Mappings of Finite Distortion

DMS 0400611PI：JANI ONNINEN密歇根大学有限位移映射摘要几何函数论主要研究解析函数理论向高维的推广。证明了具有与解析函数相同几何性质和函数论性质的映射范畴是有界偏差的映射，也称为拟正则映射，如果是内射的，则称为拟共形映射。这两种映射都在平面上一致地求解椭圆型偏微分方程组。此外，这些映射保持了在欧氏空间的子域上研究函数论和偏微分方程所产生的自然的Sobolev空间。近年来，又出现了另一种著名的映射理论，其思想触及了几何和分析的核心，即非线性弹性理论。自然出现的映射并不总是准正则的，但支配偏微分方程式是相同的。这迫使我们从经典的一致椭圆型偏微分方程组转移到退化的椭圆型方程。然而，通常情况下，需要对椭圆度界限进行一些控制才能得到具体的结果。在一些勒贝格或索博列夫空间中，这些估计通常采用积分估计的形式。这就是有限扭曲映射的理论。在这个建议中，我们主要关注欧几里得n-空间的子集之间的有限扭曲映射。我们还强调了雅可比行列式的基本作用，它已经导致了对有限变形映射的非常富有成效的研究。PI与Haj\L一起研究了n维黎曼流形之间映射的雅可比行列式(黎曼体积形式的拉回)。这项研究使得发现关于这种映射的新现象成为可能。在这个建议中，我们还研究了Sobolev空间上的Hardy-Littlewood极大算子，这是分析中最重要的工具之一。几何函数理论是一个相当成功的理论，有很多不同的应用。例如，非线性弹性理论是以数学和物理中的实际问题为基础的。为了理解某些物理现象，如分叉和相变，有必要研究非线性方程。我们在站点扭曲映射理论中的主要动机是研究退化椭圆型方程，在那里有重要的应用。

DMS 0400611PI: Jani OnninenUniversity of MichiganMappings of Finite DistortionAbstractGeometric function theory is largely concerned with generalizations of the theory of analytic functions to higher dimensions. It turns out that the category of maps with the same geometric and function theoretic properties of analytic functions are the mappings of bounded distortion, also called quasiregular mappings, or, if injective, quasiconformal mappings. Both kind of mappings solve uniformly elliptic partial differential equations in the plane. Moreover, these mappings preserve the natural Sobolev spaces which arise in the study of function theory and partial differential equations on subdomains of Euclidean n-space.In recent years there has been another well-known theory of mappings whose ideas have gotten to the core of geometry and analysis, non-linear elasticity theory. The mappings which naturally occur thereare not always quasiregular, but the governing partialdifferential equations are the same. This forces us to move from theclassical setting of uniformly elliptic partial differential equations to degenerate elliptic equations. Usually, however, some control of the ellipticity bounds will be necessary to achieve concrete results. These often take the form of integral estimates in some Lebesgue or Sobolev spaces. This is the theory of mappings of finite distortion.In this proposal we focus mainly on mappings of finite distortion between subsets of the Euclidean n-space. We also emphasize the fundamental role of the Jacobian determinant, which already has led toa very productive study of mappings of finite distortion. The PIstudies together with Haj\l asz, Iwaniec and Mal\'y, the Jacobiandeterminant (the pullback of the Riemannian volume forms) ofmappings between Riemannian n-manifolds. This study makes itpossible to discover new phenomena about such mappings. Also inthis proposal, we investigate the Hardy-Littlewood maximaloperators on Sobolev spaces, one of the most important tools inanalysis.Geometric function theory has been quite a successful theory, withmany diverse applications. The theory of non-linear elasticity forexample was based on practical problems from mathematics andphysics. It is necessary to study non-linear equations tounderstand certain physical phenomena such as bifurcation andphase transition. Our main motivation in the theory of mappings offinite distortion is to examine degenerate elliptic equationswhere important applications lie.