We revisit some problems concerning maps in Sobolev spaces between Riemannian manifolds. A classical example is the extension of the trace, where one wants to extend a map defined on a fractional Sobolev space of the boundary of a domain, to a map defined on the whole domain and belonging to a Sobolev space with the same homogeneity. It is known that such problems can encompass topological obstructions, a typical example being the extension to the disk of a non-contractible boundary datum into a supercritical Sobolev space, which embeds into the space of continuous maps. We will see that if no such obstruction occurs for a given problem (extension, lifting, composition\dots) in a given nonlinear Sobolev space, then this comes together with a linear energy bound. This result gives a nonlinear counterpart to the classical Uniform Boundedness Principle (Banach-Steinhaus theorem).

Joint work with Jean Van Schaftingen (UCLouvain, Belgium).