Finite perimeter sets II: A bit about Radon measures

My first post dealt with the definition of sets of finite perimeter. Originally I wanted to discuss minimal surfaces right away, but I think it will be better to take some time to discuss the main properties of sets of finite perimeter, and then maybe in a post or two give I will give an outline of De Giorgi’s proof of the regularity of minimal surfaces. The other thing I talked about (besides the definition itself), was how the good compactness properties of BV sets guaranteed the existence of minima to a variety of variational problems (most notable among them, the minimal surface problem).

Before going further, I want to make a remark about the definition of the perimeter, which I should have done right after I introduced them. Recall that I defined the perimeter of a set E by

\int_\Omega |\nabla \chi_E| = \sup \{ \int_E div gdx : g \in C^1_c(\Omega), ||g||_\infty \leq 1 \} (1)

Why is this a good definition? Well, let us work in 3 dimensions and pick a set E with a nice boundary, e.g. a C^1 boundary that lies a positive distance away from \partial \Omega, then there exists a C^1 vector field g that when restricted to \partial E it agrees with its exterior normal, so that

\int_{\partial E} g \cdot n d\sigma =\int_{\partial E}d\sigma=Area(\partial E)

on the other hand, by Stokes’ theorem

\int_{\partial E} g \cdot n d\sigma =\int_E div gdx

Thus one can see that for a set with smooth boundary the supremum in (1) is achieved and it is equal to the area of its boundary (or its perimeter). Taking this into account, De Giorgi’s definition is quite natural and reasonable (it is based on something we are all comfortable with: integration by parts!).

One further comment, whenever a set of (locally) finite perimeter is also a Borel set of \mathbb{R}^n it is called a Caccioppoli set, we will use this terminology from now on.

A flash course on Radon measures

We just saw that for sets with smooth boundary De Giorgi’s definition of perimeter agrees with the usual one, but now, what can be said about Caccioppoli sets in general?

It turns out that one can define a weak notion of normal vector (and thus a tangent hyperplane) at some points of the boundary of a Caccioppoli set, at least as a (vector) Radon measure. So I am going to use this post to talk a bit about such measures

As further motivation, let me state the following basic fact (first noted I think by De Giorgi):

Let \chi be the characteristic function of a Borel E, then E is a Caccioppoli set if and only its gradient \nabla \chi (in the sense of distributions) is a vector valued Radon measure

Let me give a sketch of the proof (in one direction): take the convolution of \chi with a smooth kernel, thus obtaining a smooth function \chi_\epsilon, if E has finite perimeter then using (1) above it is not hard to show that the gradients \nabla \chi_\epsilon (0<\epsilon < 1) define vector valued Radon measures  that have uniformly bounded variation, thus by a classical compactness theorem of De La Vallée Poussin (or if you are one of those people who like really abstract theorems, by the Banach-Alaouglu-Bourbaki theorem) a subsequence \nabla \chi_{\epsilon_k} converges in the sense of measures to some \mu, which actually agrees with the distribution \nabla \chi.

If you felt a bit dizzy when you read “vector valued Radon measure”, do not worry, it is just the same old thing as a Radon measure: to every Borel set in \mathbb{R}^n it assigns a vector, having all the properties of a usual measure (additivtiy, regularity with respect to Borel sets). Alternatively, you may think of a vector valued Radon measure as a list of n different real-valued Radon measures (i.e. the components of the vector measure). Recall that Radon measures are just measures in \mathbb{R}^n that are locally finite and Borel regular (and a classical theorem of Riesz says that the dual of C^0_c(\mathbb{R}^n) is composed entirely of such measures).

Moreover, one may define the total variation of any vector Radon measure: given such a measure \mu define a new (scalar, positive) measure |\mu| as follows, for any Borel set A let

|\mu|(A) = sup \{ \int g(x) \cdot d\mu(x) : g \in C^0_c(A,\mathbb{R}^n), |g(x)|\leq 1\}

The definition above looks more threatening than it really is!, just recall what the total variation of a real valued measure is and you should see at once the meaning in the definition above. At once one notices that given any vector measure \mu then it is absolutely continuous with respect to its total variation, then, by the Radon-Nykodim theorem, there exists a measurable vector valued function v(x) which is uniquely defined everywhere except for a set of zero |\mu|-measure, and such that for any Borel set A one has

\mu(A) = \int_A v(x)d|\mu|

Next time I will explain how this is used to study sets of finite perimeter.


3 Responses to “Finite perimeter sets II: A bit about Radon measures”

  1. 1 mimi May 29, 2009 at 3:49 am

    I see, thank you and expect the next part.

  2. 2 Gilbert Bernstein June 16, 2009 at 8:07 pm

    Hey Nestor, the “first post” link here is broken.

  3. 3 Nestor June 16, 2009 at 10:11 pm

    Fixed, thanks for pointing that out Gilbert!

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