摘要
预定平均曲率方程一直以来都是数学中的热点问题,其正则性问题更是得到大量数学家的关注。对线性椭圆型偏微分方程组-Δu(x)=Ω(x)·u(x),x∈B,B是平面上的有界光滑区域,Ω=(Ω_(j)^(i))∈L^(2)(B,M_(m)■R^(2))是以二维向量为分量的m阶矩阵值平方可积函数;u=(u^(1),…,u^(m))∈W^(1,2)(B,R^(m))(m>1)是弱解。其解具有内部H lder连续性,该结果可进一步应用到预定平均曲率方程的正则性问题。利用局部最大值原理和Morrey量关于半径的衰减估计,研究了线性椭圆型偏微分方程组-Δu(x)=Ω(x)·u(x),x∈B在Dirichlet边值下的边界正则性问题:如果给定的边值是连续的,则该方程组的弱解也连续到边界。并给出了单位圆盘上弱解边界正则性的又一个较为直接的证明。
The prescribed mean curvature equation has always been a hot issue in mathematics,and its regularity problem has attracted the attention of many mathematicians.For the linear elliptic partial differential equation-Δu(x)=Ω(x)·u(x),x∈B,B is a bounded smooth domain in the plane,Ω=(Ω_(j)^(i))∈L^(2)(B,M_(m)■R^(2))is a matrix-valued functions with entries in R^(2)and u=(u^(1),…,u^(m))∈W^(1,2()B,R^(m))(m>1)is a weak solution.It is proven that u is locally Holder continuous in B,which can be applied to the regularity problem of the prescribed mean curvature equation.Using the local maximum principle and Morrey decay estimate of radius,the boundary regularity theory of the above equation was studied with given Dirichlet boundary value,i.e.,if the given boundary data is continuous,the weak solution of the equation is also continuous up to the boundary.In the case B is a unit ball,a more direct proof for the boundary regularity of weak solution was also given.
作者
杜厚维
向长林
DU Houwei;XIANG Changlin(School of Information and Mathematics,Yangtze University,Jingzhou 434023,Hubei;Three Gorges Mathematical Research Center,China Three Gorges University,Yichang 443022,Hubei)
出处
《长江大学学报（自然科学版）》
2022年第4期119-126,共8页
Journal of Yangtze University(Natural Science Edition)
基金
国家自然科学基金项目“Kirchhoff方程奇异摄动问题研究”(11701045)。
