Notes of Francesco Bigolin’s lecture

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 {f:H^n\rightarrow{\mathbb R}} admit continuous first horizontal derivatives. Then, at points where the horizontal gradient does not vanish, {S=f^{-1}(0)} is the graph of some continuous function {\phi} on a plane open set.

We shall give a characterization of the functions {\phi} which arise in this manner.

1. Conservation laws

A conservation law is a PDE of the form

\displaystyle \begin{array}{rcl} u_t +f(u)_x =g(t,x), \quad u(0,x)=u_0 (x). \end{array}

Existence of solutions is more or less understood: one uses characteristics, i.e. integral lines of the vector field {\partial_t +f'(u)\partial_x}. 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 {C^1} along characteristics.

2. H-regular hypersurfaces in Heisenberg groups

We call level sets of horizontally <{C^1} 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 {\phi} is H-regular iff {\phi} is a uniform limit of smooth solutions of {\nabla^{\psi} \psi=\omega}, {\omega} converging uniformly to {w}.

3. Broad* solutions

We want to define weak solutions of {\nabla^\phi \phi=w}.

Definition 5 A continuous function {\phi} is a broad* solution of {\nabla^\phi \phi=w} if

  1. Characteristics exist.
  2. Along characteristic lines, {\phi} is {C^1}.
  3. Chain rule holds.

Theorem 6 (B, S C) The graph of {\phi} is H-regular iff {\phi} is a broad* solution of {\nabla_j^\phi \phi=w_j} for some continuous functions {w_j}.

Theorem 7 (B, S C) The graph of {\phi} is H-regular iff {\phi} is a distributional solution of {\nabla_j^\phi \phi=w_j} for some continuous functions {w_j}.

Proof relies on Dafermos theorem.

4. Regularity and uniqueness

Theorem 8 (B, S C) If the last source term {w_{n+1}} is Lipschitz, then {\phi} is Lipschitz if {n=1}, {C^1} for {n>1}.

This is sharp when {n=1}. The improved regularity when {n>1} 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 {\nabla_j^\phi \phi=w_j} 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 {1} 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 {\phi} is continuous on {{\mathbb R}^{2n}}. The graph of {\phi} is an intrinsic submanifold iff {\phi} is a distributional solution of {\nabla_j^\phi \phi=w_j} for some bounded functions {w_j}.

5.1. Continuous weak solutions

Can we extend the notion of a broad* solution to bounded sources {w} ?

Example 2 {\phi(y,t)=\sqrt{|t|}}. Changing the source term at only one point ({0}) suffices for the chain rule to be satisfied or not.

Say {\phi} is a Lagrangian (resp. broad) solution of there is at least one (resp. all) characteristic (for a suitable choice of a {\hat{w}=w} a.e.) along which the chain rule holds.

Theorem 13 (Bigolin, Caravenna, Serra Cassano) Assume {\phi} is continuous on {{\mathbb R}^{2n}} and {w_j} are bounded. The following are equivalent:

  1. {\phi} is a distributional solution of {\nabla_j^\phi \phi=w_j}.
  2. {\phi} is a Lagrangian solution of {\nabla_j^\phi \phi=w_j}.
  3. {\phi} is a broad solution of {\nabla_j^\phi \phi=w_j}.

Proof by approximation: approximate characteristic vectorfield with smooth vectorfields first. This produces in the limit the correct a.e. version of the source term.

5.2. Discussion

Le Donne: How much of this extends to general Carnot groups ?

Kozhevnikov: Step {2} makes things easier. I doubt that for higher step groups, characteristic curves (i.e. provide such a good control

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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/
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