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研究生：	黃秀娟 Huang, Hsiu-chuan
論文名稱：	變分法與橢圓型方程及迭代法與色散型方程 Variational Methods for Elliptic Equations and Iterative Methods for Dirac-Klein-Gordon Equations
指導教授：	吳宗芳 Wu, Tsung-fang 方永富 Fang, Yuang-fu
學位類別：	博士 Doctor
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2008
畢業學年度：	96
語文別：	英文
論文頁數：	187
中文關鍵詞：	固定點定理、零核結構的估計、Nehari 流形、Dirac-Klein-Gordon 方程、雙線性估計、變號一次的解、半線性橢圓型方程
外文關鍵詞：	semilinear elliptic equation, Nehari manifold, 2-nodal solution, bilinear estimate, Dirac-Klein-Gordon equation, null-form estimate, fixed-point theorem
相關次數：	點閱：203 下載：1
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這篇論文分成兩大主題，第一部份研究橢圓型方程在挖洞的有限柱體上的多解性，我們利用能量不太高的 Nehari 流形，當我們把柱體拉長時，可分解成能量分別集中在洞的上下部份，彼此不相交兩子流形；分別建立四種找變號函數最低能量的極值問題，再利用映射的次數 (mapping degree) 及變形定理 (deformation theorem) 得到上述極值問題的最小值，也就是會變號一次的解 (2-nodal solution)。
第二部份研究Dirac-Klein-Gordon 方程解的存在性，我們建立一些估計，包括線性、雙線性及零核結構的估計，利用這些估計及迭代法，得到 Dirac-Klein-Gordon 方程在電荷類空間上解短時間的存在性、唯一性，利用電荷保守律及估計，得出解可在任何時間存在。

In this thesis, we investigate two different types partial differential equations one is an elliptic type and the other is dispersive type (or hyperbolic type). We take different approaches to these two equations one is a variational method and the other is an iterative method.
In part one: we study the multiplicity and the existence of the solutions of the superlinear subcritical elliptic problem. We use the decomposition of the filtration of the Nehari manifold via the variation of domain shape to prove that the semilinear elliptic equation in a large enough finite strip with a hole has at least four 2-nodal solutions which with precisely 2 nodal domains. Futhermore, we can decribe the bump location of these solutions.
In part two: We establish local and global existence results for Dirac-Klein-Gordon equations in one space dimension in the charge class space, employing a null form estimate, a bilinear estimate and a fixed point argument.

PartI: Four 2-Nodal Solutions for a Semilinear Elliptic
       Equation in a Finite Strip with a Hole ........7
Introduction .......................................8
Functional Analysis ...............................14
1 convergences . . . . .  . . . . . . . .......... 15
2 Calculus on Banach Spaces . . . . . . . . . . . .16
3 Revisit Convergences in Lp Spaces . .  . . ..... 21
4 Sobolev Spaces . . . .  . . . . . . . .......... 24
5 Calculus of Variations . . . .   . . . . . ..... 29
The Palais-Smale Theory ...........................35
1 Palais-Smale Sequences . . .  . . . . . ........ 35
2 The Nehari Manifold . . .  . . . ............... 43
3 The Indexes of Domains . . . .  . . . . ........ 48
4 Palais-Smale Conditions . . . .  . . . . ....... 52
The Influence of the Variation of the Domains..... 54
1 The Decomposition of the Filtration of the 
    Nehari  Manifold ................................55
2 Nodal Solutions .   . . . . . . ................ 58
The Existence of 2-Nodal Solutions ................64
1 The Fixed-Point Index and Mapping Degree . . ... 64
2 The Deformation Theorem . . . . .  . . . ....... 69
Bibliography ........................................77
PartII: A Critical Case on the Dirac-Klein-Gordon
        Equations in One Space Dimension ............82
Introduction ......................................83
1 Physical Meaning of (DKD) Systems . .  . . . ... 83
2 Conservation Laws of (DKG) Systems . . ......... 86
3 History of GWP for (DKG) Systems in 1+1 Dimension . 88
Linear Dirac and Wave Equations ....................94
1 Fourier Transform . .  . . . . . . . ............ 95
2 The Linear Dirac Equation in one space dimension . 100
3 The Linear Wave Equation . . . . . . ....... . .. 110
Bilinear Estimates .................................118
1 Null Structure . . . . . . ...................... 118
2 Null Form Estimates . . . . . . . . . ........... 121
3 Two Stepts Bilinear Estimates . . . . . . . ......129
Local and Global Existence .........................140
1 The Relations of the Weighted Norms and Lebesque norms141
2 The Iteration Scheme and Existence .   .... . . . 146
The Recent Development of LWP .....................154
1 Preliminaries . .  . . . . . . . . . . . . ......155
2 Bilinear Estimates . . . .   . . . . . . . ......159
2.1 Proof of (10.19) . . . . . .   . . . . . ......161
2.2 Proof of (10.20) . . . . .  . . . . ...........163
3 Counterexamples . . . . . . . . . ...............165
3.1 Necessity of (10.37) and 
       min{α, β, γ} = α or β  ...................166
3.2 Necessity of (10.38) . . . . . . . ..... . . . 166
3.3 Necessity of (10.40) . . . . .   . . . ....... 167
3.4 Necessity of (10.37) and 
       min{α, β, γ} = γ .  .......................167
3.5 Necessity of (10.39) . . . . .  ........ . . . 168
A Appendix ...........................................170
A.1 X^{s,b}(S_T) →֒ C([0, T];H^s), b > 1  ...........170
A.2 X^{0,σ+ }· X^{0,σ− }→֒ L^2, σ > 1  . . . . ...172
A.3 Product Laws of Wave-Sobolev Norms . ............ 174
B Appendix ...........................................177
B.1 Abstract Well-posedness And Ill-Posedness Theory .177
Bibliography .........................................182
                                    

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Part II
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