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 \(p \in \mathbb{N} \setminus \{0, 1\}\).
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.
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.
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.
