Sharp Morrey regularity theory for a fourth order geometrical equation

Abstract

This paper is a continuation of recent work by Guo-Xiang-Zheng [10]. We deduce the sharp Morrey regularity theory for weak solutions to the fourth order nonhomogeneous Lamm-Rivière equation

$$\Delta^{2}u=\Delta(V\nabla u)+{\text{div}}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in}B^{4},$$

under the smallest regularity assumptions of V, ω, ω, F, where f belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the L^p type regularity theory of [10], and generalizes the work of Du, Kang and Wang [4] on a second order problem to our fourth order problems.