Let be a complete Riemannian manifold and be a non empty bounded set. Let be the set of smooth functions such that for every ,
It is easy to see that is essentially self-adjoint on . Its Friedrichs extension, still denoted , is called the Neumann Laplacian on and the semigroup it generates, the Neumann semigroup. If the boundary is smooth, then it is known from the Green’s formula that
where is the normal unit vector. As a consequence, if and only if . However, we stress that no regularity assumption on the boundary is needed to define the Neumann Laplacian and the Neumann semigroup.
Since is compact, the Neumann semigroup is a compact operator and has a discrete spectrum . We get then, the so-called Poincaré inequality on : For every ,
Our goal is in this lecture will be to try to understand how the constant depends on the size of the set . A first step in that direction was made by Poincaré himself in the Euclidean case.
Theorem: If is a bounded open convex set then for a smooth with ,
where is a constant depending on only.
Proof: The argument of Poincare is beautifully simple.
We now have
By a simple change of variables, we see that
Now, we compute
As a consequence we obtain
It is known (Payne-Weinberger) that the optimal constant is .
In this Lecture, we extend the above inequality to the case of Riemannian manifolds with non negative Ricci curvature. The key point is a lower bound on the Neumann heat kernel of . From now on we assume that and consider an open set in that has a smooth and convex boundary in the sense the second fundamental form of is non negative. Due to the convexity of the boundary, all the results we obtained so far may be extended to the Neumann semigroup. In particular, we have the following lower bound on the Neumann heat kernel:
Let be the Neumann heat kernel of . There exists a constant depending only on the dimension of such that for every , ,
As we shall see, this directly implies the following Poincare inequality:
Theorem: For a smooth with ,
where is a constant depending on the dimension of only.
Proof: We denote by the diameter of . From the previous lower bound on the Neumann kernel of , we have
where only depends on . Denote now by the Neumann semigroup. We have for
By integrating over , we find then,
But on the other hand, we have
The proof is complete
In applications, it is often interesting to have a scale invariant Poincare inequality on balls. If the manifold has conjugate points, the geodesic spheres may not be convex and thus the previous argument does not work. However the following result still holds true:
Theorem: There exists a constant depending only on the dimension of such that for every and every smooth with ,
We only sketch the argument. By using the global lower bound
for the heat kernel, it is possible to prove a lower bound for the Neuman heat kernel on the ball : For ,
Arguing as before, we get
and show then that the integral on the left hand side can be taken on by using a Whitney’s type covering argument.