Controlled-norm extensions with singularities and non-rectifiable Sobolev spaces

  • Mircea Petrache (MPI MiS, Leipzig)
A3 01 (Sophus-Lie room)


A classical way to detect "holes" in a compact manifold M is to check whether every continuous function u from a k-dimensional sphere Sk to M extends to a continuous function from the (k+1)-dimensional ball to M. The obstruction to this is the k-th fundamental group of M.

The question I consider in this talk is the natural analogue of this qualitative question when we replace the qualitative property of being a continuous function by the quantitative property of having Lp-integrable gradient. After formulating it, I will describe the following features of this Sobolev extension problem:

  1. the p corresponding the conformally invariant energy ||Du||_{Lk(Sk)} is the critical one, providing the most intriguing behaviors
  2. extensions will tend to create singularities, a big difference to the C0 case
  3. the setting is complementary to that of VMO functions and to so-called epsilon-regularity results
  4. points 2, 3 are symptoms of the fact that our function spaces are "not rectifiable"
  5. * uniformization using hyperbolic geometry provides short and elegant extension constructions

The talk is based on a recent result in collaboration with Jean Van Schaftingen.

(*: if time constraints will allow it)

Katja Heid

MPI for Mathematics in the Sciences Contact via Mail

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