{"title":"Sharp Morrey regularity theory for a fourth order geometrical equation","authors":"","doi":"10.1007/s10473-024-0202-3","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>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 <span> <span>$$\\Delta^{2}u=\\Delta(V\\nabla u)+{\\text{div}}(w\\nabla u)+(\\nabla\\omega+F)\\cdot\\nabla u+f\\qquad\\text{in}B^{4},$$</span> </span> under the smallest regularity assumptions of <em>V</em>, <em>ω</em>, <em>ω</em>, <em>F</em>, where <em>f</em> belongs to some Morrey spaces. This work was motivated by many geometrical problems such as the flow of biharmonic mappings. Our results deepens the <em>L</em><sup><em>p</em></sup> 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.</p>","PeriodicalId":50998,"journal":{"name":"Acta Mathematica Scientia","volume":null,"pages":null},"PeriodicalIF":1.2000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Scientia","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10473-024-0202-3","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 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.
期刊介绍：
Acta Mathematica Scientia was founded by Prof. Li Guoping (Lee Kwok Ping) in April 1981.
The aim of Acta Mathematica Scientia is to present to the specialized readers important new achievements in the areas of mathematical sciences. The journal considers for publication of original research papers in all areas related to the frontier branches of mathematics with other science and technology.
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