一类双调和映照型偏微分方程组正则性研究  

Research on the Regularity of a Class of Biharmonic Map-Type Partial Differential Equation Systems

作　　者：刘安淇 余婷 向长林 Liu Anqi;Yu Ting;Xiang Changlin(College of Mathematics and Physics,China Three Gorges University,Hubei Yichang 443002;Three Gorges Mathematical Research Center,China Three Gorges University,Hubei Yichang 443002)

机构地区：[1]三峡大学数理学院,湖北宜昌443002 [2]三峡大学三峡数学研究中心,湖北宜昌443002

出　　处：《数学物理学报(A辑)》2025年第2期408-417,共10页Acta Mathematica Scientia

基　　金：国家自然科学基金(12271296);湖北省自然科学基金杰出青年项目(2024AFA061)。

摘　　要：双调和映照是一类重要的几何映照,但是满足的偏微分方程非常复杂,导致其正则性研究很困难.为了研究这一类问题,该文考虑一类双调和映照型四阶椭圆偏微分方程组Δ^(2)u=Q_(1)(x,u,▽u,▽^(2)u)+divQ_(2)(x,u,▽u,▽^(2)u),x∈B_(1),其中B_(1)={x∈R^(n):|x|<1},n≥4,Q_(1),Q_(2)满足关于▽u和▽^(2)u的临界增长条件.则在适当的小性条件假设下,该文证明该方程组的解均具有Holder正则性,从而推广了文献中的相关结果.该结果有助于加深对双调和映照结构的理解与正则性理论的研究.Biharmonic mappings are an important class of geometric mappings,but the partial differential equations that are satisfied are very complex,making their regularity study difficult.In order to study this class of problems,in this note we consider a class of biharmonic map-type fourth order elliptic partial differential equation systemΔ^(2)u=Q_(1)(x,u,▽u,▽^(2)u)+divQ_(2)(x,u,▽u,▽^(2)u),x∈B_(1),in B_(1),where B_(1)={x∈R^(n):|x|<1},with n≥4,and Q_(1),Q_(2)satisfy critical growth conditions with respect to Vu and v?u.Then,under suitable smallness assumption,this note proves that the solutions of this system of equations all have Holder regularity,thus generalising related results in the literature.This result helps to deepen the understanding of the structure of biharmonic mappings and the research on the regularity theory.

关 键 词：双调和映照 临界非线性椭圆方程组 Holder正则性 反向Poincaré不等式 衰减估计 

分 类 号：O175.25[理学—数学]