博碩士論文 88241001 詳細資訊




以作者查詢圖書館館藏 以作者查詢臺灣博碩士 以作者查詢全國書目 勘誤回報 、線上人數:6 、訪客IP:3.226.251.81
姓名 楊世光(Sze-Guang Yang)  查詢紙本館藏   畢業系所 數學系
論文名稱 高階橢圓偏微分方程解的存在性及其行為之研究
(Existence and Behavior for Solutions Of Polyharmonic Equations)
相關論文
★ 薛丁格方程式上直立波解的分類。★ Conformality of Planar Parameterization for Single Boundary Triangulated Surface Mesh
★ 一些線性矩陣方程其平滑及週期的最小 l_2-解之探討★ 關於漢米爾頓矩陣的某些平滑性分解
★ 在N維實數域之雙調和微分方程★ 一維動態系統其週期解之研究
★ 一些延滯方程其週期解之探討★ On the Blow-up solutions of Biharmonic Equation on a ball
★ 雙調和微分方程其正整域解的存在性與不存在性之探討★ 有絲分裂中染色體運動之動態分析
★ 非線性橢圓方程及系統中解的唯一性和結構性之探討★ On the Positive Solution for Grad-Shafranov Equation
★ 關於三物種間之高流動性Lotka-Vollterra競爭擴散系統的波形極限行為★ 非線性橢圓型偏微分方程系統之解結構分析
★ On the study of the Golden-Thompson inequality★ 探討源自於隨機最佳化控制問題之偏微分方程與其相關應用
檔案 [Endnote RIS 格式]    [Bibtex 格式]    [相關文章]   [文章引用]   [完整記錄]   [館藏目錄]   [檢視]  [下載]
  1. 本電子論文使用權限為同意立即開放。
  2. 已達開放權限電子全文僅授權使用者為學術研究之目的,進行個人非營利性質之檢索、閱讀、列印。
  3. 請遵守中華民國著作權法之相關規定,切勿任意重製、散佈、改作、轉貼、播送,以免觸法。

摘要(中) 本論文就研究課題而論,大致分為兩個部分。第一部分主要在闡釋當一個定義在具有奇異點區域上的高階非線性偏微分方程,其非線性項滿足特定前題時,那麼該方程式的解其實就會具有類似於「散度定理」的性質。而這樣的性質有助於了解這類方程式的解在奇異點附近的行為。其次,在第二部分之中,主要是從變分學的觀點來看特定方程式的多重解的存在性問題。其中同時探討解的正負變號數與其所對應的變分泛函值的關係。
以章節來分,前述第一部分的內容歸納在本文的第一章。其實在處理具奇異點的高階偏微分方程時,相對於二階且無奇異點情形而言,基本上有兩個困難點。第一、在二階方程式常用的極大值原(maximum principle)在高階方程式上往往失去預期的效果。第二、「散度定理」在具有奇異點的區域上,其使用受到很大的限制,而這對解的行為的估計,常造成瓶頸。無論如何,作者在這裡試著對該問題的困難點提供一個解決的途徑。除了證明解的「散度定理」的性質外,作者對此亦給了兩個應用。其一是探討Dirichlet 邊界問題解Laplacian 階數與其零根的關係;其次則為對特定方程式解的不存在性的驗證。詳述於該章各節。
在第二章作者用變分方法證明特定方程式多重解的存在性。事實上,該類方程式在二階的情形,其存在性問題早被廣泛探討過。在方法上其實不能算是新的。在這裡作者除了將二階的存在性結果推廣到高階的情形之外,值得一提的是,非負的解的存在性在高階的情形並不容易看得出來,然而,從上述所提的解的正負變號數與其變分泛函值的關係其實正好能夠彌補此一盲點。
另外,第二章所需要的正則性(regularity)定理,獨立論述於本文的第三章。至於比較冗長的證明以及較不重要的細節則收編在附錄之中。
摘要(英) This dissertation is concerned with the existence and behavior for solutions of some polyharmonic equations. It is divided into two parts according to the difference of problems to which the author has devoted. The first part includes the study of a polyharmonic problem in a punctured domain. The second contains subjects about the existence of multiple solutions of some nonlinear higher order equations whose nonlinearities are assumed to be negative near the origin.
In Chapter 1 we prove a divergence-type identity for positive solutions of a certain type of equations in punctured domains. Roughly, the usual divergence theorem is assumed to holds for functions which are defined and differentiable on a smooth domain. When the domain is punctured, the behavior of functions defined there, may be very complicated near singularities even though it is very smooth otherwhere. But if a function satisfies an equation on a domain except at some isolated singularities, its behavior near those singularities will turn to be describable. In practice, considering a positive solution in our case, its behavior near singular points is governed by divergence identities. This property is helpful to the study of some singular problems, especially when the usual maximum principle or integration by part does not work.
Applying this identity the author extends a theorem about counting zeros to its singular case. Further, a onexistence result can also be proved in this manner. The details will appear in Section 1.3.
In Chapter 2 one of the main purposes is to study the existence of multiple solutions for equations whose nonlinearities satisfy some growth conditions. The method which is applied is due to Berestycki and Lions as well as Struwe. The first result concerning existence of infinitely many radial solutions, does not seems to bring more surprise than it does in the second order cases. It is believed that one can also conclude this via the method of ordinary differential equations. The second result of this chapter is to estimate the number of nodal domains of a solution by its energy value. Even in second order problems such a result has been proved quite recently.
Finally, it is worthy to mention that in the study of higher order problems some classical tools, which is used in dealing with second order equations to construct a nonnegative solution, does not work similarly. The existence of nonnegative solution is not obvious in higher order cases. Therefore, the study of nodal structure of a solution seems to suggest a viewpoint to answer this question.
關鍵字(中) ★ 散度定理
★ 橢圓偏微分方程
★ 變分
關鍵字(英) ★ existence of solutions
★ divergence
★ higher order elliptic
★ polyharmonic
論文目次 1 A divergence-type identity in punctured domains 1
§1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§1.2 Behavior of solutions near singularity . . . . . . . . . . . . . . . . . . . . . 3
§1.3 Proofs of Theorem 1.1.2 and Theorem 1.1.3 . . . . . . . . . . . . . . . . . 6
§1.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Multiple solutions of some polyharmonic problems 12
§2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
§2.2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
§2.3 Infinitely many critical points . . . . . . . . . . . . . . . . . . . . . . . . . 22
§2.4 Nodal behavior and nonnegative solutions . . . . . . . . . . . . . . . . . . 30
3 Regularity of weak solutions 33
Appendix A 38
Appendix B 44
Bibliography 48
參考文獻 Berestycki, H. and Lions, P. L.,
Nolinear scalar field equations I and II.
Arch. Rat. Mech. Anal. 82 (1983), 313-345 and 347-375
Struwe, M.,
Multiple solutions of differential equations without the Palais-Smale condition.
Math. Ann. 261 (1982), 399-412.
Bartsch, T., Liu, Z. and Weth, T.,
Sign changing solutions of superlinear Schr{"o}dinger equations.
Comm. Partial Differential Equations} 29 (2004), 25-42.
指導教授 陳建隆(Jann-Long Chern) 審核日期 2007-7-18
推文 facebook   plurk   twitter   funp   google   live   udn   HD   myshare   reddit   netvibes   friend   youpush   delicious   baidu   
網路書籤 Google bookmarks   del.icio.us   hemidemi   myshare   

若有論文相關問題,請聯絡國立中央大學圖書館推廣服務組 TEL:(03)422-7151轉57407,或E-mail聯絡  - 隱私權政策聲明