Sharp Morrey Regularity for an Even Order Elliptic System

Abstract

In this short note, we establish a sharp Morrey regularity theory for an even order elliptic system of Rivière type:

$$\Delta^{m}u=\sum_{l=0}^{m-1}\Delta^{l}\langle{V}_{l}, du\rangle+\sum_{l=0}^{m-2}\Delta^{l}\delta(w_{l}du)+f \quad {\rm in} \; B^{2m}$$

under minimal regularity assumptions on the coefficients functions V_l, w_l and that f belongs to certain Morrey space. This can be regarded as a further extension of the recent L^p-regularity theory obtained by Guo–Xiang–Zheng [J. Math. Pures Appl. (9), 165, 286–324 (2022)], and generalizes [Proc. Amer. Math. Soc., 152(10), 4261–4268 (2024)], [Acta Math. Sci. Ser. B (Engl. Ed.), 44(2), 420–430 (2024)] for second and fourth order elliptic systems.