带边缘场的分数阶MEMS方程的研究

发布时间：2018-01-29 02:36

本文关键词： 分数阶Laplace算子 MEMS方程边缘场上下解解的存在性　出处：《华东师范大学》2017年硕士论文　论文类型：学位论文

【摘要】：在本文中,首先我们将用延拓的方法重新给出分数阶Laplace算子的定义,接下来会叙述并给出分数阶方程上下解方法的证明,最后将讨论带有边缘场的分数阶MEMS方程解的存在性.在第三章中,我们主要讨论1/2阶Laplace算子,首先将N维空间内的区域Ω延拓为N+1维空间内的柱形区域D,然后通过迹算子定义一个新的空间∧(Ω)并定义空间内函数的调和延拓.通过一系列等价性证明,我们可将带有Dirichlet边界条件的非局部问题转化为带有Neumann边界条件的局部问题,由此得到1/2阶方程弱解的定义.在证明了分数阶方程解的存在性上下解方法之后,我们将用上下解方法重点讨论带有边缘场的分数阶MEMS方程解的存在性,在讨论λ足够小时,我们会转而讨论一个近似方程的解的存在性.
[Abstract]:In this paper, we first give the definition of fractional Laplace operator by extension method, then we describe and prove the method of upper and lower solutions of fractional order equation. Finally, we discuss the existence of solutions for fractional MEMS equations with edge fields. In chapter 3, we mainly discuss the Laplace operators of order 1/2. Firstly, the domain 惟 in N-dimensional space is extended to the cylindrical region D in N-dimensional space. Then we define a new space A (惟) by trace operator and define the harmonic extension of functions in the space, and prove it by a series of equivalence. We can transform a nonlocal problem with Dirichlet boundary condition into a local problem with Neumann boundary condition. The definition of the weak solution of the 1/2 order equation is obtained. After the existence of the upper and lower solutions of the fractional order equation is proved. We will focus on the existence of solutions for fractional MEMS equations with edge fields by using the upper and lower solution method. When we discuss the sufficient number of 位, we will turn to the existence of solutions for an approximate equation.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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