Geometric Measure Theory, Harmonic Analysis, and Calculus of Variations

几何测度论、调和分析和变分法摘要本文主要讨论几何测度论、调和分析和变分法这三个相互交叉的领域中的三个问题。第一个问题是Besicovitch关于（a.e.）欧氏空间（或更一般的空间）中纯不可求长1-集的较低球面密度。第二个是调和Lip_1-容量问题（Mattila猜想和相关的David-Semmes问题），关于余维1的集合的特征（在可求正性方面），它允许在集合外定义的Lipschitz调和函数扩展到所有的欧几里得空间。这正是解析容量问题的高维模拟。第三个问题是关于问题的规律性极小的放松能源功能（介绍了Bethuel，Brezis，和科龙），这基本上是通常的狄利克雷能源加上成本函数的奇异性。在这三个问题上，PI都做了大量的工作（第三个问题是与R。（一）知识的进步。我们关于第一个问题的计划在最近几个月取得了重大进展，超出了PI的预期，似乎接近完成。我们关于另外两个问题的工作似乎也朝着取得进一步进展的方向发展。第一个问题解决了一个最根本的问题，关于几何性质的纯粹unrectifiable集可以追溯到贝西科维奇的两个经典文件，建立了基础的主题（1928年，1938年）。这些集合是分形的，出现在物理世界的几个上下文中，因此也出现在科学中（例如动力系统，数论等）。我们在这个问题上的工作需要找到新的算法来处理非常一般的闭集，我们希望它能对理解密度和可求正性等其他相关几何问题产生影响，更不用说我们对这些自然集的性质的理解了。我们的第二个问题结合了几何和谐波分析在一个自然的方式，有影响的几个领域的分析，也对物理，其中谐波函数有一个特殊的作用（例如在静电）。我们的第三个问题重叠几何，拓扑，偏微分方程，几何测度论。它涉及其领域的深层问题。它在液晶理论中具有特殊的重要性，因为被最小化的量类似于这种物理系统的能量

Geometric measure theory, harmonic analysis, and calculus of variations AbstractThis proposal is focused on three problems in three somewhat overlapping areas, namely geometric measure theory, harmonic analysis, and the calculus of variations. The first problem is Besicovitch's 1/2- conjecture regarding the sharpest bound for the (a.e.) lower spherical density for purely unrectifiable 1-sets in Euclidean space (or more general spaces as well). The second is the Harmonic Lip_1-Capacity problem (Mattila's conjecture and the related David-Semmes problem), regarding the characterization (in terms of rectifiability) of sets of co-dimension one which allow a Lipschitz harmonic function defined off the set to be extended to all of Euclidean space. This is precisely the higher dimensional analogue of the analytic capacity problem. The third problem is regarding the question of regularity of minimizers of the relaxed energy functional (introduced by Bethuel, Brezis, and Coron) which is basically the usual Dirichlet energy plus a cost function on singularities. On all three problems the PI has done a good deal of work (the third problem is joint with R. Hardt) advancing the state of knowledge. Our program on the first problem has met significant progress in the recent months exceeding the PI's expectations and appears close to completion. Our work on the other two appears to also be heading in direction of further progress as well. The first problem addresses one of the most fundamental questions regarding the geometrical properties of purely unrectifiable sets going back to Besicovitch's two classic papers which established the foundation of the subject (1928, 1938). These sets are fractal-like, appearing in several contexts in the physical world and hence the sciences (e.g. dynamical systems, number theory etc.). Our work on the problem has required finding new algorithms dealing with very general closed sets, and we expect it to have an impact on the understanding of other related geometrical questions of densities and rectifiability not to mention our understanding of the properties of these natural sets.. Our second problem combines geometry and harmonic analysis in a natural way, having impact on several areas of analysis and also to Physics, where harmonic functions have a special role (e.g. in electrostatics). Our third problem overlaps geometry, topology, partial differential equations, and geometric measure theory. It addresses deep questions in its area. It is of special importance in the theory of liquid crystals since the quantities being minimized are similar to the energies of such physical systems