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.

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 \(\partial E\). Then, suppose the normals point approximately in the same directions, then the reduced boundary is locally a Lipschitz graph. Moreover, if the normal is continuous, the graph is actually \(C^1\).

We continue with the Lipschitz regularity of the reduced boundary: if the normals point approximately in the same directions, the reduced boundary is locally a Lipschitz graph. Moreover, if the normal is even continuous, the graph is actually \(C^1\).

In this talk, we will explore two distinct methods for proving interior gradient estimates for prescribed mean curvature equations on hypersurface graphs in Euclidean space: the Bombieri-De Giorgi-Miranda (BDM) integral method and Korevaar's differentiation method. The BDM method leverages the mean value inequality for hypersurfaces with bounded mean curvature, while Korevaar’s approach is based on the maximum principle. We will first outline the essential geometric identities and present the logic of the BDM proof, as detailed in Chapter 16 of [Gi-Tr]. The focus will then shift to Korevaar’s maximum principle method, which was later used in Colding-Minicozzi’s interior gradient estimate for mean curvature flow, and the speaker’s generalization to arbitrary co-dimension.

References:

[GiTr] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order.

[Kor] N. Korevaar, An easy proof of the interior gradient bound for solutions to the prescribed mean curvature equation.

In this talk, we delve into the application of Korevaar's maximum principle within the context of the graphical mean curvature flow (GMCF). We focus on presenting the sharp interior gradient estimate for hypersurface GMCF developed by Colding and Minicozzi, achieved using Korevaar’s maximum principle. To conclude, we will comment on what one can do in higher codimension case. 

References:

[CM] Colding-Minicozzi, Sharp estimates for mean curvature flow of graphs

We set up our plan for proving De Giorgio’s lemma. We start by motivating minimality, and introduce simplifications in the case of C^1 bounary. We first sketch and prove its analogue for harmonic functions, then prove a version for C^1 functions close to harmonic functions (main result for the seminar). Finally we generalize to Caccioppoli sets with C^1 boundaries. This concludes the case for \partial E nearly flat and C^1.

With preparation work in the previous session, we can finally prove the De Giorgi Lemma. Under smallness assumption, the normal is provminimizercontinuous 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.