We motivate the use of BV functions and introduce the fundamental properties of them, including semicontinuity, smooth approximations, isoperimetric inequalities. Making use of these characteristics of BV functions, we construct the trace using some basic measure theory and show its compatibility with Green’s identities.

This session is devoted to the construction of the trace of BV functions and its properties. We first consider BV functions defined on an open cylinder and construct a trace on the cylinder base, and then extend such definition to general Lipschitz domains. Next, we shall show that it is well-defined, compatible with Green’s identities, and possesses many other desirable properties.

This is the first step in approaching analyticity of minimal surfaces. We introduce the notion of reduced boundary \(\partial^*E\) and present some smooth behaviors upon blowing up the neighborhood of points on \(\partial^*E\). Then, we show that \(\partial^*E\) can be covered by countably many \(C^1\) hypersurfaces up to a null set, and that \(\partial^*E\) is dense in \(\partial E\).

One of the most crucial tools in achieving analyticity of minimal surfaces is the De Girogi Lemma, which ensures flattening of surfaces upon blowing up. We first prove the Lemma for \(C^1\) sets, and then approximate general Caccioppoli sets with \(C^1\) sets to finally show that boundary pieces which are flat enough initially shall be blown up to a plane.

With preparation work in the previous section, we can finally prove the De Giorgi Lemma. Under smallness assumption, the normal is proved to be continuous and thus the reduced boundary is \(C^1\)-regular. Noting that the reduced boundary is locally a minimizer to a convex functional, we follow classical PDE theory to show its analyticity.

Having proved the analyticity of the reduced boundary, we are concerned with the regularity of the actual minimal surface. Since the blow-up of minimal surfaces is a minimal cone, the problem reduces to the question of the existence of minimal cones which are not hyperplanes. We first introduce the basics of minimal cones, and show that minimal cones are smooth in dimension \(\le 7\).

We begin estimating the dimension on the singular set - the actual boundary less the reduced boundary - in \(\mathbb R^n\) where \(n\ge 8\). In this section, we shall prove the singular set is of Hausdorff dimension no greater than \(n-8\) - a conclusion that actually includes the result in the previous section.

We first derive a priori gradient estimate in terms of the supremum of the solution by considering smooth solutions to the minimal surface equation. Then, we show the Hölder continuity of the solution up to the boundary, assuming the boundary has strictly negative mean curvature.

We present methods in obtaining sharp and non-sharp a priori gradient estimates of minimal graphs over the past 50 years. a priori gradient estimate takes the form of \(|\nabla u(0)|\le K_1 \exp(K_2 ||u||_\infty ^p)\). It has been proven, via the integral method, that \(p=1\) is sharp for surfaces of dimension 2 (and later \(\ge 2\)) with codimension 1. An easier method - maximum principle method - can lead to \(p=2\). Then, M.T. Wang extended these methods and results to codimensions \(>1\).

Colding and Mincozzi obtained gradient estimates for graph mean curvature flow (GMCF) where the graph is Lipschitz initially, with the exponent \(p=2\) being sharp. The speaker proved analogous results assuming the graph is initially area-decreasing, by combining ideas from the integral method and the maximum principle method. Technicalities of his approach will be presented in detail.

Abstract: In 1915, N.S. Bernstein proved his celebrated ‘Liouville theorem’ for entire minimal graphs in \(\mathbb R^2\): entire minimal surfaces are planes. We apply results presented over the semester to prove this theorem.