【作者】 王晓华；

【导师】 易法槐；

【作者基本信息】 苏州大学 ， 应用数学， 2002， 博士

【摘要】 本文集中于对一个耦合问题的古典局部解和古典整体解的研究。具体内容由以下三章组成： 第一章我们讨论的是具有固定边界的椭圆型方程问题和椭圆型方程组问题(即绕射问题)。首先，在第一节中我们对椭圆型方程问题和椭圆型方程组问题运用差分方法得到时间方向上的h(?)lder模估计，其结果将用在第二章。第二节讨论边界条件含有时间方向上导数的椭圆问题，并且得到相关的正则性估计，其结果在第三章中被引用。 在第二章处理的是一个高维的含一个椭圆方程和两个抛物方程的耦合问题的局部古典解的存在性和唯一性。这里主要用到了第一章相关的结果，并且通过Schauder不动点原理我们得到相应的结论。 第三章是对具有三个椭圆方程组成的高维自由边界问题的整体解的存在唯一性的研究。这里同样用到第一章相关结果和不动点原理。为了得到整体解，我们首先找出一个特殊的解，然后对初-边值条件作小的扰动，最后我们证明了初-边值条件作小的扰动后的问题仍然存在唯一的古典整体解。更多还原

【Abstract】 This dissertation focuses on the researches on the local classical solution and global classical solution of the problem with a coupled system. We organize it along the following lines:In Chapter 1 we discuss the fixed boundary problem with elliptic equation or elliptic equations (diffraction problem) system. First in Section 1 we obtain the estimate of holder norm in time direction about the system of elliptic equation and elliptic equations by difference method and the result will be used in Chapter 2. Then in Section 2 we discuss the elliptic problem which has time derivative on the boundary condition and obtain the corresponding regularity estimate which is quoted in Chapter 3.In Chapter 2 we deal with the existence and uniqueness of the local classical solution about the multidimensional coupled problem with one parabolic equation and two elliptic equations. Here we mainly use the result in Chapter 1 and through Schauder fixed point principle we prove the result.In Chapter 3 we study the global existence and uniqueness of the multidimensional free boundary problem which include three elliptic equations. Here we also used the result in Chapter I and Schauder fixed point principle . In order to find the global solution we first, find a special solution. Then we make some small perturbation on the initial-boundary condition. And in the end we prove that the problem which has been made some perturbation on the initial-boundary condition also has unique global classical solution.更多还原