Mappings of Finite Distortion

DMS 0400611 PI：贾尼Onninen的里根大学映射有限失真AbstractGeometric函数理论在很大程度上是与概括的理论分析功能，以更高的层面。结果表明，具有与解析函数相同的几何和函数论性质的映射范畴是有界畸变映射，也称为拟正则映射，或者，如果是内射，则称为拟共形映射。这两类映射都能解平面上的一致椭圆型偏微分方程。此外，这些映射保持了自然的Sobolev空间，这些自然的Sobolev空间是在欧几里得n-空间的子域上的函数论和偏微分方程的研究中出现的。近年来，出现了另一个著名的映射理论，其思想已经成为几何和分析的核心，即非线性弹性理论。 自然发生在那里的映射并不总是拟正则的，但控制偏微分方程是相同的。这迫使我们从经典的一致椭圆型偏微分方程的设置退化椭圆型方程。 然而，通常需要对椭圆度界限进行一些控制才能获得具体结果。这些通常采取的形式，积分估计在一些勒贝格或索伯列夫空间。这就是有限偏差映射理论，在这个建议中，我们主要关注欧氏n-空间子集之间的有限偏差映射。我们还强调了雅可比行列式的基本作用，它已经导致了一个非常富有成效的研究映射的有限失真。PI与Haj lasz，Iwaniec和Mal 'y一起研究了黎曼n-流形之间映射的Jacobian行列式（黎曼体积形式的拉回）.这一研究使得发现这类映射的新现象成为可能。同样在这个方案中，我们研究了Sobolev空间上的Hardy-Littlewood极大算子，Sobolev空间是分析中最重要的工具之一。例如非线性弹性理论就是建立在数学和物理实际问题的基础上的。研究非线性方程组对于理解某些物理现象如分岔和相变是必要的。我们在有限变形映射理论中的主要动机是研究退化椭圆型方程的重要应用所在。

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.