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 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.
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.
Together with Mircea Petrache, 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.
Alexandra Convent and Jean Van Schaftingen, 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
Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, 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
Jean Van Schaftingen, 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)
Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, 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
Mircea Petrache and Jean Van Schaftingen, 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
Alexandra Convent and Jean Van Schaftingen, 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
Alexandra Convent and Jean Van Schaftingen, 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
Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, 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
Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, 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
Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, 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
Augusto C. Ponce and Jean Van Schaftingen, 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
Pierre Bousquet, Augusto C. Ponce and Jean Van Schaftingen, A case of density in \(W^{2,p}(M;N)\), C. R. Math. Acad. Sci. Paris 346 (2008), no. 13–14, 735–740.
