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椭圆方程解的正则性研究,目录第一章引言11.1椭圆方程介绍及解的正则性的研究进展11.2预备知识11.3主要结论2第二章定理证明过程42.1 定理1.1的证明42.2 定理1.2的证明112.3 定理1.3的证明18结论23致谢24参考文献25摘要 椭圆型方程是数学物理中一类非常重要的方程，在弹性力学、流体力学、几何学、...
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椭圆方程解的正则性研究
目 录
第一章引言………………………………………………………………………………………1
1.1 椭圆方程介绍及解的正则性的研究进展……………………………………………1
1.2 预备知识………………………………………………………………………………1
1.3 主要结论………………………………………………………………………………2
第二章定理证明过程…………………………………………………………………………4
2.1 定理1.1的证明………………………………………………………………………4
2.2 定理1.2的证明………………………………………………………………………11
2.3 定理1.3的证明………………………………………………………………………18
结论……………………………………………………………………………………………23
致谢………………………………………………………………………………………………24
参考文献…………………………………………………………………………………………25
摘要 椭圆型方程是数学物理中一类非常重要的方程，在弹性力学、流体力学、几何学、电磁学和变分法中都有应用。本论文关心椭圆方程的角点正则性，由椭圆方程的凝固系数法知，本质上我们只需要研究椭圆型方程最典型的代表——Laplace方程。就椭圆方程在非光滑区域中的解的正则性问题方向，已有许多研究结果，但是其理论涉及到椭圆方程中许多复杂和深入的理论和技巧。本论文则采用完全初等的技巧，如分离变量法、Sturm-Liuville定理及一些简单的分部积分的技巧，得到角点正则性的相关结论。本文讨论了二维Laplace方程在相同的角状区域中，由于Dirichlet条件，混合边界条件，Neumann边界条件的不同，分别形成问题（D）（M）（N）,其角点正则性也是不同的。问题（D）中，为得到解的最佳正则性，我们首先考虑原问题的一个简化问题，运用极坐标变换和分离变量法得到相应的特征值和特征向量，从而得到解的最佳正则性。然后同样运用极坐标变换和分离变量法，得到解。接下来论证级数一致收敛，级数允许关于两个变量逐项微分两次，证明了级数为定解问题在角形区域上的古典解。最后利用标准的Scaling技巧，得到解的估计，这里。问题（M）类似求解得到估计，这里。问题（N）类似得，这里。
关键词：椭圆方程 正则性 分离变量法 加权空间
The regularity of the solutions of elliptic equations
Abstract Elliptic equation is a very important equation in mathematical physics, which has many applications in elasticity, fluid mechanics , geometry, electromagnetism and calculus of variations. This paper is about the most typical exception elliptic equations - Laplace equation expanded the study of the regularity of its solution. There are many studies on issues about regularity of elliptic equations in non-smooth regions of solutions, but the theory of elliptic equations involving many complex and in-depth theory and techniques, this paper is a fully elementary skills, such as separation variable method,the Sturm-Liuville theorem and some simple tips segment integral to obtain the relevant conclusions about the regularity on the corner. This article discusses the two-dimensional equation in the same angular region, due to the different conditions, mixed boundary conditions, boundary conditions, namely the formation of the problem (D)(M)(N), whose corner regularity is different, but the solution of the problem of the three regular discuss ways are similar. In the problem (D), in order to get the best solution regularity, firstly,to consider the deformation of the original problem, use the polar coordinate transformation and separation of variables corresponding eigenvalues and eigenvectors, to get the best the regularity of solution. Then use the same polar coordinate transformation and separation of variables, get the solution. The next argument is consistent series converges, the series allows itemized differential twice on two variables, the series proved to be the classical solution of the problem solution on the angular region. Finally using Scaling standard techniques to obtain estimates of the solution , here. Problem (M) obtained similar estimates here. Problem (N) obtained similar estimates, here.
Key words：Elliptic equations; Regularity; Separation variable method; The weighted space
目 录
第一章引言………………………………………………………………………………………1
1.1 椭圆方程介绍及解的正则性的研究进展……………………………………………1
1.2 预备知识………………………………………………………………………………1
1.3 主要结论………………………………………………………………………………2
第二章定理证明过程…………………………………………………………………………4
2.1 定理1.1的证明………………………………………………………………………4
2.2 定理1.2的证明………………………………………………………………………11
2.3 定理1.3的证明………………………………………………………………………18
结论……………………………………………………………………………………………23
致谢………………………………………………………………………………………………24
参考文献…………………………………………………………………………………………25
摘要 椭圆型方程是数学物理中一类非常重要的方程，在弹性力学、流体力学、几何学、电磁学和变分法中都有应用。本论文关心椭圆方程的角点正则性，由椭圆方程的凝固系数法知，本质上我们只需要研究椭圆型方程最典型的代表——Laplace方程。就椭圆方程在非光滑区域中的解的正则性问题方向，已有许多研究结果，但是其理论涉及到椭圆方程中许多复杂和深入的理论和技巧。本论文则采用完全初等的技巧，如分离变量法、Sturm-Liuville定理及一些简单的分部积分的技巧，得到角点正则性的相关结论。本文讨论了二维Laplace方程在相同的角状区域中，由于Dirichlet条件，混合边界条件，Neumann边界条件的不同，分别形成问题（D）（M）（N）,其角点正则性也是不同的。问题（D）中，为得到解的最佳正则性，我们首先考虑原问题的一个简化问题，运用极坐标变换和分离变量法得到相应的特征值和特征向量，从而得到解的最佳正则性。然后同样运用极坐标变换和分离变量法，得到解。接下来论证级数一致收敛，级数允许关于两个变量逐项微分两次，证明了级数为定解问题在角形区域上的古典解。最后利用标准的Scaling技巧，得到解的估计，这里。问题（M）类似求解得到估计，这里。问题（N）类似得，这里。
关键词：椭圆方程 正则性 分离变量法 加权空间
The regularity of the solutions of elliptic equations
Abstract Elliptic equation is a very important equation in mathematical physics, which has many applications in elasticity, fluid mechanics , geometry, electromagnetism and calculus of variations. This paper is about the most typical exception elliptic equations - Laplace equation expanded the study of the regularity of its solution. There are many studies on issues about regularity of elliptic equations in non-smooth regions of solutions, but the theory of elliptic equations involving many complex and in-depth theory and techniques, this paper is a fully elementary skills, such as separation variable method,the Sturm-Liuville theorem and some simple tips segment integral to obtain the relevant conclusions about the regularity on the corner. This article discusses the two-dimensional equation in the same angular region, due to the different conditions, mixed boundary conditions, boundary conditions, namely the formation of the problem (D)(M)(N), whose corner regularity is different, but the solution of the problem of the three regular discuss ways are similar. In the problem (D), in order to get the best solution regularity, firstly,to consider the deformation of the original problem, use the polar coordinate transformation and separation of variables corresponding eigenvalues and eigenvectors, to get the best the regularity of solution. Then use the same polar coordinate transformation and separation of variables, get the solution. The next argument is consistent series converges, the series allows itemized differential twice on two variables, the series proved to be the classical solution of the problem solution on the angular region. Finally using Scaling standard techniques to obtain estimates of the solution , here. Problem (M) obtained similar estimates here. Problem (N) obtained similar estimates, here.
Key words：Elliptic equations; Regularity; Separation variable method; The weighted space
