for every $ 0 < s < 1 $ and $ p \geq \ell $, a bounded $ W^{s,p} $ map into $ \mathbb{R}^{\ell} $ whose singular projections onto the sphere $ \mathbb{S}^{\ell-1} $ all fail to belong to $ W^{s,p} $. As a byproduct of our method, a similar conclusion is obtained for the closely related method of almost retraction, devised by Haj\l asz, for which we also prove a more stringent threshold of applicability when $ 0 < s < 1 $.

math.FA↗

arXiv2025

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↗

arXiv2024

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↗

arXiv2024

Generic non-uniqueness of minimizing harmonic maps from a ball to a sphere

In this note, we study non-uniqueness for minimizing harmonic maps from $B^3$ to $\mathbb{S}^2$. We show that every boundary map can be modified to a boundary map that admits multiple minimizers of the Dirichlet energy by a small $W^{1,p}$-change for $p<2$. This strengthens a remark by the second-named author and Strzelecki. The main novel ingredient is a homotopy construction, which is the answer to an easier variant of a challenging question regarding the existence of a norm control for homotopies between $ W^{1,p} $ maps.

math.AP↗

arXiv2024

An improved dense class in Sobolev spaces to manifolds

We consider the strong density problem in the Sobolev space $ W^{s,p}(Q^{m};\mathscr{N}) $ of maps with values into a compact Riemannian manifold $ \mathscr{N} $. It is known, from the seminal work of Bethuel, that such maps may always be strongly approximated by $ \mathscr{N} $-valued maps that are smooth outside of a finite union of $ (m -\lfloor sp \rfloor - 1) $-planes. Our main result establishes the strong density in $ W^{s,p}(Q^{m};\mathscr{N}) $ of an improved version of the class introduced by Bethuel, where the maps have a singular set without crossings. This answers a question raised by Brezis and Mironescu. In the special case where $ \mathscr{N} $ has a sufficiently simple topology and for some values of $ s $ and $ p $, this result was known to follow from the method of projection, which takes its roots in the work of Federer and Fleming. As a first result, we implement this method in the full range of $ s $ and $ p $ in which it was expected to be applicable. In the case of a general target manifold, we devise a topological argument that allows to remove the self-intersections in the singular set of the maps obtained via Bethuel's technique.

math.FA↗

arXiv2023

A complete answer to the strong density problem in Sobolev spaces with values into compact manifolds

We consider the problem of strong density of smooth maps in the Sobolev space $ W^{s,p}(Q^{m};\mathcal{N}) $, where $ 0 < s < +\infty $, $ 1 \leq p < +\infty $, $ Q^{m} $ is the unit cube in $ \mathbb{R}^{m} $, and $ \mathcal{N} $ is a smooth compact connected Riemannian manifold without boundary. Our main result fully answers the strong density problem in the whole range $ 0 < s < +\infty $: the space $ \mathcal{C}^{\infty}(\overline{Q}^{m};\mathcal{N}) $ is dense in $ W^{s,p}(Q^{m};\mathcal{N}) $ if and only if $ \pi_{[sp]}(\mathcal{N}) = \{0\} $. This completes the results of Bethuel ($ s=1 $), Brezis and Mironescu ($ 0 < s < 1 $), and Bousquet, Ponce, and Van Schaftingen ($ s = 2 $, $ 3 $, ...). We also consider the case of more general domains $ \Omega $, in the setting studied by Hang and Lin when $ s = 1 $.

math.FA↗

arXiv2020

A decomposition for Borel measures $\mu \le \mathcal{H}^{s}$

We prove that every finite Borel measure $\mu$ in $\mathbb{R}^N$ that is bounded from above by the Hausdorff measure $\mathcal{H}^s$ can be split in countable many parts $\mu\lfloor_{E_k}$ that are bounded from above by the Hausdorff content $\mathcal{H}_\infty^s$. Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to show the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

math.CA↗