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

TL;DR

This paper investigates the regularity of solutions to a fourth-order PDE system motivated by biharmonic mappings, establishing optimal regularity results and applications to weak compactness in four dimensions.

Contribution

It provides new regularity results for the Lamm-Riviere system, including optimal higher order regularity and sharp Hölder continuity, extending weak convergence theory for biharmonic mappings.

Findings

Optimal higher order regularity of solutions

Sharp Hölder continuity results

Weak compactness for bounded energy solutions

Abstract

Motived by the heat flow and bubble analysis of biharmonic mappings, we study further regularity issues of the fourth order Lamm-Riviere system $$\Delta^{2}u=\Delta(V\cdot\nabla u)+{\rm div}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f$$ in dimension four, with an inhomogeneous term $f$ which belongs to some natural function space. We obtain optimal higher order regularity and sharp Holder 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.