Rectifiability via curvature and regularity in anisotropic problems

Abstract

Understanding the geometry of rectifiable sets and measures has led to a fascinating interplay of geometry, harmonic analysis, and PDEs. Since Jones' work on the Analysts' Traveling Salesman Problem, tools to quantify the flatness of sets and measures have played a large part in this development. In 1995, Melnikov discovered and algebraic identity relating the Menger curvature to the Cauchy transform in the plane allowing for a substantially streamlined story in $\mathbb{R}^{2}$. It was not until the work of Lerman and Whitehouse in 2009 that any real progress had been made to generalize these discrete curvatures in order to study higher-dimensional uniformly rectifiable sets and measures. Since 2015, Meurer and Kolasinski began developing the framework necessary to use discrete curvatures to study sets that are countably rectifiable. In Chapter 2 we bring this part of the story of discrete curvatures and rectifiability to its natural conclusion by producing multiple classifications of countably rectifiable measures in arbitrary dimension and codimension in terms of discrete measures. Chapter 3 proceeds to study higher-order rectifiability, and in Chapter 4 we produce examples of $1$-dimensional sets in $\mathbb{R}^{2}$ that demonstrate the necessity of using the so-called ``pointwise'' discrete curvatures to study countable rectifiability. Since at least the 1930s with the work of Douglas and Rad{`o}, Plateau's problem has been at the heart of many developments in geometric measure theory and PDEs. After the pioneering work by De Giorgi on the regularity of area minimizing surfaces, studying the regularity of anisotropic energy minimizing surfaces has been an extremely active area of research leading to new interactions between geometry and the fields of PDEs, Calculus of Variations, and Optimal Transport. For anisotropic minimal surfaces all known higher-order regularity results, that is, regularity beyond countable rectifiability, require an assumption on the energy that is similar in effect to the ``ellipticity'' condition of Almgren in so far as the condition forces the PDE that eventually arises to be a linear elliptic PDE. The second part of this thesis discusses regularity results for anisotropic problems in geometry and PDEs which do not benefit from a naturally arising elliptic PDE.{ Chapter 5} begins with results in low-dimensions, including a quick proof of regularity in dimension $2$ for all anisotropic minimal surfaces, and a Bernstein-type theorem. { Chapter 6} studies $\| \cdot \|_{p}$-minimal surfaces by a monotonicity formula, allowing one to consider types of surfaces not covered in Chapter 5. The monotonicity formula ensures that blow-ups of $\| \cdot \|_{p}$-stationary surfaces are stationary cones, motivating an exploration of what anisotropic stationary cones in $\mathbb{R}^{2}$ can look like. {Chapter 7} produces regularity results for a general family of PDEs, including a maximum principle, a Harnack inequality, and a Liouville theorem.