作者：（越）阮明智
页数：284
出版社：哈尔滨工业大学出版社
出版日期：2021
ISBN：9787560395142
电子书格式：pdf/epub/txt
内容简介
本书是一部英文版的微分方程方面的专著。本书由3个部分,8个章节,参考文献和索引构成。章通过构造相应齐次方程的非光滑(无界)解的方法研究了退化椭圆算子的亚椭圆性的必要条件，它包含3小节；第二章专门研究退化椭圆方程的亚椭圆性的充分条件,它包含5小节；第三章讨论了非线性椭圆方程解的解析性.本书中出现的不是新的研究结果.但是,得到它们的方法是经过改进的.我们遵循 Friedman提出的证明方法,但在很后阶段应用了新的论点,它包含3小节；第四章专门研究半线性退化椭圆型方程解的无穷可微性和 Gevrey正则性,其主要部分是[136]中考虑的Mizohata算子的幂.它包含3小节；第五章研究了半线性方程解的解析性和 Gevrey正则性,该半线性方程的主要部分是章介绍的 Gilioli Treves的线性算子.它包含2小节；第六章考虑了 Heisenberg群上的Kohn- Laplace型半线性方程组.它包含3小节；在第七章中,我们研究了退化椭圆微分方程的边值问题的非平凡解的存在性.它包含2小节；很后一章涉及上一章所讨论的边值问题的非平凡解的不存在性问题以及边界附近解的正则性问题.它包含2小节。
目录
PART 1. INFINITE SMOOTHNESS OF SOLUTIONS
CHAPTER 1. FUNDAMENTAL SOLUTIONS AND NONBOUNDED
SOLUTIONS OF LINEAR HOMOGENEOUS
DEGENERATE ELLIPTIC EQUATIONS
1.1. Basic definitions, hypergeometric functions
1.2. The case of finite degeneracy
1.3. The case of infinite degeneracy
CHAPTER 2. SMOOTHNESS OF SOLUTIONS OF SEMILINEAR
DEGENERATE ELLIPTIC EQUATIONS
2.1. Auxiliary results
2.2. Semilinear equations of HSrmander’s type
2.3. Higher order semilinear hypoelliptic equations
2.4. Semilinear hypoelliptic equations of the Gilioli – Treves type
2.5. Equations with infinite degeneracy
PART 2. ANALYTICITY OF SOLUTIONS
CHAPTER 3. A NEW APPROACH FOR PROVING GEVREY REGULARITY
OF SOLUTIONS OF ELLIPTIC EQUATIONS
3.1. Friedman’s Lemmas
3.2. Weighted Holder spaces
3.3. Douglis-Nirenberg estimates
3.4. New argument in the proof of analyticity and Gevrey
regularity of solutions
CHAPTER 4. ANALYTICITY AND GEVREY REGULARITY OF SOLUTIONS
OF SEMILINEAR EQUATIONS OF MIZOHATA TYPE
4.1. Uniform fundamental solutions and representation formula
4.2. Infinite smoothness of solutions
4.3. Analyticity and Gevrey regularity of solutions
CHAPTER 5. ANALYTICITY AND GEVREY REGULARITY OF
SOLUTIONS OF SEMILINEAR EQUATIONS
OF THE GILIOLI – TREVES TYPE
5.1. A Model of Grushin’s type equations
5.2. Equation of tile Gilioli – Treves type
CHAPTER 6. ANALYTICITY AND GEVREY REGULARITY OF SOLUTIONS
OF SEMILINEAR KOHN-LAPLACIAN ON THE
HEISENBERG GROUP
6.1. Semilinear Kohn- Laplacian.
6.2. Fundamental solution of the Kohn – Laplacian
6.3. Oevrey regularityof solutions
PART 3. GLOBAL PROPERTIES OF SOLUTIONS
CHAPTER 7. EXISTENCE OF NONTRIVIAL SOLUTIONS TO BOUNDARY
VALUE PROBLEMS FOR. SEMILINEAR DEGENERATE
ELLIPTIC EQUATIONS
7.1. Embedding theorems of Sobolev’s type
7.2. Theorems on existence of solutions
CHAPTER 8. NON EXISTENCE OF NONTRIVIAL SOLUTIONS TO
BOUNDARY VALUE PROBLEMS AND REGULARITY
UP TO THE BOUNDARY
8.1 Generalized Pohozaev’s identity and theorems
on nonexistence of solutions
8.2. Regularity of solutions to boundary value problems
up to the boundary
BIBLIOGRAPHY
LIST OF NOTATIONS AND ABBREVIATIONS
编辑手记
