The Lamm–Rivière system I: $$L^p$$ regularity theory

Abstract

Motivated by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm–Rivière system $$\begin{aligned} \Delta ^{2}u=\Delta (V\cdot \nabla u)+\mathrm{div}(w\nabla u)+(\nabla \omega +F)\cdot \nabla u+f \end{aligned}$$in dimension four, with an inhomogeneous term f which belongs to some natural function space. We obtain optimal higher order regularity and sharp Hölder continuity of weak solutions. Among several applications, we derive weak compactness for sequences of weak solutions with uniformly bounded energy, which generalizes the weak convergence theory of approximate biharmonic mappings.