静电场电位边值问题唯一性定理的补充与完整证明

SUPPLEMENT AND COMPLETE PROOF OF THE ELECTROSTATIC BOUNDARY VALUE PROBLEM AND THE UNIQUENESS OF SOLUTIONS

本文对静电场电位边值问题与解的唯一性定理作了补充与完整的证明.首先将区域边界与衔接边界从通常的混称中区分开来,确认了静电场边值问题中第三类边界条件应有的形式,在解的唯一性定理中增加了衔接条件和无限远边界条件,并根据数学表达式的形式重新归类。然后在区域边界条件、无限远边界条件和衔接条件下电位解的唯一性的证明中,讨论了第一、第三类边值问题电位解的唯一性与全二类边界条件下电位存在常数差的问题,解除了第三类边界条件系数为正的限制,说明了整个求解空间为无限大时适用的边值问题。最后通过例题说明了区域、无限远和衔接3种边界条件在解题中的应用。补充后的定理可以更好地作为解题和后续学习的依据和基础。

The electrostatic boundary value problem and the uniqueness of solutions are supplemented and proved in this paper.At first,the region condition and the convergence boundary are distinguished from the usual mixed singularity.The form of Robin Problem in electrostatic field boundary value problem is confirmed.The convergence condition and the infinite boundary condition are added to the uniqueness theorem of solutions.These boundary conditions are re-classified according to the form of mathematical expressions.Then in the proof of the uniqueness of the potential solutions under boundary conditions,infinite boundary conditions and convergence conditions,the problem of the coefficient of the third kind of boundary condition and the applicative boundary value problem with infinite space are solved.We also demonstrate the uniqueness of potential solutions for Dirichlet and Robin Problem and constant differences in the potential of Neumann Problem.Finally,the application of region,infinity and convergence boundary conditions in problems solving is illustrated by an example.The supplemented theorem can be better used as the basis for solving problems and follow-up learning.