## 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 ${w}$.

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