**1. The isoperimetric problem **

I want to show how PDE results can be used to solve geometric problems.

** 1.1. The isoperimetric inequality **

I will prove the isoperimetric inequality in Carnot groups,

It has lots of applications, see the conference in Paris at the end of september.

** 1.2. Doubling metric spaces **

A metric space is *doubling* if it admits a Borel measure such that for all balls, . On can define a dimension by .

**Exercise**: Prove that this implies for all .

** 1.3. Weak spaces **

The weak (Marcinkiewicz) space, denoted by , is the set of functions such that

It contains (Cavalieri’s principle) strictly. For instance, belongs to but not to . The standard operators of analysis often fail to send to , but send to weak . The loss is not so serious since Marcinkiewicz’ interpolation theorem tells us that interpolating and weak spaces gives spaces.

** 1.4. Fractional integration **

The *Riesz fractional integration operator* is

Theorem 1If , then is bounded , provided . Morover, its norm is at most

In fact, the theorem holds for doubling metric spaces.

Theorem 2 (Nagel-Stein-Wainger 1984)Carnot manifolds are locally doubling.

** 1.5. Fundamental solutions, again **

In this section, we deal with a bracket-generating family of vectorfields , the corresponding sub-Laplacian , and the correponding gradient

Everything is local.

Theorem 3 (NSW, Sanchez-Calle 1984)There exists a fundamental solution of , it satisfiesFurthermore,

An integration by parts gives

Corollary 4 (Citti-Garofalo-Lanconelli)For compactly supported functions ,

Indeed,

Corollary 5For compactly supported functions ,

This easily follows from previous results. Combining the last two corollaries yields

Theorem 6For , for compactly supported functions ,

** 1.6. From weak to strong Sobolev inequality **

Fleming and Richel observed in 1971 that, thanks to coarea formula, the weak Sobolev inequality implies the strong one. This works only for , the geometric case, which is equivalent to the isoperimetric inequality, since one uses this equivalence.

** 1.7. Perimeter **

To give a precise statement of the isoperimetric inequality, we need to define perimeter. The following definition, in case , is due to de Giorgi.

The norm of a vectorfield is if , and if .

The *total variation* of an function is

The space of functions of bounded variation has norm

Note that ( functions with ) is strictly contained in . It does not contain indicators of sets , for instance, although they are often in .

Definition 7

In , for smooth sets, one gets back the surface measure.

** 1.8. Proof of the isoperimetric inequality **

Theorem 8

\proof

Let be a smooth domain. The idea is to apply the weak Sobolev inequality to the indicator . plays the role of on the right hand side. On the left hand side,

hence

To justify replacement of perimeter with , approximate with smooth functions and apply the coarea formula as in next subsection.

** 1.9. Proof of the strong Sobolev inequality **

Theorem 9

\proof

Assume is smooth and compactly supported. By Sard’s theorem, for a.e. , is a smooth manifold. In general, Federer’s coarea formula states that, for a Lipschitz function,

We apply it to and .

Finally, express the -norm of as an integral,

which concludes the proof. We have used the easy fact that, for every nondecreasing function and ,

is a non decreasing function of (differentiate !) and thus nonnegative.