Nonlocal Differential Equations in Analysis and Geometry
Final Report Abstract
The project lead to new results in all the proposed areas. With respect to the most analytical Part A (Analysis of non-local energies and PDE) substantial progress was obtained on the regularity theory of scalar PDEs involving so-called fractional p-Laplacian. Also a new aspect of commutator estimates via harmonic extensions was introduced. With respect to Part B (Geometric PDEs and harmonic maps) within this project major progress was obtained in the regularity theory and singularity analysis of (classical and fractional) harmonic maps into manifolds, as well as existence theory (in homotopy). A new framework for studying nonlocal harmonic maps was introduced, which provides the notion and applicability of fractional div-curl–type theory. In particular we obtained the most natural proof for regularity theory for fractional harmonic maps (which mimicks the local theory almost verbatim). Major progress was also obtained for the part of knot- and curvature energies, the regularity theory has been extended to the full class of the so-called O’Hara energies (which were completely out of reach before this project). We are preparing our angle of attack for the surface version, and as model cases for nonlocal surface-curvature type energies we obtained a version of a conformal parametrization theorem (similar to the Müller-Sverak theory) and an obstacle problem. In Part C (Analysis of free boundary problems) we made progress in three directions. First we obtained boundary regularity for Rivière’s antisymmetric system under a free boundary condition. Secondly, somewhat suprisingly and certainly not envisioned when proposing this project, we found an application to the half-wave equation and were able to classify constant-speed solutions. Finally, we considered a free boundary problem for Willmore surfaces which also features conformal invariance. We proved a reflection principle for weak solutions which may be useful when constructing periodic surfaces. An existence result for minimizers in a global setting is in preparation, for this estimates in fractional Sobolev space are important.
