Characterizations of intrinsic regular graphs in Heisenberg groups
joint with L. Caravenna and F. Serra Cassano.
The following theorem is a sub-Riemannian version of the implicit function theorem.
Theorem 1 (Franchi, Serapioni, Serra Cassano) Let admit continuous first horizontal derivatives. Then, at points where the horizontal gradient does not vanish, is the graph of some continuous function on a plane open set.
We shall give a characterization of the functions which arise in this manner.
1. Conservation laws
A conservation law is a PDE of the form
Existence of solutions is more or less understood: one uses characteristics, i.e. integral lines of the vector field . However, solutions are not unique in general.
Theorem 2 (Kruzhkov) Entropy solutions exist and are unique.
Theorem 3 (Dafermos 1977) Characteristics exist and have Lipschitz derivatives. The entropy solution is along characteristics.
2. H-regular hypersurfaces in Heisenberg groups
We call level sets of horizontally < functions (as in Theorem 1) regular intrinsic graphs.
Example 1 (Kirchheim, Serra Cassano) There exists a regular intrinsic graph whose Euclidean Hausdorff dimension is 2.5.
Theorem 4 (Ambrosio, Serra Cassano, Vittone 2006) The graph of is H-regular iff is a uniform limit of smooth solutions of , converging uniformly to .
3. Broad* solutions
We want to define weak solutions of .
Definition 5 A continuous function is a broad* solution of if
- Characteristics exist.
- Along characteristic lines, is .
- Chain rule holds.
Theorem 6 (B, S C) The graph of is H-regular iff is a broad* solution of for some continuous functions .
Theorem 7 (B, S C) The graph of is H-regular iff is a distributional solution of for some continuous functions .
Proof relies on Dafermos theorem.
4. Regularity and uniqueness
Theorem 8 (B, S C) If the last source term is Lipschitz, then is Lipschitz if , for .
This is sharp when . The improved regularity when comes from the fact that integral curves of characteristic vectorfields allow to visit all of the domain (the induced distribution on the hypersurface is Carnot).
Theorem 9 (B, S C) Solutions of Burger’s equations are unique.
5. Intrinsic Lipschitz graphs
Definition 10 We consider cones defined in terms of distance to a horizontal line. Say a submanifolds is an intrinsic graph if at each point, it is outside a cone with vertex at that point.
Theorem 11 (Franchi, Serapioni, Serra Cassano) Codimension intrinsic submanifolds are graphs of a.e. differentiable functions.
Proof relies of De Giorgi’s theory of finite perimeter sets.
Theorem 12 (Bigolin, Caravenna, Serra Cassano) Assume is continuous on . The graph of is an intrinsic submanifold iff is a distributional solution of for some bounded functions .
5.1. Continuous weak solutions
Can we extend the notion of a broad* solution to bounded sources ?
Example 2 . Changing the source term at only one point () suffices for the chain rule to be satisfied or not.
Say is a Lagrangian (resp. broad) solution of there is at least one (resp. all) characteristic (for a suitable choice of a a.e.) along which the chain rule holds.
Theorem 13 (Bigolin, Caravenna, Serra Cassano) Assume is continuous on and are bounded. The following are equivalent:
- is a distributional solution of .
- is a Lagrangian solution of .
- is a broad solution of .
Proof by approximation: approximate characteristic vectorfield with smooth vectorfields first. This produces in the limit the correct a.e. version of the source term.
Le Donne: How much of this extends to general Carnot groups ?
Kozhevnikov: Step makes things easier. I doubt that for higher step groups, characteristic curves (i.e. provide such a good control