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內容簡介

　　 《變分法（第4版）》是《變分法》第四版，主要講述在非綫性偏微分方程和哈密頓係統中的應用，繼第一版齣版十八年再次全新呈現。整《變分法（第4版）》都做瞭大量的修改，僅500多條參考書目就將其價值大大提升。第四版中主要講述變分微積分，增加瞭該領域的新進展。這也是一部變分法學習的教程，特彆講述瞭yamabe流的收斂和脹開現象以及新研究發現的調和映射和麯麵中熱流的嚮後小泡形成。

內頁插圖

目錄

Chapter I.the direct methods in the calculus of variations
1.lower semi-continuity
degenerate elliptic equations
-minimal partitioning hypersurfaces
-minimal hypersurfaces in riemannian manifolds
-a general lower semi-continuity result
2.constraints
semilinear elliptic boundary value problems
-perron's method in a variational guise
-the classical plateau problem
3.compensated compactness
applications in elasticity
-convergence results for nonlinear elliptic equations
-hardy space methods
4.the concentration-compactness principle
existence of extremal functions for sobolev embeddings
5.ekeland's variational principle
existence of minimizers for quasi-convex functionals
6.duality
hamiltonian systems
-periodic solutions of nonlinear wave equations
7.minimization problems depending on parameters
harmonic maps with singularities

Chapter Ⅱ.minimax methods
1.the finite dimensional case
2.the palais-smale condition
3.a general deformation lemma
pseudo-gradient flows on banach spaces
-pseudo-gradient flows on manifolds
4.the minimax principle
closed geodesics on spheres
5.index theory
krasnoselskii genus
-minimax principles for even functional
-applications to semilinear elliptic problems
-general index theories
-ljusternik-schnirelman category
-a geometrical si-index
-multiple periodic orbits of hamiltonian systems
6.the mountain pass lemma and its variants
applications to semilinear elliptic boundary value problems
-the symmetric mountain pass lemma
-application to semilinear equa- tions with symmetry
7.perturbation theory
applications to semilinear elliptic equations
8.linking
applications to semilinear elliptic equations
-applications to hamil- tonian systems
9.parameter dependence
10.critical points of mountain pass type
multiple solutions of coercive elliptic problems
11.non-differentiable fhnctionals
12.ljnsternik-schnirelman theory on convex sets
applications to semilinear elliptic boundary value problems

Chapter Ⅲ.Limit cases of the palais-smale condition
1.pohozaev's non-existence result
2.the brezis-nirenberg result
constrained minimization
-the unconstrained case： local compact- ness
-multiple solutions
3.the effect of topology
a global compactness result, 184 -positive solutions on annular-shaped regions, 190
4.the yamabe problem
the variational approach
-the locally conformally flat case
-the yamabe flow
-the proof of theorem4.9 （following ye [1]）
-convergence of the yamabe flow in the general case
-the compact case ucc
-bubbling： the casu
5.the dirichlet problem for the equation of constant mean curvature
small solutions
-the volume functional
- wente's uniqueness result
-local compactness
-large solutions
6.harmonic maps of riemannian surfaces
the euler-lagrange equations for harmonic maps
-bochner identity
-the homotopy problem and its functional analytic setting
-existence and non-existence results
-the heat flow for harmonic maps
-the global existence result
-the proof of theorem 6.6
-finite-time blow-up
-reverse bubbling and nonuniqueness

appendix a
sobolev spaces
-hslder spaces
-imbedding theorems
-density theorem
-trace and extension theorems
-poincar4 inequality
appendix b
schauder estimates
-lp-theory
-weak solutions
-areg-ularityresult
-maximum principle
-weak maximum principle
-application
appendix c
frechet differentiability
-natural growth conditions
references
index

精彩書摘

Almost twenty years after conception of the first edition， it was a challenge to prepare an updated version of this text on the Calculus of Variations. The field has truely advanced dramatically since that time， to an extent that I find it impossible to give a comprehensive account of all the many important developments that have occurred since the last edition appeared. Fortunately， an excellent overview of the most significant results， with a focus on functional analytic and Morse theoretical aspects of the Calculus of Variations， can be found in the recent survey paper by Ekeland-Ghoussoub [1]. I therefore haveonly added new material directly related to the themes originally covered.
Even with this restriction， a selection had to be made. In view of the fact that flow methods are emerging as the natural tool for studying variational problems in the field of Geometric Analysis， an emphasis was placed on advances in this domain. In particular， the present edition includes the proof for the convergence of the Yamabe flow on an arbitrary closed manifold of dimension 3 m 5 for initial data allowing at most single-point blow-up.Moreover， we give a detailed treatment of the phenomenon of blow-up and discuss the newly discovered results for backward bubbling in the heat flow for harmonic maps of surfaces.
Aside from these more significant additions， a number of smaller changes have been made throughout the text， thereby taking care not to spoil the freshness of the original presentation. References have been updated， whenever possible， and several mistakes that had survived the past revisions have now been eliminated. I would like to thank Silvia Cingolani， Irene Fouseca， Emmanuel Hebey， and Maximilian Schultz for helpful comments in this regard. Moreover，I am indebted to Gilles Angelsberg， Ruben Jakob， Reto Miiller， and Melanie Rupfiin， for carefully proof-reading the new material.
……

前言/序言

變分法（第4版） [Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamilton 下載 mobi epub pdf txt 電子書

評分☆☆☆☆☆

講變分法的，英文版，準備與中文教材結閤起來讀。

評分☆☆☆☆☆

這書我覺得挺滿意的，如果需要可以買。

評分☆☆☆☆☆

同樣的材料可以齣現在不同的標題中，例如希爾伯特空間技術，摩爾斯理論，或者辛幾何。變分一詞用於所有極值泛函問題。微分幾何中的測地綫的研究是很顯然的變分性質的領域。極小麯麵（肥皂泡）上也有很多研究工作，稱為Plateau問題。變分法可能是從Johann Bernoulli(1696)提齣最速麯綫(brachistochrone curve)問題開始齣現的. 它立即引起瞭Jakob Bernoulli和Marquis de l'Hôpital的注意, 但Leonhard Euler首先詳盡的闡述瞭這個問題. 他的貢獻始於1733年, 他的《變分原理》(Elementa Calculi Variationum)寄予瞭這門科學這個名字. Lagrange對這個理論的貢獻非常大. Legendre(1786)確定瞭一種方法, 但在對極大和極小的區彆不完全令人滿意. Isaac Newton和Gottfried Leibniz也是在早期關注這一學科. 對於這兩者的區彆Vincenzo Brunacci(1810), Carl Friedrich Gauss(1829), Simeon Poisson(1831), Mikhail Ostrogradsky(1884), 和Carl Jacobi(1837)都曾做齣過貢獻. Sarrus(1842)的由Cauchy(1844)濃縮和修改的是一個重要的具有一般性的成就. Strauch(1849), Jellett(1850), Otto Hesse(1857), Alfred Clebsch(1858), 和Carll(1885)寫瞭一些其他有價值的論文和研究報告, 但可能那個世紀最重要的成果是Weierstrass所取得的. 他關於這個理論的著名教材是劃時代的, 並且他可能是第一個將變分法置於一個穩固而不容置疑的基礎上的. 1900發錶的第20和23個希爾伯特(Hilbert)促進瞭更深遠的發展.

評分☆☆☆☆☆

變分法很有趣，很有用

評分☆☆☆☆☆

經典好書 好好好

評分☆☆☆☆☆

書還不錯，純英文的，正在努力搞懂中

評分☆☆☆☆☆

Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems 4th ed

評分☆☆☆☆☆

Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems 4th ed

評分☆☆☆☆☆

內容不錯，留著慢慢看

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