Lp regularity theory for even order elliptic systems with antisymmetric first order potentials

Abstract

Motivated by a challenging expectation of Rivière [24], in the recent interesting work [5], de Longueville and Gastel proposed the following geometrical even order elliptic systemΔmu=∑l=0m−1Δl〈Vl,du〉+∑l=0m−2Δlδ(wldu) in B2m which includes polyharmonic mappings as special cases. Under minimal regularity assumptions on the coefficient functions and an additional algebraic antisymmetry assumption on the first order potential, they successfully established a conservation law for this system, from which everywhere continuity of weak solutions follows. This beautiful result amounts to a significant advance in the expectation of Rivière.In this paper, we seek for the optimal interior regularity of the above system, aiming at a more complete solution to the aforementioned expectation of Rivière. Combining their conservation law and some new ideas together, we obtain optimal Hölder continuity and sharp Lp regularity theory, similar to that of Sharp and Topping [27], for weak solutions to a related inhomogeneous system. Our results can be applied to study heat flow and bubbling analysis for polyharmonic mappings.