一类分数阶非线性偏微分方程解的研究
发布时间：2018-09-14 09:21
【摘要】：偏微分方程理论广泛应用于一些数学分支,物理学,自然科学等领域中,国内外许多学者对偏微分方程解的性质进行了研究。Laplace算子作为偏微分方程起源之一,有着重要的应用。由对任意整数阶Laplace算子的研究,结合实际情况,分数阶Laplace算子也开始被讨论,分数阶指的是微分次数非整数。分数阶算子不仅在数学领域,在其他方面,例如力学、物理学、生物医学工程、金融等领域都发挥着重要的作用。实际生活中,遇到的不少问题都是非线性的,所以对分数阶非线性偏微分方程的研究十分必要。本文主要对一类分数阶非线性偏微分方程Dirichlet问题的解的性质进行了探讨。第一部分,给出了偏微分方程的发展,分数阶Laplace算子相关背景知识及国内外研究进展,并叙述本文的主要研究内容。给出与本文相关的定义、引理及符号表示。第二部分,给出一类带有分数阶算子的非线性偏微分方程,先利用反证法给出R~n空间中该方程古典解的比较原理,然后讨论满足一定边值条件的古典解的Lipschitz连续性。第三部分,给出简单的最大值定理,反对称函数的最大值定理,狭窄区域定理及无穷远处衰减定理,介绍它们及移动平面法在正解的对称性证明中的应用,给出B_1(0)中和R~n空间中正解的径向对称性。本文最后将移动平面法应用到上半空间R_+~n中,讨论在该空间中正解的不存在性。
[Abstract]:The theory of partial differential equations is widely used in some fields such as mathematics physics and natural sciences. Many scholars at home and abroad have studied the properties of solutions of partial differential equations. Laplace operator is one of the origins of partial differential equations and has important applications. Based on the study of Laplace operator of arbitrary integer order, the fractional order Laplace operator is also discussed, and the fractional order refers to the non-integer of differential degree. Fractional order operators play an important role not only in the field of mathematics, but also in other fields, such as mechanics, physics, biomedical engineering, finance and so on. In real life, many problems are nonlinear, so it is necessary to study fractional nonlinear partial differential equations. In this paper, the properties of the solutions of Dirichlet problems for a class of fractional nonlinear partial differential equations are discussed. In the first part, the development of partial differential equations, the background knowledge of fractional Laplace operators and the research progress at home and abroad are given, and the main research contents of this paper are described. The definition, Lemma and symbolic representation of this paper are given. In the second part, we give a class of nonlinear partial differential equations with fractional operators. First, we give the comparison principle of the classical solution of the equation in rn space by using the counter-proof method, and then discuss the Lipschitz continuity of the classical solution satisfying certain boundary value conditions. In the third part, a simple maximum value theorem, a maximum value theorem for antisymmetric functions, a narrow region theorem and an infinite attenuation theorem are given. The applications of these theorems and the moving plane method in the proof of symmetry of positive solutions are introduced. The radial symmetry of positive solutions in B _ s _ 1 (0) and R _ n spaces is given. In the end, the moving plane method is applied to the upper half space Rn, and the nonexistence of positive solutions in this space is discussed.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29
本文编号：2242288
[Abstract]:The theory of partial differential equations is widely used in some fields such as mathematics physics and natural sciences. Many scholars at home and abroad have studied the properties of solutions of partial differential equations. Laplace operator is one of the origins of partial differential equations and has important applications. Based on the study of Laplace operator of arbitrary integer order, the fractional order Laplace operator is also discussed, and the fractional order refers to the non-integer of differential degree. Fractional order operators play an important role not only in the field of mathematics, but also in other fields, such as mechanics, physics, biomedical engineering, finance and so on. In real life, many problems are nonlinear, so it is necessary to study fractional nonlinear partial differential equations. In this paper, the properties of the solutions of Dirichlet problems for a class of fractional nonlinear partial differential equations are discussed. In the first part, the development of partial differential equations, the background knowledge of fractional Laplace operators and the research progress at home and abroad are given, and the main research contents of this paper are described. The definition, Lemma and symbolic representation of this paper are given. In the second part, we give a class of nonlinear partial differential equations with fractional operators. First, we give the comparison principle of the classical solution of the equation in rn space by using the counter-proof method, and then discuss the Lipschitz continuity of the classical solution satisfying certain boundary value conditions. In the third part, a simple maximum value theorem, a maximum value theorem for antisymmetric functions, a narrow region theorem and an infinite attenuation theorem are given. The applications of these theorems and the moving plane method in the proof of symmetry of positive solutions are introduced. The radial symmetry of positive solutions in B _ s _ 1 (0) and R _ n spaces is given. In the end, the moving plane method is applied to the upper half space Rn, and the nonexistence of positive solutions in this space is discussed.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29
【参考文献】
中国期刊全文数据库 前5条
1 Yan LI;;NONEXISTENCE OF POSITIVE SOLUTIONS FOR A SEMI-LINEAR EQUATION INVOLVING THE FRACTIONAL LAPLACIAN IN R~N[J];Acta Mathematica Scientia(English Series);2016年03期
2 赵金虎;刘白羽;徐尔;;一类完全非线性椭圆型方程组解的对称性[J];数学物理学报;2015年02期
3 宋利梅;;具p-Laplace算子的分数阶微分方程边值问题的正解[J];高校应用数学学报A辑;2014年04期
4 关丽红;常晶;赵昕;;一类分数阶椭圆型方程解的存在性[J];吉林大学学报(理学版);2014年04期
5 施小丁,顾永耕,陈京;Symmetry and monotonicity for systems of semilinear elliptic equations in a half space[J];Chinese Science Bulletin;1996年21期
，本文编号：2242288
本文链接：https://www.wllwen.com/kejilunwen/yysx/2242288.html
