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Jean Van Schaftingen

Sobolev maps

Approximation

If \(M\) is a compact manifold and \(N \subset \mathbb{R}^\nu\) is an imbedded compact manifold, one considers for \(k \in \mathbb{N}_*\) and \(p \ge 1\) the Sobolev space \[ W^{k, p}(M, N)=\bigl\{ u \in W^{k, p}(M, \mathbb{R}^\nu)\;:\; u \in N \text{ almost everywhere in } M\bigr\}. \] One wonders whether the class of smooth functions \(C^\infty(M, N)\) is dense in \(W^{k, p}(M, N)\). In general, the answer is negative. For \(k=1\), Bethuel, and Hang and Lin have proved that the answer is positive if and only if \(M\) and \(N\) satisfy some topological condition.

With Pierre Bousquet and Augusto Ponce, we have solved the corresponding problem for \(k \ge 2\) by developping suitable tools for higher-order Sobolev spaces. We have developped a technique of topological screening to characterise the mappings that have a strong approximation.

We have treated some density problems concerning fractional Sobolev spaces. We have also considered the density problem for Sobolev spaces of maps into a noncompact complete Riemannian manifold, where a new geometric obstruction arises when \(p\) is an integer for both the strong and the weak approximation problems.

With Antoine Detaille, we have provided counterexamples to the weak approximation for every \(p \in \mathbb{N} \setminus \{0, 1\}\).

and , Heterotopic energy for Sobolev mappings.

arXiv:2506.16204

, and , Generic topological screening and approximation of Sobolev maps.

arXiv:2501.18149

and , Analytical obstructions to the weak approximation of Sobolev mappings into manifolds.

arXiv:2412.12889

, and , Weak approximation by bounded Sobolev maps with values into complete manifolds, C. R. Math. Acad. Sci. Paris 356 (2018), no. 3, 264–271.

doi:10.1016/j.crma.2018.01.017 DIAL:196135 arXiv:1701.07627

, and , Density of bounded maps in Sobolev spaces into complete manifolds, Ann. Mat. Pura Appl. (4) 196 (2017), no. 6, 2261–2301.

doi:10.1007/s10231-017-0664-1 SharedIt DIAL:185227 arXiv:1501.07136

, and , Strong density for higher order Sobolev spaces into compact manifolds, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 763–817.

doi:10.4171/JEMS/518 DIAL:158312 arXiv:1203.3721

, and , Strong approximation of fractional Sobolev maps, J. Fixed Point Theory Appl. 15 (2014), no. 1, 133–153.

doi:10.1007/s11784-014-0172-5 SharedIt DIAL:153456 arXiv:1310.6017

, and , Density of smooth maps for fractional Sobolev spaces \(W^{s, p}\) into \(\ell\)–simply connected manifolds when \(s \ge 1\), Confluentes Math. 5 (2013), no. 2, 3–22.

doi:10.5802/cml.5 DIAL:135962 arXiv:1210.2525

and , Closure of Smooth Maps in \(W^{1,p}(B^3;S^2)\), Differential Integral Equations 22 (2009), no. 9–10, 881–900.

euclid.die/1356019513 DIAL:58605 arXiv:0901.4491

, and , A case of density in \(W^{2,p}(M;N)\), C. R. Math. Acad. Sci. Paris 346 (2008), no. 13–14, 735–740.

doi:10.1016/j.crma.2008.05.006 MR:2427072 DIAL:36411

Extension of traces

I have characterised the manifolds for which the trace operator is surjectif through the construction of extensions in the cases where no anatytical or topological obstructions was known.

With Katarzyna Mazowiecka, we have given a characterisation of traces connected to topological screening. I have proved the local character of the extension of traces.

Benoît Van Vaerenbergh and myself have constructed extensions directly when \(p=1\).

Together with Mircea Petrache and Bohdan Bulanyi, we have constructed extensions of maps from the sphere \(\mathbb{S}^{n}\) into a manifold \(N\) to maps from the ball \(\mathbb{B}^{n + 1}\) in such a way that Marcinkiewicz weak \(L^{n +1}\) norm of the extension is controlled by the critical trace norm \(W^{n + 1, n/(n + 1)}\) of the original map.

and , Stationary \(p\)-harmonic maps approaching planar singular harmonic maps to the circle.

arXiv:2505.10424

and , Extensions of traces for Sobolev mappings into manifolds at the endpoint \(p=1\).

arXiv:2502.17245

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

arXiv:2412.12713

, The extension of traces for Sobolev mappings between manifolds.

arXiv:2403.18738

and , Singular extension of critical Sobolev mappings under an exponential weak-type estimate, J. Funct. Anal. 288 (2025), no. 1, 110681.

doi:10.1016/j.jfa.2024.110681 DIAL:293678 arXiv:2309.12874

and , Quantitative characterization of traces of Sobolev maps, Commun. Contemp. Math. 25 (2023), no. 02, 2250003 (31 pages).

doi:10.1142/S0219199722500031 DIAL:259271 arXiv:2101.10934

and , Trace theory for Sobolev mappings into a manifold, Ann. Fac. Sci. Toulouse Math. (6) 30 (2021), no. 2, 281–299.

doi:10.5802/afst.1675 DIAL:249396 arXiv:2001.02226

and , Controlled singular extension of critical trace Sobolev maps from spheres to compact manifolds, Int. Math. Res. Not. IMRN 2017 (2017), no. 12, 3467–3683.

doi:10.1093/imrn/rnw109 DIAL:186080 CVGMT:2784 arXiv:1508.07813

Lifting

With Petru Mironescu, we have completed the solution of the lifting of fractional mappings by constructing liftings over a compact covering spaces.

When the covering space is not compact, the lifting is known to fail. For the circle, Petru Mironescu had showed that liftings could be taken in a sum of Sobolev spaces. I have proved a nonlinear counterpart; the notion is connected to the nonlinear characterisation of sums of fractionals Sobolev spaces that I have given with Rémy Rodiac.

, Lifting of fractional Sobolev mappings to noncompact covering spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 42 (2025), no. 1, 41–84.

doi:10.4171/AIHPC/98 DIAL:298828 arXiv:2301.07663

and , Lifting in compact covering spaces for fractional Sobolev mappings, Anal. PDE 14 (2021), no. 6, 1851–1871.

doi:10.2140/apde.2021.14.1851 DIAL:250778 arXiv:1907.01373

and , Metric characterization of the sum of fractional Sobolev spaces, Stud. Math. 258 (2021), 27–51.

doi:10.4064/sm190408-21-4 DIAL:243396 arXiv:1904.03946

Other results

My student Alexandra Convent and myself have proposed and studied an intrinsic definition of Sobolev spaces between Riemannian manifolds. This definition relies on a new concept of weak differentiability adapted to the nonlinear setting of manifolds and to higher-order Sobolev maps.

In a joint work with Antoin Monteil, we have proved a general uniform boudedness principle for the analytical obstructions for the lifting, extension and approximation problem in Sobolev spaces of mappings between manifolds.

I proved general estimates on the homotopy of a critical Sobolev application. With Armin Schikorra we proved fractional estimates on the Hopf degree.

and , An estimate of the Hopf degree of fractional Sobolev mappings, Proc. Amer. Math. Soc. 148 (2020), no. 7, 2877–2891.

doi:10.1090/proc/15026 DIAL:230076 arXiv:1904.12549

, Estimates by gap potentials of free homotopy decompositions of critical Sobolev maps, Adv. Nonlinear Anal. 9 (2019), no. 1, 1214–1250.

doi:10.1515/anona-2020-0047 DIAL:223840 arXiv:1811.01706

and , Higher order intrinsic weak differentiability and Sobolev spaces between manifolds, Adv. Calc. Var. 12 (2019), no. 3, 303–332.

doi:10.1515/acv-2017-0008 DIAL:197260 arXiv:1702.07171

and , Uniform boundedness principles for Sobolev maps into manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 2, 417–449.

doi:10.1016/j.anihpc.2018.06.002 DIAL:214140 CVGMT:3593 arXiv:1709.08565

, Sobolev mappings: from liquid crystals to irrigation via degree theory, Lecture notes of the Godeaux Lecture delivered at the 9th Brussels Summer School of Mathematics (2018)

arXiv:1702.00970

and , Intrinsic colocal weak derivatives and Sobolev spaces between manifolds, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), no. 1, 97–128.

doi:10.2422/2036-2145.201312_005 DIAL:173889 arXiv:1312.5858

and , Geometric partial differentiability on manifolds: the tangential derivative and the chain rule, J. Math. Anal. Appl. 435 (2016), no. 2, 1672–1681.

doi:10.1016/j.jmaa.2015.11.036 DIAL:167763 arXiv:1501.01223