一类偶数阶几何偏微分方程组弱解的正则性理论  

Regularity theory for weak solutions to a class of even order geometric partial differential equations

作　　者：文竹 汪继秀 向长林 Zhu Wen;Jixiu Wang;Chang-Lin Xiang

机构地区：[1]长江大学信息与数学学院,荆州434023 [2]湖北文理学院数学与统计学院,襄阳441053 [3]三峡大学三峡数学研究中心,宜昌443002

出　　处：《中国科学：数学》2022年第11期1267-1282,共16页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11701045);湖北文理学院开放基金(批准号:XK2021023)资助项目。

摘　　要：de Longueville和Gastel(2021)提出了下述非常一般的高阶线性椭圆型方程组:△^(m)u=∑^(m-1)_(t=0)△^(l)(V_(l),du)+∑^(m-2)_(t=0)△^(l)δ(ω_(l)du),并以多调和映照方程为其典型例子.通过给系数函数以最少的光滑性假设和一阶位势的代数反对称性假设,他们成功建立了该方程组的守恒律,从而得到弱解的处处连续性,推广了Rivière(2007)及Lamm和Rivière(2008)关于2阶和4阶方程组的相应理论.最近,Guo和Xiang(2021)证明了上述方程组解的局部Holder连续性,改进了de Longueville和Gastel(2021)的连续性结果.本文使用另一种方法证明对任意的0<α<1,该方程组的弱解都是局部α-Holder连续的,进一步改进了Guo和Xiang(2021)的局部Holder连续性结果.在标准的Dirichlet边界条件下,本文还得到上述方程组弱解直到边界的连续性,推广了Guo和Xiang(2020)关于4阶方程的边界正则性结果.de Longueville and Gastel(2021)proposed the very general higher-order elliptic equations △^(m)u=∑^(m-1)_(t=0)△^(l)(V_(l),du)+∑^(m-2)_(t=0)△^(l)δ(ω_(l)du),of which polyharmonic mapping equations are a typical example.Under the smallest regularity assumption on the coefficient functions and the algebraic antisymmetric assumption on the first-order potential,they succeeded in establishing a conservation law for this system,from which the weak solutions are continuous everywhere.Recently,Guo and Xiang(2021)proved the localα-Holder continuity of weak solutions to the above system for someα∈(0,1),improving the result of de Longueville and Gastel(2021).In this work,we use another method to prove that,for anyα∈(0,1),the weak solution to the above system is locallyα-Holder continuous.This further improves the result of Guo and Xiang(2021).Moreover,under the standard Dirichlet boundary value condition,we obtain the boundary continuity,which extends the boundary regularity theory of Guo and Xiang(2020)in the fourth-order case.

关 键 词：偶数阶椭圆型方程组 守恒律 HOLDER连续性 Lorentz-Sobolev空间 RIESZ位势 

分 类 号：O175[理学—数学]