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

作者：Guo, Chang -Yu; Xiang, Chang -Lin^*; Zheng, Gao-Feng

来源：Journal des Mathematiques Pures et Appliquees, 2022, 165: 286-324.

DOI：10.1016/j.matpur.2022.07.010

• Even order elliptic system
• Conservation law
• Polyharmonic mappings
• L-p theory
• Riesz potential theory

Motivated by a challenging expectation of Riviere [24], in the recent interesting work [5], de Longueville and Gastel proposed the following geometrical even order elliptic system delta(m)u = sigma(m-1)(t=0) delta l<V-l, du>+ sigma(m-2)(t=0) delta(l)delta<V-l, du>(wldu) in B-2m 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 Riviere. 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 Riviere. Combining their conservation law and some new ideas together, we obtain optimal Holder continuity and sharp L-p 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.

• 单位
  山东大学