Maximal principles and Harnack inequalities for PDO’s in divergence form
CR geometry (sub-Laplacians), stochastic PDE’s.
2.1. Standing assumptions
- Total nondegeneracy.
- Smooth hypoellipticity.
Sometimes, we require that 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.
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).
3.1. Hopf Lemma
Let be the set where achieves its maximum. Let and be orthogonal to (meaning that the interior of some ball centered at and passing through is disjoint from ). Then…
3.2. From Hopf lemma to maximum principle
Theorem 2 Non total degeneracy and hypoellipticity imply strong maximum principle.
Principal vectorfields have to be tangent to . This implies has to be invariant under . 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 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 imply strong a Harnack inequality where, however, the constant depends on the shape of the domain and of the considered subdomain.
Follows Bony’s approach. Solve the Dirichlet for (based on maximal principle). Prove existence of the Green kernel of . Get a weak Harnack inequality. Use potential theory to get Harnack from weak Harnack.
By maximum principle, the Green kernel of is positive. Then for such that , Bony proves that
Since , this allows to locally bound from below with the -norm of . On the space of -harmonic functions, the and topologies coincide. This way, we get the weak Harnack inequality
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 -subharmonic functions ?
Use balls defined by Green’s function (-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 -harmonic functions on -balls, with a correction term. The corresponding inequality characterizes sub-harmonicity. So does monotonicity of mean values on -balls.