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

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