Archive for the 'Geometric measure theory' Category

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.

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.