Notes of Andrea Bonfiglioli’s lecture

Maximal principles and Harnack inequalities for PDO’s in divergence form

1. Motivation

CR geometry (sub-Laplacians), stochastic PDE’s.

2. Introduction

2.1. Standing assumptions

  1. Total nondegeneracy.
  2. Smooth hypoellipticity.

Sometimes, we require that {L-\epsilon} is hypoelliptic as well. Or even existence of a global, positive fundamental solution (unfortunately, this is known only for special classes, like homogeneous operators on nilpotent groups, Nagel Stein 1990.

2.2. Earlier work

Theorem 1 (Bony 1969) Maximum principle and Harnack inequality for a class of degenerate elliptic operators (sums of squares of Hörmander vectorfields).

Bony uses a Hopf-type lemma and maximum propagation to get maximum principle. Then is used to get Harnack inequality.

Huge litterature in the 1980’s : Fabes, Jerison, Serapioni Franchi, Lanconelli, Chanillo, Wheeden, Sanchez-Calle. All assume hypo-ellipticity.

Nowadays, the framework has been enlarged : doubling metric spaces satisfying Poincaré inequality.

2.3. Examples

Sub-Laplacians.

Fedii 1971 : sum of squares of non Hörmander vectorfields (a constant basis, whose vectors are multiplied with flat functions). This can be hypo-elliptic but not sub-elliptic (Fefferman-Phong 1981).

3. Results

3.1. Hopf Lemma

Let {F} be the set where {u} achieves its maximum. Let {y\in F} and {\nu} be orthogonal to {F} (meaning that the interior of some ball centered at {y+\epsilon \nu} and passing through {y} is disjoint from {F}). Then…

3.2. From Hopf lemma to maximum principle

Theorem 2 Non total degeneracy and hypoellipticity imply strong maximum principle.

\proof

Principal vectorfields {X} have to be tangent to {F}. This implies {F} has to be invariant under {X}. How can one build them ? Use columns of the matrix defining the operator. Note that Hörmander’s condition need not hold for these vectorfields.

Amano 1979 observed that non total degeneracy and hypoellipticity imply connectivity of {{\mathbb R}^n} with respect to such vectorfields plus a drift vectorfields. Thus maximum principle follows.

3.3. Harnack inequality

Theorem 3 Non total degeneracy and hypoellipticity of {L-\epsilon} imply strong a Harnack inequality where, however, the constant depends on the shape of the domain and of the considered subdomain.

\proof

Follows Bony’s approach. Solve the Dirichlet for {L-\epsilon} (based on maximal principle). Prove existence of the Green kernel of {L-\epsilon}. Get a weak Harnack inequality. Use potential theory to get Harnack from weak Harnack.

By maximum principle, the Green kernel {k_\epsilon} of {L-\epsilon} is positive. Then for {u\geq 0} such that {Lu=0}, Bony proves that

\displaystyle  \begin{array}{rcl}  u(x)\geq \epsilon\int u(y)k_\epsilon(x,y)\,d\nu(y). \end{array}

Since {k_\epsilon>0}, this allows to locally bound {u(x)} from below with the {L^1_{loc}}-norm of {u}. On the space of {L}-harmonic functions, the {L^1_{loc}} and {C^\infty} topologies coincide. This way, we get the weak Harnack inequality

\displaystyle  \begin{array}{rcl}  \sup_K u \leq C(x_0)u(x_0). \end{array}

3.4. Role of potential theory

Theorem 4 (Mokobodzki-Brelot 1964) Very abstract setting. Assume weak Harnack inequality holds and that Dirichlet problem on small open sets has a solution, then strong Harnack inequality holds.

4. More on potential theory

How can one characterize {L}-subharmonic functions ?

Use balls defined by Green’s function ({\Gamma}-balls) to define inradius of a domain. Then a representation formula follows, based on the divergence theorem, with kernel expressible in terms of Green’s function. A mean value formula holds for {L}-harmonic functions on {\Gamma}-balls, with a correction term. The corresponding inequality characterizes sub-harmonicity. So does monotonicity of mean values on {\Gamma}-balls.

Advertisements

About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
This entry was posted in seminar and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s