Stable Solutions of Elliptic Partial Differential Equations
Stable Solutions of Elliptic Partial Differential Equations
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DOI：
10.1201/b10802
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发表时间：
2011-03
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通讯作者：
L. Dupaigne
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文献类型：
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作者：
L. Dupaigne
定义稳定性和能量的变化线性化稳定性稳定解的基本性质紧集外的动力稳定性解二义性Gelfand问题动机维N=1维N=2维N=>3总结极值解弱解稳定弱解稳定解的稳定分支正则性稳定解的径向情形返回Gelfand问题维N=1，2，3几何Poincare公式维N=4有界Morse指数奇异稳定解的正则性摄动球平坦区域中的Gelfand问题稳定解的部分正则性高维稳定解的刘维尔定理分类径向稳定整体解分类稳定整体解分类在紧集外稳定的解E de Giorgi猜想猜想的动机陈述维度N=2维N=3进一步阅读稳定解的区域对称性抛物方程以外的稳定解的稳定性与几何其他能量泛函附录A：极大值原理附录B：椭圆算子的正则性附录C：几何工具参考指数
Defining Stability Stability and the variations of energy Linearized stability Elementary properties of stable solutions Dynamical stability Stability outside a compact set Resolving an ambiguity The Gelfand Problem Motivation Dimension N = 1 Dimension N = 2 Dimension N => 3 Summary Extremal Solutions Weak solutions Stable weak solutions The stable branch Regularity Theory of Stable Solutions The radial case Back to the Gelfand problem Dimensions N = 1, 2,3 A geometric Poincare formula Dimension N = 4 Regularity of solutions of bounded Morse index Singular Stable Solutions The Gelfand problem in the perturbed ball Flat domains Partial regularity of stable solutions in higher dimensions Liouville Theorems for Stable Solutions Classifying radial stable entire solutions Classifying stable entire solutions Classifying solutions that are stable outside a compact set A Conjecture of E De Giorgi Statement of the conjecture Motivation for the conjecture Dimension N = 2 Dimension N = 3 Further Readings Stability versus geometry of the domain Symmetry of stable solutions Beyond the stable branch The parabolic equation Other energy functional Appendix A: Maximum Principles Appendix B: Regularity Theory for Elliptic Operators Appendix C: Geometric Tools References Index