LOCAL BEHAVIOR OF SOLUTIONS OF QUASI-LINEAR EQUATIONS
LOCAL BEHAVIOR OF SOLUTIONS OF QUASI-LINEAR EQUATIONS
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DOI：
10.1007/bf02391014
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发表时间：
1964-01-01
期刊：
影响因子：
3.7
通讯作者：
SERRIN, J
中科院分区：
文献类型：
--
作者：
SERRIN, J
本文讨论了n/> 2个自变量的二阶拟线性偏微分方程解的局部性态。re应特别关注解决方案的先验优化，可移动奇点的性质，以及孤立奇点附近正解的行为。相应的结果在很大程度上对于拉普拉斯方程的情况是众所周知的;粗略地说，我们的工作构成了这些结果对广泛一类非线性方程的扩展。在整个文件中，我们关注的是真实的拟线性方程的一般形式div。，4（x，u，uz）=~（x，u，ux）.（1）这里~是变量x，u，ux的给定向量函数，~是相同变量的给定标量函数，并且u~=（Ou/Ox 1. Ou/Oxn）表示因变量u= u（x）的梯度，其中x=（xl，.，xn）。（1）的结构由函数决定，4（x，u，p）和B（x，u，p）。我们假设它们是对欧氏数空间En的某个连通开集（域）~中的所有点x以及对u和p的所有值定义的。
This paper deals with the local behavior of solutions of quasi-linear partial differential equations of second order in n/> 2 independent variables.~ re shall be concerned specifically with the a priori majorization of solutions, the nature of removable singularities, and the behavior of a positive solution in the neighborhood of an isolated singularity. Corresponding results are for the most part well known for the case of the Laplace equation; roughly speaking, our work constitutes an extension of these results to a wide class of non-linear equations. Throughout the paper we are concerned with real quasi-linear equations of the general form div., 4 (x, u, uz)=~(x, u, ux).(1)Here~ is a given vector function of the variables x, u, ux,~ is a given scalar function of the same variables, and u~=(Ou/Ox 1..... Ou/Oxn) denotes the gradient of the dependent variable u= u (x), where x=(xl,..., xn). The structure of (1) is determined by the functions., 4 (x, u, p) and B (x, u, p). We assume that they are defined for all points x in some connected open set (domain)~ of the Euclidean number space E n, and for all values of u and p. Furthermore, they are to satisfy inequalities of the form