A Nash-Moser implicit function theorem with Whitney regularity and applications
A Nash-Moser implicit function theorem with Whitney regularity and applications
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具有惠特尼正则的纳什-莫泽隐函数定理及其应用
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发表时间：
2002
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通讯作者：
J. Vano
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作者：
J. Vano
感谢我的父母，他们教会了我理解的价值;感谢我的妻子，她确保我有规律地吃饭和睡觉;感谢我所有的猫，她们偶尔激发了我对数学的洞察力;感谢所有帮助我实现这一奋进的人。我要特别感谢我在古斯塔夫·阿道夫学院的本科论文导师杰夫·罗索夫博士，我在德克萨斯大学奥斯汀分校的“预备选项“导师汉斯·科赫博士，我所有的研究生伙伴，包括尼古拉·彼得罗夫，以及我的博士学位。论文导师拉斐尔德拉Llave博士也在德克萨斯大学奥斯汀分校。这些人是我的老师、导师和朋友，我对他们的帮助和鼓励深表感谢。我也非常感谢委员会成员的帮助和耐心。最后，在个人方面，我想感谢我的父母安德鲁和萨莉·瓦诺以及我的妻子安蒂·瓦诺，感谢他们多年来的爱和支持。监督人：本文从Nash-Moser隐函数定理出发，建立了隐函数关于参数的Whitney正则性。作为这一结果的应用，我们研究了谐振腔中波的传播问题。使用[Zeh 75]中一般设置的修改，我们考虑func-（为了简洁起见，这里省略了尺度参数）和C U是一个任意参数集（在应用中C通常是康托集）。在对F的适当假设下，我们证明了：给定（x 0，y 0），F（x 0，y 0）= 0，x ∈ C在x 0附近，存在函数g（x），关于x的Whitney正则，满足F（x，g（x））= 0.波在具有（准周期性）运动边界的空腔中的传播问题可以归结为对一族环面映射的研究。由于其极端简并的性质，这个族没有被已知的KAM理论所覆盖。然而，我们的隐函数定理的方法使我们能够克服这些问题，并证明退化KAM理论。我们的方法也可以应用于当前感兴趣的其他问题。
To my parents Who taught me the value of understanding To my wife Who makes sure I eat and sleep somewhat regularly And to all my cats Who occasionally inspire mathematical insight Acknowledgments I would like to acknowledge the support of all who helped make this endeavor possible. I would specifically like to thank my undergraduate thesis adviser Dr. Jeff Rosoff at Gustavus Adolphus College, my " prelim option " adviser Dr. Hans Koch at University of Texas at Austin, all my graduate student cohorts including Nikola Petrov, and my Ph.D. thesis adviser Dr. Rafael de la Llave also at University of Texas at Austin. These individuals have served as teachers, mentors and friends and I am deeply grateful for their help and encouragement. I am also deeply grateful for help and patience of the members of my committee. Finally, on a personal note I would like to thank my parents Andrew and Sally Vano and my wife Jennie Vano for all their love and support over the years. Supervisor: Rafael de la Llave This dissertation establishes the Whitney regularity with respect to parameters of implicit functions obtained from a Nash-Moser implicit function theorem. As an application of this result, we study the problem of wave propagation in resonating cavities. Using a modification of the general setup in [Zeh75], we consider func-(scale parameters are suppressed here for brevity) and C ⊆ U is an arbitrary set of parameters (in applications C is often a Cantor set). Under appropriate hypothesis on F, which are natural extensions of [Zeh75], we show that given (x 0 , y 0) with F(x 0 , y 0) = 0 for x ∈ C near x 0 there exists a function g(x), Whitney regular with respect to x, which satisfies F(x, g(x)) = 0. The problem of wave propagation in a cavity with (quasi-periodically) moving boundary can be reduced to the study of a family of torus maps. Because of their extremely degenerate nature, this family is not covered by known versions of KAM theory. However, our implicit function theorem approach allows us to overcome these problems and prove a degenerate KAM theory. Our approach can also be applied to other problems of current interest.