Armin Schikorra

Regularity of n/2-harmonic maps into spheres

We prove Holder continuity for n/2-harmonic maps from arbitrary subsets of Rn into a sphere. This extends a recent one-dimensional result by F. Da Lio and T. Riviere to arbitrary dimensions. The proof relies on compensation effects which we quantify adapting an approach for Wenteʼs inequality by L. Tartar, instead of Besov space arguments which were used in the one-dimensional case. Moreover, fractional analogues of Hodge decomposition and higher order Poincare inequalities as well as several localization effects for nonlocal operators similar to the fractional laplacian are developed and applied. This work was the authorʼs PhD thesis, written in March 2010.

A remark on gauge transformations and the moving frame method

Abstract In this note we give a shorter proof of recent regularity results on elliptic partial differential equations with antisymmetric structure presented in Riviere (2007) [23] , Riviere and Struwe (2008) [24] . We differ from the mentioned articles in using the direct method of Heleins moving frame, i.e. minimizing a certain variational energy-functional, in order to construct a suitable gauge transformation. Though this is neither new nor surprising, it enables us to describe a proof of regularity using elementary arguments of calculus of variations and algebraic identities. Moreover, we remark that in order to prove Hildebrandts conjecture on regularity of critical points of 2D-conformally invariant variational problems one can avoid the application of the Nash–Moser imbedding theorem.

Boundary regularity via Uhlenbeck–Rivière decomposition

Abstract We prove that weak solutions of systems with skew-symmetric structure, which possess a continuous boundary trace, have to be continuous up to the boundary. This applies, e.g., to the H-surface system Δu = 2H(u)∂x1u∧∂x2u with bounded H and thus extends an earlier result by P. Strzelecki and proves the natural counterpart of a conjecture by E. Heinz. Methodically, we use estimates below natural exponents of integrability and a recent decomposition result by T. Rivière.

A Note on Regularity for the n-dimensional H-System assuming logarithmic higher Integrability

Summary We prove Hölder continuity for solutions to the n-dimensional H-System assuming logarithmic higher integrability of the solution.

n/p-harmonic maps: regularity for the sphere case

Abstract. We introduce n/-harmonic maps as critical points of the energy where pointwise , for the N-sphere and . This energy combines the non-local behaviour of the fractional harmonic maps introduced by Rivière and the first author with the degenerate arguments of the n-Laplacian. In this setting, we will prove Hölder continuity.

Nonlinear commutators for the fractional p-Laplacian and applications

We prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p-Laplacian. This implies that solutions to certain degenerate nonlocal equations are higher differentiable. Also, weakly fractional p-harmonic functions which a priori are less regular than variational solutions are in fact classical. As an application we show that sequences of uniformly bounded

Integro-Differential Harmonic Maps into Spheres

\frac{n}{s}

Fractional div-curl quantities and applications to nonlocal geometric equations

ns-harmonic maps converge strongly outside at most finitely many points.