【摘要】：偏微分方程理論廣泛應(yīng)用于一些數(shù)學(xué)分支,物理學(xué),自然科學(xué)等領(lǐng)域中,國內(nèi)外許多學(xué)者對偏微分方程解的性質(zhì)進(jìn)行了研究。Laplace算子作為偏微分方程起源之一,有著重要的應(yīng)用。由對任意整數(shù)階Laplace算子的研究,結(jié)合實(shí)際情況,分?jǐn)?shù)階Laplace算子也開始被討論,分?jǐn)?shù)階指的是微分次數(shù)非整數(shù)。分?jǐn)?shù)階算子不僅在數(shù)學(xué)領(lǐng)域,在其他方面,例如力學(xué)、物理學(xué)、生物醫(yī)學(xué)工程、金融等領(lǐng)域都發(fā)揮著重要的作用。實(shí)際生活中,遇到的不少問題都是非線性的,所以對分?jǐn)?shù)階非線性偏微分方程的研究十分必要。本文主要對一類分?jǐn)?shù)階非線性偏微分方程Dirichlet問題的解的性質(zhì)進(jìn)行了探討。第一部分,給出了偏微分方程的發(fā)展,分?jǐn)?shù)階Laplace算子相關(guān)背景知識及國內(nèi)外研究進(jìn)展,并敘述本文的主要研究內(nèi)容。給出與本文相關(guān)的定義、引理及符號表示。第二部分,給出一類帶有分?jǐn)?shù)階算子的非線性偏微分方程,先利用反證法給出R~n空間中該方程古典解的比較原理,然后討論滿足一定邊值條件的古典解的Lipschitz連續(xù)性。第三部分,給出簡單的最大值定理,反對稱函數(shù)的最大值定理,狹窄區(qū)域定理及無窮遠(yuǎn)處衰減定理,介紹它們及移動平面法在正解的對稱性證明中的應(yīng)用,給出B_1(0)中和R~n空間中正解的徑向?qū)ΨQ性。本文最后將移動平面法應(yīng)用到上半空間R_+~n中,討論在該空間中正解的不存在性。
[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.
【學(xué)位授予單位】：哈爾濱工業(yè)大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175.29

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