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Motived by the heat flow and bubble analysis of biharmonic mappings, we studyfurther regularity issues of the fourth order Lamm-Riviere system$$\Delta^{2}u=\Delta(V\cdot\nabla u)+{\rm div}(w\nablau)+(\nabla\omega+F)\cdot\nabla u+f$$ in dimension four, with an inhomogeneousterm $f$ which belongs to some natural function space. We obtain optimal higherorder regularity and sharp Holder continuity of weak solutions. Among severalapplications, we derive weak compactness for sequences of weak solutions withuniformly bounded energy, which generalizes the weak convergence theory ofapproximate biharmonic mappings.

The Lamm-Riviere system I: $L^p$ regularity theory