Sharp Morrey regularity theory for a fourth order geometrical equation

摘要

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-Riviere equation Delta 2u=Delta(V backward difference u)+div(w backward difference u)+( backward difference omega+F)& sdot; backward difference u+fin B4,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta<^>{2}u=\Delta(V\nabla u)+{\text{div}}(w\nabla u)+(\nabla\omega+F)\cdot\nabla u+f\qquad\text{in}B<^>{4},$$\end{document} under the smallest regularity assumptions of V, omega, omega, 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 Lp 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.

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