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 1 If , 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 satisfies
An integration by parts gives
Corollary 4 (Citti-Garofalo-Lanconelli) For compactly supported functions ,
Corollary 5 For compactly supported functions ,
This easily follows from previous results. Combining the last two corollaries yields
Theorem 6 For , 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.
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 .
In , for smooth sets, one gets back the surface measure.
1.8. Proof of the isoperimetric inequality
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,
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
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.