|
|
|
包含了大量科研人員學習多重積分變分問題和雙曲偏微分方程問題的材料。書中不僅是作者科研成果的總結，更是同時代該領域或者相關領域的專業人士的巨大貢獻。《變分學中的多重積分(英文)》無疑會成為該領域科研人員的標准差參考書。《變分學中的多重積分(英文)》的主要讀者是對數學分析有一定基礎的人員，分析專業的學生將會受益匪淺。
Chapter 1 Introduction
1.1. Introductory remarks
1.2. The plan of the book: notation
1.3. Very brief historical remarks
1.4. The euler equations
1.5. Other classical necessary conditions
1.6. Classical sufficient conditions
1.7. The direct methods
1.8. Lower semicontinuity
1.9. Existence
1.10. The differentiability theory. introduction
1.11. Differentiability; reduction to linear equations
Chapter 2 Semi-classical results
2.1. Introduction
2.2. Elementary properties of harmonic functions
2.3. Weyl’’s lemma
2.4. Poisson’’s integral formula; elementary functions; GREEN’’S functions
2.5. Potentials
2.6. Generalized potential theory; singular integrals
2.7. The calderon-zygmund inequalities
2.8. The maximum principle for a linear elliptic equation of the second order
Chapter 3 The spaces Hmp and Hmp0
3.1. Definitions and first theorems
3.2. General boundary values; the spaces hmp0 (g); weak convergence
3.3. The dirichlet problem
3.4. Boundary values
3.5. Examples; continuity; some sobolev lemmas
3.5. Miscellaneous additional results
3.7. Potentials and quasi-potentials; generalizations
Chapter 4 Existence theorems
4.1. The lower-semicontinuity theorems of SERRIN
4.2. Variational problems with f = f(p); the equations (i. t 0. t 3) with n = i, bi=0, aα = aα(p)
4.3. The borderline cases k = v
4.4. The general quasi-regular integral
Chapter 5 Differentiability of weak solutions
5.1. Introduction
5.2. General theory; V >2
5.3. Extensions Of The DE giorgi-nash-moser results; V >2
5.4. The case V=2
5.5. Lp and SCHAUDER estimates
5.6. The Equation a.▽2u+b.▽u+cu-λu=f
5.7. Analyticity of the solutions of analytic linear equations
5.8. Analyticity of the solutions of analytic, non-linear, elliptic equations
5.9. Properties of the extremals; regular cases
5.10. The extremals in the case 1
