Minimal surfaces I: De Giorgi’s finite perimeter sets

For our first post I am going to talk about one of the many ways to represent a surface. At least since the works of Plateau, mathematicians have been busy studying minimal surfaces and their properties, they appear all over nature: in phase transition models as the interphase between two states, as the shapes of soap bubbles, in fluid mechanics, and they are also of great importance in the study of Riemannian geometry in higher dimensions.

There are several definitions, a very classical one is as follows: a smooth surface $S \subset \mathbb{R}^3$ (with or without boundary) is called minimal if the average of the principal curvatures of $S$ at each point is zero (for those points not lying on the boundary of course). A variational definition is: $S$ is a minimal surface if any deformation of $S$ away from its boundary will increase its area, these properties are actually equivalent!. In this post I will focus only on the second one, first we need to define precisely what we mean by area.

One of the many frameworks for these is the one of functions of bounded variation and sets of finite perimeter, which were introduced first by Ennio DeGiorgi in his work on minimal surfaces. This post is pretty much a quick survey of Chapter 1 of the book Minimal Surfaces and Functions of Bounded Variation, by Enrico Giusti.

Sets of finite perimeter

Let $f \in L^1(\Omega)$, we define its BV-seminorm by

$\int_{\Omega}|\nabla f|=\sup \left \{ \int_\Omega f \mbox{div} g dx \right \}$

where the supremum is taken over all vector fields $g$ that are in $C^\infty_c(\Omega,\mathbb{R}^n)$ and such that $|g(x)|\leq 1$ for all $x \in \Omega$. Then we define the BV-norm by

$||f||_{BV(\Omega)}=||f||_{L^1(\Omega)}+\int_\Omega|\nabla f|$

BV stands for “Bounded variation”, so a function for which the above norm is finite is called a function of bounded variation, and the class of such functions will be denoted by $BV(\Omega)$. Note further that we haven’t given any meaning to the symbol $|\nabla f|$, although this has a precise meaning I won’t discuss yet, but for now we shall use the suggesting notation $\int_\Omega |\nabla f|$ to refer only to the supremum defined above.

The case that we are interested in is when $f$ is the characteristic function of a set $E$, thus we define a measurable set $E \subset \Omega$ to be a set of finite perimeter if the characteristic function of $E$ has bounded variation. Accordingly, we define the perimeter of $E$ as the quantity:

$P(E)=\int_\Omega|\nabla \chi_E|$

Compactness of BV sets

Why bother with this weaker definition of set of finite perimeter? Because we want to find minima to a functional, and as done in calculus one would like to pick a minimizing subsequence and get from it a converging subsequence. In calculus one has the Bolzano-Weierstrass theorem, for sets of finite perimeter one has the following result, which is also valid in Sobolev spaces and is due to Rellich

Assume $\Omega$ has a Lipschitz boundary, and let
$\{f_k\}$ be a sequence of functions in $\Omega$ which are uniformly bounded in the BV-norm. Then there exists a subsequence $\{f_{k'}\}$ and a  BV function $f$ such that $f_{k'} \to f$ in $L^1$

In particular, if one has a sequence of sets $E_k$ with bounded perimeter then one can always pick a subsequence converging to a set of finite perimeter $E$ in the $L^1$ norm. A (very vague) sketch of the of the proof goes like this: one first approximates the sequence of sets by a sequence of set with smooth boundaries,  then the assumption of bounded BV-norm allows one to show that the approximating smooth sequence is equi-continuous and thus by the Ascoli-Arzela theorem one has a converging subsequence.

Finding minimizers to variational problems

As an illustration of these concepts let us consider the following variational problem: Given is a reference set $E_0$ and a domain $\Omega$, then among all sets $E \in BV(\Omega)$ that agree with $E_0$ outside $\Omega$, one would like to find that with the smallest possible perimeter

$P(E)=\int_\Omega |\nabla \chi_E|$

Since $P(E) \geq 0$ for all $E$ one can consider the infimum $p_0$ of all such numbers, then we can pick a sequence $E_k$ of finite perimeter sets all of which agree with $E_0$ outside of $\Omega$ and such that $\lim P(E_k) = p_0$. So in particular the sequence $\{P(E_k)\}$ is bounded and thus by Rellich’s theorem we can pick a subsequence which we will also call $E_k$, and a BV set $E$, such that

$E_k \to E$ in $L^1$

Moreover, since the functional $P(.)$ is lower-semicontinuous in the $L^1$ topology (as can be checked easily from the definition of $P$), we have

$\liminf_{k\to \infty} P(E_k)\geq P(E)$

Therefore the infimum $p_0$ is achieved by the set $E$. Hopefully this shows how simple it is to prove existence of minima to functionals similar to this, e.g. functionals of the form

$F(E)=\int_\Omega|\nabla \chi_E| +\int_{E}f(x)dx$

or isoperimetric problems: given $V>0$, find $E$ such that $P(E)$ is the smallest among all sets with volume $V$.

Once one knows that there is at least one solution, one can start dealing with its geometrical properties (as it is well known for the last problem, the solution is a ball of volume $V$). I will talk about this in my next post.

3 Responses to “Minimal surfaces I: De Giorgi’s finite perimeter sets”

1. 1 Veronica Quitalo February 10, 2009 at 4:44 pm

I liked very much your first entry! It is very clear and a good way to explain the direct methods of CV also!
I am looking for the next!
Good idea!

2. February 13, 2011 at 12:52 pm

hey, is there any embedding theorem like sobolev spaces for BV? because BV looks like H^1.

3. March 15, 2011 at 11:36 am

Yes there is, the embedding theorem is actually equivalent to the isoperimetric inequality (you control the L^1 norm in terms of the BV norm)