Student PDE Seminar (Fall 2024): The Theory of Bounded Variation Functions & Regularity Analysis of the Minimal Surface PDE
Organizers: Kunyi Ma, Chun Szeto, Jingbo Wan
Time: Wednesday 4:30 pm  6:30 pm
Location: Room 507 at Columbia Math Department
This seminar aims to introduce the regularity theory for the minimal surface PDE. We begin with presenting fundamental properties of functions of bounded variation, laying the functional foundation for the notion of minimality, which allows a Dirichlet energy approach. Then, we would in sequence study the existence, Lipschitz regularity, \(C^{1}\) regularity and analyticity of the minimizers. Estimations on the dimension of the singular set will also be discussed. Our attention is then turned to the Dirichlet problem, with the focus on interior and boundary regularity subject to different boundary conditions. If time permits, we will also discuss their application to the Bernstein problem and their recent development.
References: Minimal Surfaces and Functions of Bounded Variation (E. Giusti)
Date 
Speaker 
Title and abstract 

Sep 4th 

Organizational Meeting 
Sep 11th 
Kunyi Ma 
Basic properties of functions of bounded variation Show/hide AbstractsWe 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.

Sep 18th 
Chun Szeto 
Trace of functions of bounded variation Show/hide AbstractsThis 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 welldefined, compatible with Green’s identities, and possesses many other desirable properties.

Sep 25th 
Kunyi Ma 
Regularity of the reduced boundary Show/hide AbstractsThis 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 \(\partial E\).

Oct 2nd 
Chun Szeto 
Preparation for proving De Giorgi Lemma Show/hide AbstractsOne 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.

Oct 9th 
Kunyi Ma 
Analyticity of the reduced boundary Show/hide AbstractsWith 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.

Oct 16th 
Chun Szeto 
Minimal cones are smooth in dimension \(\le 7\) Show/hide AbstractsHaving proved the analyticity of the reduced boundary, we are concerned with the regularity of the actual minimal surface. Since the blowup 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 \(\le 7\).

Oct 23rd 
Kunyi Ma 
Dimension of the singular set Show/hide AbstractsWe 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 \(n8\)  a conclusion that actually includes the result in the previous section.

Oct 30th 
Chun Szeto 
a priori gradient estimate and boundary regularity of solutions to the minimal surface equation Show/hide AbstractsWe 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.

Nov 6th 
Jingbo Wan 
Methods in proving a priori gradient estimate in low and high codimensions Show/hide AbstractsWe present methods in obtaining sharp and nonsharp 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 \(>1\).

Nov 13th 
Jingbo Wan 
Sharp interior gradient estimate for areadecreasing GMCF in higher codimensions Show/hide AbstractsColding 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 areadecreasing, by combining ideas from the integral method and the maximum principle method. Technicalities of his approach will be presented in detail.

Nov 20th 
Chun Szeto 
Bernstein’s ‘Liouville theorem’ on entire minimal graphs Show/hide AbstractsAbstract: 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.
