# Computations of Gromov–Witten invariants of toric varieties

Giosuè Muratore  CMAFcIO, Faculdade de Ciências da ULisboa, Campo Grande
1749-016 Lisboa, Portugal<EMAIL_ADDRESS>https://sites.google.com/view/giosue-muratore

(Date: August 28, 2024)

###### Abstract.

We present the Julia package `ToricAtiyahBott.jl`, providing an easy way to
perform the Atiyah–Bott formula on the moduli space of genus $0$ stable maps
$\overline{M}_{0,m}(X,\beta)$ where $X$ is any smooth projective toric
variety, and $\beta$ is any effective $1$-cycle. The list of the supported
cohomological cycles contains the most common ones, and it is extensible. We
provide a detailed explanation of the algorithm together with many examples
and applications. The toric variety $X$, as well as the cohomology class
$\beta$, must be defined using the package `Oscar.jl`.

###### Key words and phrases:

Toric varieties, Julia, Gromov–Witten, scientific computing, stable maps,
rational curve

###### 2020 Mathematics Subject Classification:

14-04, 14M25, 14N35, 14N10

##  1\. Introduction

The Gromov–Witten invariants of a variety $X$ are the degrees of certain
cycles in the moduli space of stable maps $\overline{M}_{g,m}(X,\beta)$, where
$\beta\in H_{2}(X,\mathbb{Z})$ is an effective class. They are widely used to
solve a large class of enumerative problems. For this reason, there is a great
interest in finding explicit algorithms to compute Gromov–Witten invariants.
Some algorithms are very efficient and can be applied to a wide range of cases
(see, for example, the recent articles [ABPZ23, Hu22a, Hu22b]).

In this article, we present an implementation of the Atiyah–Bott formula to
compute any Gromov–Witten invariant where $g=0$ and $X$ is a smooth projective
toric variety. Our main references are [Spi99, Liu13].

In Section 2 we recall the main properties of toric varieties and
Gromov–Witten invariants. In Section 3 we describe the algorithm together with
the equivariant classes of the most common cohomological cycles. In the last
section, we present some explicit computations and show some possible
applications.

Our package is written in Julia ([BEKS17]) and is freely available at:

https://github.com/mgemath/ToricAtiyahBott.jl

It requires `Oscar.jl` ([OSC23, DEF+24]), in particular its toric geometry
part (see [BK23]). We think that our package could be a useful tool to the
mathematical community.

Acknowledgments. The author learned toric varieties in the school BAGS in
Nancy in 2018. He is truly grateful to the organizers, in particular to
Gianluca Pacienza. Moreover, he thanks Csaba Schneider and Xiaowen Hu for
mathematical support. Finally, he thanks Martin Bies, Janko Boehm, Lars
Kastner, and Yue Ren for their help with `Oscar.jl`. This project is supported
by FCT - Fundação para a Ciência e a Tecnologia, under the project:
UIDP/04561/2020. The author is a member of GNSAGA (INdAM).

##  2\. Gromov–Witten invariants on toric varieties

In this section we briefly review toric varieties and Gromov–Witten
invariants, in order to fix notation.

###  2.1. Toric varieties

A toric variety $X$ is an algebraic variety containing a torus
$T=(\mathbb{C}^{*})^{n}$ as a dense open subset, in such a way that the action
of $T$ on itself extends to the whole $X$. The classification of toric
varieties of dimension $n$ is equivalent to the classification of fans of
cones in $\mathbb{Z}^{n}$. In particular, for each toric variety there is a
fan $\Sigma$, uniquely defined modulo isomorphism. We refer to [CLS11] for a
complete account on the correspondence between fans $\Sigma$ and toric
varieties $X=X_{\Sigma}$. In the following examples, we denote by
$\\{e_{1},\ldots,e_{n}\\}$ the standard generators of $\mathbb{Z}^{n}$.

######  Example 2.1.

The projective space $\mathbb{P}^{n}$ is a toric variety with maximal cones
$\\{\sigma_{0},\ldots,\sigma_{n}\\}$ where $\sigma_{i}$ is the cone generated
by the rays $\\{e_{0},\ldots,\hat{e}_{i},\ldots,e_{n}\\}$ and
$e_{0}:=\sum_{i=1}^{n}-e_{i}$.

Some natural constructions (direct product, blow-up along a toric invariant
subvariety,…) are stable in the toric context. If $\mathcal{E}$ is a vector
bundle of a smooth complete toric variety, then the projectivization
$\mathbb{P}(\mathcal{E})$ is toric if and only if $\mathcal{E}$ is the direct
sum of line bundles [CP16, Proposition A.2]. The general strategy to find the
fan of $\mathbb{P}(\mathcal{E})$ is described in [Oda78, Chapter 1 Section 7].

######  Example 2.2.

Any Hirzebruch surface
$\mathbb{F}_{m}=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(m))$
is toric. The maximal cones of its fan are

$\\{\\{e_{1},e_{2}\\},\\{e_{1},-e_{2}\\},\\{e_{2},-e_{1}-me_{2}\\},\\{-e_{2},-e_{1}-me_{2}\\}\\}.$

Analogously,
$\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(m))$
is toric. The maximal cones of its fan are

$\bigcup_{\mathfrak{e}=\pm
e_{3}}\\{\\{e_{1},e_{2},\mathfrak{e}\\},\\{e_{2},\mathfrak{e},-e_{1}-e_{2}-me_{3}\\},\\{e_{1},\mathfrak{e},-e_{1}-e_{2}-me_{3}\\}\\}.$

######  Notation 2.3.

From now on, for any toric variety $X$ relative to a fan $\Sigma$, we use the
following notation.

  * •

$\Sigma^{(k)}$ is the subset of $k$-dimensional cones of $\Sigma$. So,
$\Sigma^{(n)}$ are the maximal cones and they are usually denoted by $\sigma$.

  * •

For each $\sigma\in\Sigma^{(n)}$, we denote by $\sigma^{*}$ the set of all
$\sigma^{\prime}\in\Sigma^{(n)}$ such that
$\sigma\cap\sigma^{\prime}\in\Sigma^{(n-1)}$.

  * •

$\Sigma^{(1)}=\\{\rho_{1},\ldots,\rho_{r}\\}$ are the $1$-dimensional cones
(rays) of $\Sigma$.

  * •

For each cone $\tau\in\Sigma^{(k)}$, $V(\tau)$ is the $T$-invariant subvariety
of dimension $(n-k)$ defined by $\tau$.

###  2.2. Moduli space of maps

Let us briefly recall the main properties of the moduli space of stable maps.
The toric variety $X$ is smooth and projective.

  * •

$\overline{M}_{0,m}(X,\beta)$ is the coarse moduli space of genus $0$ stable
maps to $X$ of class $\beta$ with $m$ marked points.

  * •

$\mathrm{ev}_{j}\colon\overline{M}_{0,m}(X,\beta)\rightarrow X$ is the
evaluation map at the $j^{\mathrm{th}}$ marked point.

  * •

$\delta\colon\overline{M}_{0,m+1}(X,\beta)\rightarrow\overline{M}_{0,m}(X,\beta)$
is the forgetful map of the last point.

Usually, the moduli points are denoted by $(C,f,p_{1},\ldots,p_{m})$ where
$f\colon C\rightarrow X$ is the morphism and $p_{1},\ldots,p_{m}\in C$ are the
marked points. We denote by $\mathbb{L}_{i}$ the line bundle whose fiber at
$(C,f,p_{1},\ldots,p_{m})$ is $\Omega_{C,p_{i}}$. Finally, $\psi_{i}$ is the
first Chern class of $\mathbb{L}_{i}$. The virtual dimension of
$\overline{M}_{0,m}(X,\beta)$ is

(2.1)
$\mathrm{virdim}(\overline{M}_{0,m}(X,\beta))=X-\deg(c_{1}(\Omega_{X})\cdot\beta)+m-3.$

Given a vector bundle $\mathcal{E}$ or rank $r$, cohomology classes
$\gamma_{1},\ldots,\gamma_{m}\in H^{*}(X,\mathbb{C})$, and non negative
integers $a_{1},\ldots,a_{m}$, we define a Gromov–Witten invariant the
following degree of a cohomology class of maximal codimension:

(2.2)
$\int_{\overline{M}_{0,m}(X,\beta)}\mathrm{ev}_{1}^{*}(\gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\gamma_{m})\cdot\psi_{1}^{a_{1}}\cdots\psi_{m}^{a_{m}}\cdot
c_{r}(\mathcal{E}).$

The action of $T$ on $X$ lifts to $\overline{M}_{0,m}(X,\beta)$ by
composition. The fixed point locus of this new action is the disjoint union of
subvarieties of $\overline{M}_{0,m}(X,\beta)$ parameterized by all possible
decorated graphs of $\overline{M}_{0,m}(X,\beta)$.

######  Definition 2.4.

A decorated graph of $\overline{M}_{0,m}(X,\beta)$ is the isomorphism class of
the tuple $\Gamma=(g,\mathbf{c},\mathbf{w},\nu)$, such that

  1. (1)

$g$ is a tree; that is $g$ is a simple undirected, connected graph without
cycles. We denote by $V_{g}$ and $E_{g}$ the sets of vertices and edges of
$g$.

  2. (2)

The vertices of $g$ are colored by the maximal cones of $\Sigma$; the coloring
is given by a map $\mathbf{c}\colon V_{g}\rightarrow\Sigma^{(n)}$ such that:
if $v$ and $v^{\prime}$ are two vertices in the same edge, then
${\mathbf{c}(v)}\cap{\mathbf{c}(v^{\prime})}\in\Sigma^{(n-1)}$.

  3. (3)

Each edge $e=(v,v^{\prime})$ is weighted by a map $\mathbf{w}\colon
E_{g}\rightarrow\mathbb{Z}$, such that $\mathbf{w}(e)>0$ and $\sum_{e\in
E_{g}}\mathbf{w}(e)\cdot[V({\mathbf{c}(v)}\cap{\mathbf{c}(v^{\prime})})]=\beta$.

  4. (4)

The marks of the vertices of $g$ are given by a map $\nu\colon A\rightarrow
V_{g}$, where $A=\\{1,\ldots,m\\}$ if $m>0$, and $A=\emptyset$ if $m=0$.

An automorphism of $\Gamma$ is an automorphism of $g$ compatibles with
$\mathbf{c},\mathbf{w},\nu$. We denote by $a_{\Gamma}$ the integer

$a_{\Gamma}:=|\mathrm{Aut}(\Gamma)|\prod_{e\in E_{g}}\mathbf{w}(e),$

and by $F_{\Gamma}$ the fixed point locus relative to $\Gamma$.

Following [AB84], given a cohomology class $\alpha\in
H^{*}(\overline{M}_{0,m}(X,\beta),\mathbb{C})$, we denote by
$\alpha^{T}(\Gamma)$ the $T$-equivariant cohomology class induced by $\alpha$.
Finally, we have the integration formula.

######  Theorem 2.5.

Let $P$ be a symmetric polynomial in Chern classes of equivariant vector
bundle of $\overline{M}_{0,m}(X,\beta)$. Then

(2.3)
$\int_{\overline{M}_{0,m}(X,\beta)}P=\sum_{\Gamma}\frac{1}{a_{\Gamma}}\frac{P^{T}(\Gamma)}{c_{\mathrm{top}}^{T}(N_{\Gamma})(\Gamma)},$

where $N_{\Gamma}$ is the normal tangent bundle of $F_{\Gamma}$.

See [Kon95, GP99, Spi99, Liu13]. Finally, we use Gromov–Witten invariants to
define the small quantum ring of $X$, denoted by $QH^{s}(X)$. Let
$\\{\theta_{1},\ldots,\theta_{t}\\}$ be a basis of the vector space
$H^{*}(X,\mathbb{Q})$ and let $\\{\vartheta_{1},\ldots,\vartheta_{t}\\}$ be
its dual basis with respect to the cup product. Given any two classes
$\alpha_{1},\alpha_{2}\in H^{*}(X,\mathbb{Z})$ we define the product

(2.4)
$\alpha_{1}\star\alpha_{2}=\alpha_{1}\cdot\alpha_{2}+\sum_{\beta}q^{\beta}\sum_{i=1}^{t}\vartheta_{i}\int_{\overline{M}_{0,3}(X,\beta)}\mathrm{ev}^{*}_{1}(\alpha_{1})\cdot\mathrm{ev}^{*}_{2}(\alpha_{2})\cdot\mathrm{ev}^{*}_{3}(\theta_{i}),$

and extend by linearity to the whole $QH^{s}(X)$.

##  3\. The algorithm

In this section we will describe the main features of our implementation of
the equation (2.3). The core part of the algorithm consists of four nested
`for` cycles that generates the decorated graphs
$\Gamma=(g,\mathbf{c},\mathbf{w},\nu)$. We used the algorithm of [WROM86] in
order to generate all trees. So that, the first `for` cycle runs among all
trees with at most $p$ edges. The number $p$ corresponds to the maximal number
of $T$-invariant curves $\\{C_{i}\\}_{i=1}^{p}$ such that

$\beta=[C_{1}]+[C_{2}]+\cdots+[C_{p}].$

Using the fact that the Mori cone is generated by $T$-invariant curves and it
is dual to the cone of nef divisors [CLS11, §6.3], we have that
$p=\beta\cdot(M_{1}+\cdots+M_{k})$ where $M_{i}$ are the classes of the
generators of the nef cone.

The generation of $\mathbf{c}$ and $\nu$ modulo isomorphisms is made using
standard properties of trees. In order to generate the functions
$\mathbf{w}\colon E_{g}\rightarrow\mathbb{Z}$, we proceed in the following
way. Let us fix a tree $g$ and a coloration $\mathbf{c}\colon
V_{g}\rightarrow\Sigma^{(n)}$. By definition, for any map $\mathbf{w}\colon
E_{g}\rightarrow\mathbb{Z}$ we have

(3.1) $\beta=\sum_{e\in
E_{g}}\mathbf{w}(e)\cdot[V(\mathbf{c}(v)\cap\mathbf{c}(v^{\prime}))].$

Thus, if $\bar{e}\in E_{g}$ is a fixed edge, since $\mathbf{w}(e)>0$ we have
that

(3.2)
$\beta-\mathbf{w}(\bar{e})\cdot[V(\mathbf{c}(v)\cap\mathbf{c}(v^{\prime}))]-\sum_{e\in
E_{g}\setminus\\{\bar{e}\\}}[V(\mathbf{c}(v)\cap\mathbf{c}(v^{\prime}))]$

is an effective $1$-cycle. Thus, an upper bound for $\mathbf{w}(\bar{e})$ is
given by the number $b_{\bar{e}}$ that is the maximal number that, in place of
$\mathbf{w}(\bar{e})$, makes (3.2) an effective cycle. The package `Oscar.jl`
provides all divisors that span the extremal rays of the nef cone111I thank
Martin Bies for this feature., so (3.2) is effective if and only if it
intersects non negatively those nef divisors [CLS11, Theorem 6.3.20]. Hence we
can find $b_{\bar{e}}$. Now, we generate all possible functions $\mathbf{w}$
such that $1\leq\mathbf{w}(e)\leq b_{e}$ for all $e\in E_{g}$ and verifying
(3.1).

So that, we generate all decorated graphs. The equivariant cycles are
implemented as Julia functions that take a graph as input and return a
rational number. In order to implement (2.3), we sum all these rational
numbers to a variable `ans`. At the end of all sums, the value of `ans` will
be the value of the desired Gromow–Witten invariant.

For completeness, we list the explicit formulae of the equivariant classes we
used, whose proofs can be found in [Kon95, CK99, Spi99, Spi02, Liu13, Mur23,
MS22]. Some proofs are worked out only for $\mathbb{P}^{n}$, but they extend
tautologically for any smooth projective toric variety. We try to be as clear
and simple as possible. We use the notation introduced in 2.3. The following
is Lemma 6.10 in Spielberg’s thesis.

######  Lemma 3.1.

Let $\sigma_{1},\sigma_{2}\in\Sigma^{(n)}$ be such that
$\tau=\sigma_{1}\cap\sigma_{2}\in\Sigma^{(n-1)}$.

Let $\\{\rho_{i_{1}},\ldots,\rho_{i_{n-1}}\\}$ be the rays of $\tau$.
Moreover, let $\rho^{\prime}$ be the ray of $\sigma_{1}$ that is not in
$\tau$. Fix $\omega_{1},\ldots,\omega_{r}$ weights of the action of
$(\mathbb{C}^{*})^{r}$ on $\mathbb{C}^{r}$. The induced
$\mathbb{C}^{*}$-action on the subvariety $V(\tau)$ has weight
$\omega^{\sigma_{1}}_{\sigma_{2}}$ at the point $V(\sigma_{1})$ given by:

$\omega^{\sigma_{1}}_{\sigma_{2}}:=\sum_{i=1}^{r}\langle\rho_{i},u_{n}\rangle\omega_{i}$

where $\\{u_{1},\ldots,u_{n}\\}$ is the $\mathbb{Z}$-dual basis of
$\\{\rho_{i_{1}},\ldots,\rho_{i_{n-1}},\rho^{\prime}\\}\subset\mathbb{Z}^{r}$.

Suppose that we fix a decorated graph $\Gamma$. A flag of a vertex $v$ is the
pair $F=(v,e)$ where $e$ is an edge of $g$ containing $v$. We denote by
$F_{v}$ the set of flags of $v$. Let $d=\mathbf{w}(e)$, $\sigma_{1}$ be the
color of $v$, and $\sigma_{2}$ be the color of the other vertex of $e$. We
will denote by

$\omega_{F}:=\frac{1}{d}\omega^{\sigma_{1}}_{\sigma_{2}}.$

Given $\gamma\in\sigma_{1}^{*}$ we denote by $\lambda^{\gamma}_{e}$ the degree
of the class $d[V(\sigma_{1}\cap\sigma_{2})]\cdot[V(\rho))]$, where $\rho$ is
the ray of $\sigma_{1}$ that is not a ray of $\gamma$. We define by

$\displaystyle\Delta(e)$ $\displaystyle:=$
$\displaystyle\frac{(-1)^{d}d^{2d}}{(d!)^{2}(\omega^{\sigma_{1}}_{\sigma_{2}})^{2d}}\prod_{\gamma\in\sigma_{1}^{*}\setminus\\{\sigma_{2}\\}}\prod_{i\in
K}\left(\omega^{\sigma_{1}}_{\gamma}-\frac{i}{d}\omega^{\sigma_{1}}_{\sigma_{2}}\right)^{-\mathrm{sgn}(\lambda^{\gamma}_{e}+1)}$
$\displaystyle\Xi(\Gamma)$ $\displaystyle:=$ $\displaystyle\left(\prod_{v\in
V_{g}}\prod_{\gamma\in\mathbf{c}(v)^{*}}\omega^{\mathbf{c}(v)}_{\gamma}\right)\prod_{e\in
E_{g}}\Delta(e),$

where $K=\\{0,\ldots,\lambda^{\gamma}_{e}\\}$ if $\lambda^{\gamma}_{e}\geq 0$,
$K=\\{\lambda^{\gamma}_{e}+1,\ldots,-1\\}$ if $\lambda^{\gamma}_{e}\leq-2$,
and $K=\emptyset$ otherwise. Let $\\{a_{i}\\}_{i=1}^{m}$ be non-negative
integers. Let $S_{v}:=\nu^{-1}(v)=\\{j_{1},...,j_{k}\\}$ denote the set of
marks mapped to $v$, and let $\overline{S_{v}}:=\sum_{i\in S_{v}}a_{i}$ and
$n(v):=|F_{v}|+|S_{v}|$. If $m=0$ or $S_{v}=\emptyset$, we set
$\overline{S_{v}}:=0$. If $n(v)-3\geq\overline{S_{v}}$, we recall the
multinomial coefficients

$\binom{n(v)-3}{n(v)-3-\overline{S_{v}},a_{j_{1}},...,a_{j_{k}}}=\frac{(n(v)-3)!}{(n(v)-3-\overline{S_{v}})!\cdot
a_{j_{1}}!\cdots a_{j_{k}}!}.$

We may extend the multinomial in the following way:

  * •

if $\overline{S_{v}}>n(v)-3\geq 0$, then its value is $0$;

  * •

if $\overline{S_{v}}=0$, then its value is $1$;

  * •

if $S_{v}=\\{a_{j_{1}}\\}$ and $n(v)=2$, then its value is $(-1)^{a_{j_{1}}}$.

Let us denote by $\Psi$ the following class

(3.3) $\displaystyle\Psi(\Gamma)$ $\displaystyle:=$ $\displaystyle\prod_{v\in
V_{g}}\binom{n(v)-3}{n(v)-3-\overline{S_{v}},a_{j_{1}},...,a_{j_{k}}}\left(\sum_{F\in
F_{v}}\omega_{F}^{-1}\right)^{-\overline{S_{v}}}.$

Finally, the equivariant class of the inverse of
$c_{\mathrm{top}}^{T}(N_{\Gamma})$ times the $\psi$-classes is

(3.4)
$\displaystyle\frac{[\psi_{1}^{a_{1}}\cdots\psi_{m}^{a_{m}}]^{T}(\Gamma)}{c_{\mathrm{top}}^{T}(N_{\Gamma})(\Gamma)}$
$\displaystyle=$ $\displaystyle\Psi(\Gamma)\Xi(\Gamma)\prod_{v\in
V_{g}}\prod_{F\in F_{v}}\omega_{F}^{-1}\left(\sum_{F\in
F_{v}}\omega_{F}^{-1}\right)^{n(v)-3}.$

In the package, we split (3.4) in two different functions called `Psi` and
`Euler_inv`. The first one implements (3.3), the latter implements (3.4) when
all $a_{i}$ are trivial.

So, we clarify the contribution of $\psi$ classes and
$c_{\mathrm{top}}(N_{\Gamma})$ to (2.3). Let us see the other classes.

For any maximal cone $\sigma\in\Sigma^{(n)}$, for any ray
$\rho\in\Sigma^{(1)}$, and for any non negative integer $k\in\mathbb{Z}$, we
define the following

$\lambda(\sigma,\rho,k):=\begin{cases}0&\text{if $\rho$ is not a ray of
$\sigma$, and $k>0$},\\\ 1&\text{if $\rho$ is not a ray of $\sigma$, and
$k=0$},\\\ (\omega^{\sigma}_{\gamma})^{k}&\text{if $\gamma\in\sigma^{*}$, and
$\rho$ is a ray of $\sigma$ but not of $\gamma$.}\end{cases}$

The cone $\gamma$ with the property of the last case exists and it is unique,
so the definition makes sense. Let $Z$ be the following cohomology class:

(3.5)
$Z=[V(\rho_{1})]^{k_{1}}\cdot[V(\rho_{2})]^{k_{2}}\cdots[V(\rho_{r})]^{k_{r}}.$

For any maximal cone $\sigma$, let us define

(3.6) $\Lambda(\sigma,Z):=\prod_{j=1}^{r}\lambda(\sigma,\rho_{j},k_{j}).$

Since the cohomology ring of $X$ is generated by invariant divisors [CLS11,
Chapter 12], we may extend (3.6) by linearity to any homogeneous class $Z\in
H^{*}(X,\mathbb{Q})$.

For any $i=1,\ldots,m$ the contribution of $\mathrm{ev}_{i}^{*}(Z)$ is

(3.7) $\mathrm{ev}_{i}^{*}(Z)^{T}(\Gamma)=\Lambda(\mathbf{c}(\nu(i)),Z).$

Let $M$ be a line bundle of $X$. We extend (3.6) and (3.7) to $M$ by taking
$Z=c_{1}(M)$. Finally, let us assume that $h^{1}(X,M)=0$. We define the
integer

$M_{e}:=d(c_{1}(M)\cdot[V(\mathbf{c}(v)\cap\mathbf{c}(v^{\prime}))]).$

The equivariant class of the Euler class of the bundle
$\delta_{*}(\mathrm{ev}_{m+1}^{*}(M))$ is

$c_{\mathrm{top}}^{T}(\delta_{*}(\mathrm{ev}_{m+1}^{*}(M)))(\Gamma)=\prod_{v\in
V_{g},|F_{v}|\neq 1}\Lambda(\mathbf{c}(v),M)^{1-|F_{v}|}\\\
\times\left(\prod_{e\in
E_{g},M_{e}>0}\prod_{\alpha=0}^{M_{e}}\frac{\alpha\Lambda(\mathbf{c}(v),M)-(M_{e}-\alpha)\Lambda(\mathbf{c}(v^{\prime}),M)}{M_{e}}\right).$

######  Remark 3.2.

The algorithm we described, as well as the examples in the next section, are
relative to Version 1.0.0 of the package and Version 0.13.0 of `Oscar.jl`.
Some part of the code may be improved in the future. We refer to the internal
documentation of the package for an up-to-date explanation of its features.

######  Remark 3.3.

Csaba Schneider and the author created a similar package for computations of
Gromov–Witten invariants in projective spaces, see [MS22]. The projective
spaces are toric, so all computations of that package are supported by this
one. However, that package does not require `Oscar.jl`, so it is faster and
easier to install.

##  4\. Explicit computations

In this section we use our package to perform some explicit computations, in
order to show its functionalities. For more examples, please consult the
internal documentation of the package. The typical use of the package is the
following. First, one defines the variety $X$ and the class $\beta$ using
`Oscar.jl`, then defines the function `P` by combining the functions that are
built-in the package. Finally, one calls `IntegrateAB(X, beta, m, P)` to get
the result of the integration. We provided our package with the following
functions.

  * •

`a_point(X)` returns the cohomology class $[\mathrm{pt}]$ of a point of $X$.

  * •

`moment_graph(X)` returns the moment graph of $X$.

For example, if `mg=moment_graph(X)`, then `mg[1,2]` is the cohomology class
of the $T$-invariant curve $V(\sigma)\cap V(\sigma^{\prime})$, where
$\sigma,\sigma^{\prime}$ are the first two elements of $\Sigma^{(n)}$.

###  4.1. Enumeration of curves

Let $h=c_{1}(\mathcal{O}_{\mathbb{P}^{3}}(1))$. The following computation

$\int_{\overline{M}_{0,5}(\mathbb{P}^{3},2)}\mathrm{ev}_{1}^{*}(h^{3})\cdot\mathrm{ev}_{2}^{*}(h^{3})\cdot\mathrm{ev}_{3}^{*}(h^{3})\cdot\mathrm{ev}_{4}^{*}(h^{2})\cdot\mathrm{ev}_{5}^{*}(h^{2})$

equals the number of conics in $\mathbb{P}^{3}$ passing through three points
and two lines in general position (see [FP97, Lemma 14]). It can be computed
in the following way

    P3 = projective_space(NormalToricVariety, 3); # the space P^3.
    h = cohomology_class(toric_line_bundle(P3, [1]));
    P = ev(1, h^3)*ev(2, h^3)*ev(3, h^3)*ev(4, h^2)*ev(5, h^2);
    IntegrateAB(P3, 2*h^2, 5, P);
    Result: 1

Let $d\geq 0$ and $e\leq 0$ such that $2d+e>0$. We want to compute the the
numbers of degree $d$ rational curves in $\mathbb{P}^{3}$ meeting $2d+e$
generic points, and in addition passing through a fixed point with global
multiplicity $-e$. By [Gat01, Example 8.2] we know that such a number is

$\int_{\overline{M}_{0,2d+e}(X,\beta)}\prod_{i=1}^{2d+e}\mathrm{ev}_{i}^{*}([\mathrm{pt}]),$

where $\pi\colon X\rightarrow\mathbb{P}^{3}$ is the blow-up at one point,
$H=\pi^{*}(h^{2})$, $E$ is the class of a line in the exceptional divisor, and
finally $\beta=dH+eE$. This invariant can be computed using the package in the
following way.

    X = domain(blow_up(P3, ideal(gens(cox_ring(P3))[1:3])));
    mg = moment_graph(X);
    H = mg[4,5];
    E = mg[1,2];
    d = 1; # this can be any non negative integer
    e = -1; # this can be any non positive integer
    beta = d*H+e*E;
    P = prod(i -> ev(i, a_point(X)), 1:(2*d+e)); # we require 2d+e>0
    IntegrateAB(X, beta, 2*d+e, P);
    Result: 1

In the Gromov–Witten invariant (2.2), the cohomology class
$\psi_{1}^{a_{1}}\cdots\psi_{m}^{a_{m}}$ is implemented using the command
`Psi(a_1,...,a_m)`. We may use this to compute the number of curves with
tangency conditions. For example, the number of lines in $\mathbb{P}^{3}$
passing through a general point of a quartic surface with multiplicity $3$ is

$\int_{\overline{M}_{0,1}(\mathbb{P}^{3},1)}\frac{1}{4}\left(\mathrm{ev}_{1}^{*}(4h)^{3}\cdot
3\mathrm{ev}_{1}^{*}(4h)^{2}\psi_{1}\cdot
2\mathrm{ev}_{1}^{*}(4h)\psi^{2}_{1}\right)\mathrm{ev}_{1}^{*}(h)^{2}=2,$

see [CM18, MS21, Mur21]. We compute it using the following code.

    l = 4*h;
    P = 1//4*(ev(1,l)^3+3*ev(1,l)^2*Psi(1)+2*ev(1,l)*Psi(2))*ev(1,h)^2;
    IntegrateAB(P3, h^2, 1, P);
    Result: 2

###  4.2. Quantum cohomology ring of a threefold

In [Spi99, Section 9.3], Spielberg computed the small quantum cohomology ring
of
$X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{2}}\oplus\mathcal{O}_{\mathbb{P}^{2}}(2))$.
His strategy was the following. Among all Gromov–Witten invariants that appear
in the definition of quantum product, only a finite number are non zero. Using
some clever arguments he can list all of them. After that, he computes each
one by hand, using Equation (2.3). In this subsection we will use our code in
order to get the computations he made by hand. In order to define the toric
variety $X$ using `Oscar.jl`, first we list its rays and maximal cones, and
then we use the command `normal_toric_variety`. We use Spielberg’s notation
instead of that of Example 2.2.

    e1 = [1,0,0]; e2 = [0,1,0]; e3 = [0,0,1];
    ray_gens = [e1, -e1, e2, e3, -e2-e3-2*e1];
    max_cones = [[1,3,4],[1,3,5],[1,4,5],[2,3,4],[2,3,5],[2,4,5]];
    X=normal_toric_variety(ray_gens,max_cones;non_redundant=true);

Let us define an array containing the generators of $H^{*}(X,\mathbb{C})$,

    Z = [cohomology_class(X, Z) for Z in gens(cohomology_ring(X))];

With this notation, the cohomology ring of $X$ is

$H^{*}(X,\mathbb{C})=\frac{\mathbb{C}[Z_{1},Z_{2},Z_{3},Z_{4},Z_{5}]}{(Z_{1}-Z_{2}-2Z_{5},Z_{3}-Z_{4},Z_{3}-Z_{5},Z_{1}Z_{2},Z_{3}Z_{4}Z_{5})}$

The generators of the Mori cone, that in the paper are denoted by
$\lambda_{1}$ and $\lambda_{2}$, are

    mg = moment_graph(X);
    lambda1 = mg[1,4];
    lambda2 = mg[4,5];

The first non zero invariant computed by Spielberg is

$\int_{\overline{M}_{0,3}(X,\lambda_{2})}\mathrm{ev}_{1}^{*}(Z_{3})\cdot\mathrm{ev}_{2}^{*}(Z_{3})\cdot\mathrm{ev}_{3}^{*}(Z_{4}Z_{5})=-1.$

In his paper it is denoted by $\Phi^{\lambda_{2}}(Z_{3},Z_{3},Z_{4}Z_{5})$. In
our package this is computed using the following lines:

    P = ev(1, Z[3])*ev(2, Z[3])*ev(3, Z[4]*Z[5]);
    IntegrateAB(X, lambda2, 3, P);

The result is as expected. The next non zero invariant is

$\Phi^{\lambda_{1}}(Z_{1},Z_{1},Z_{2}Z_{3}Z_{4})=\int_{\overline{M}_{0,3}(X,\lambda_{2})}\mathrm{ev}_{1}^{*}(Z_{1})\cdot\mathrm{ev}_{2}^{*}(Z_{1})\cdot\mathrm{ev}_{3}^{*}(Z_{2}Z_{3}Z_{4})=1.$

Again, the same computation with our code gives the same result:

    P = ev(1, Z[1])*ev(2, Z[1])*ev(3, Z[2]*Z[3]*Z[4]);
    IntegrateAB(X, lambda1, 3, P);
    Result: 1

We tested all computations in Spielberg’s tesis. Those computations, as well
as all computations in this article, are publicly available for reader’s
convenience222The repository is
https://github.com/mgemath/ToricAB_computations..

###  4.3. Quantum cohomology ring of a fourfold

Spielberg’s strategy works for any toric Fano variety even if the number of
necessary computations grows. Take the toric Fano fourfold
$X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{3}}\oplus\mathcal{O}_{\mathbb{P}^{3}}(1))$,
which appears in Batyrev’s classification [Bat99, Sat00]. We implements $X$ in
the following way.

    X = proj(toric_line_bundle(P3, [0]), toric_line_bundle(P3, [1]));

Take two classes $\alpha_{1},\alpha_{2}\in H^{2}(X,\mathbb{Z})$ and the class
of a curve $\beta$. By Equation (2.1), a necessary condition for the invariant

(4.1)
$\int_{\overline{M}_{0,3}(X,\beta)}\mathrm{ev}_{1}^{*}(\alpha_{1})\cdot\mathrm{ev}_{2}^{*}(\alpha_{2})\cdot\mathrm{ev}_{3}^{*}(\theta)$

to be non zero is that the codimension of the class $\theta$ is equal to
$2-\deg(c_{1}(\Omega_{X})\cdot\beta)$. Thus the codimension of $\theta$ is $3$
or $4$. If it is $3$, then $-\deg(c_{1}(\Omega_{X})\cdot\beta)=1$. But this is
impossible since the Mori cone of $X$ is generated by the classes
$\lambda_{1}$ and $\lambda_{2}$ where

    mg = moment_graph(X);
    lambda1 = mg[1,2];
    lambda2 = mg[4,8];

and a direct computation shows $-\deg(c_{1}(\Omega_{X})\cdot\lambda_{1})=2$,
and $-\deg(c_{1}(\Omega_{X})\cdot\lambda_{2})=3$.
So that, the invariant (4.1) is non zero necessary if $\theta$ has codimension
$4$ (in particular, it is the class of a point) and $\beta=\lambda_{1}$. The
usual cohomology ring of $X$ is

$H^{*}(X,\mathbb{C})=\frac{\mathbb{C}[Z_{1},Z_{2},Z_{3},Z_{4},Z_{5},Z_{6}]}{(Z_{1}-Z_{6},Z_{2}-Z_{6},Z_{3}-Z_{6},Z_{1}+Z_{4}-Z_{5},Z_{4}Z_{5},Z_{1}Z_{2}Z_{3}Z_{6})}$

As before, we implement it using the following command.

    Z = [cohomology_class(X, Z) for Z in gens(cohomology_ring(X))];

We want to compute $Z_{4}\star Z_{5}$. By Equation (2.4), we have

$Z_{4}\star Z_{5}=Z_{4}\cdot
Z_{5}+q^{\lambda_{1}}\int_{\overline{M}_{0,3}(X,\lambda_{1})}\mathrm{ev}_{1}^{*}(Z_{4})\cdot\mathrm{ev}_{2}^{*}(Z_{5})\cdot\mathrm{ev}_{3}^{*}([\mathrm{pt}]).$

The code to compute it is the following:

    P = ev(1, Z[4])*ev(2, Z[5])*ev(3, a_point(X));
    IntegrateAB(X, lambda1, 3, P);
    Result: 1

Thus $Z_{4}\star Z_{5}=q^{\lambda_{1}}$. Following this path one can recover
the entire multiplication table of the ring $QH^{s}(X)$.

###  4.4. Twisted Gromov–Witten invariants

Let $M$ be a line bundle on $X$ with $h^{1}(X,M)=0$. The integral

$\int_{\overline{M}_{0,m}(X,\beta)}\mathrm{ev}_{1}^{*}(\gamma_{1})\cdots\mathrm{ev}_{m}^{*}(\gamma_{m})\cdot\psi_{1}^{a_{1}}\cdots\psi_{m}^{a_{m}}\cdot
c_{\mathrm{top}}(\delta_{*}(\mathrm{ev}^{*}_{m+1}(M)))$

gives Gromov–Witten invariants of a subvariety of $X$ cut out by a section of
$M$ (see, e.g., [CCIT09, Example B]). In the package, in order to insert
$c_{\mathrm{top}}(\delta_{*}(\mathrm{ev}^{*}_{m+1}(M)))$ we use the command
`push_ev(M)`.

For example, take
$X=\mathbb{P}(\mathcal{O}_{\mathbb{P}^{3}}\oplus\mathcal{O}_{\mathbb{P}^{3}}(1))$
that we used in the last subsection, and $M=\det(\Omega_{X})^{\vee}$ the
anticanonical bundle. A section of $M$ cuts out a Calabi–Yau threefold and its
Gromov–Witten invariants

$\displaystyle\int_{\overline{M}_{0,1}(X,\lambda_{1})}\psi_{1}^{1}c_{3}(\delta_{*}(\mathrm{ev}^{*}_{2}(M)))$
$\displaystyle=-120,$
$\displaystyle\int_{\overline{M}_{0,0}(X,\lambda_{2})}c_{4}(\delta_{*}(\mathrm{ev}^{*}_{1}(M)))$
$\displaystyle=27,$

can be computed in the following way.

    M = anticanonical_bundle(X);
    P = push_ev(M);
    IntegrateAB(X, lambda1, 1, Psi(1)*P);
    Result: -120
    IntegrateAB(X, lambda2, 0, P);
    Result: 27

We refer to the page on Github for a list of all implemented classes.

## References

  * [AB84] M. F. Atiyah and R. Bott, _The moment map and equivariant cohomology_ , Topology 23 (1984), no. 1, 1–28. MR 721448
  * [ABPZ23] Hülya Argüz, Pierrick Bousseau, Rahul Pandharipande, and Dimitri Zvonkine, _Gromov-Witten Theory of Complete Intersections via Nodal Invariants_ , 2023.
  * [Bat99] V. V. Batyrev, _On the classification of toric Fano $4$-folds_, J. Math. Sci. (New York) 94 (1999), no. 1, 1021–1050. MR 1703904
  * [BEKS17] Jeff Bezanson, Alan Edelman, Stefan Karpinski, and Viral B. Shah, _Julia: a fresh approach to numerical computing_ , SIAM Rev. 59 (2017), no. 1, 65–98. MR 3605826
  * [BK23] Martin Bies and Lars Kastner, _Toric Geometry in OSCAR_ , 2023, arXiv 2303.08110.
  * [CCIT09] Tom Coates, Alessio Corti, Hiroshi Iritani, and Hsian-Hua Tseng, _Computing genus-zero twisted Gromov-Witten invariants_ , Duke Math. J. 147 (2009), no. 3, 377–438. MR 2510741
  * [CK99] David A. Cox and Sheldon Katz, _Mirror symmetry and algebraic geometry_ , Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117
  * [CLS11] David A. Cox, John B. Little, and Henry K. Schenck, _Toric varieties_ , Graduate Studies in Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322
  * [CM18] K. Cieliebak and K. Mohnke, _Punctured holomorphic curves and Lagrangian embeddings_ , Invent. Math. 212 (2018), no. 1, 213–295. MR 3773793
  * [CP16] Suyoung Choi and Seonjeong Park, _Projective bundles over toric surfaces_ , Internat. J. Math. 27 (2016), no. 4, 1650032, 30. MR 3491049
  * [DEF+24] Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, and Michael Joswig (eds.), _The OSCAR book_ , 2024.
  * [FP97] W. Fulton and R. Pandharipande, _Notes on stable maps and quantum cohomology_ , Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 45–96. MR 1492534
  * [Gat01] Andreas Gathmann, _Gromov-Witten invariants of blow-ups_ , J. Algebraic Geom. 10 (2001), no. 3, 399–432. MR 1832328
  * [GP99] T. Graber and R. Pandharipande, _Localization of virtual classes_ , Invent. Math. 135 (1999), no. 2, 487–518. MR 1666787
  * [Hu22a] Xiaowen Hu, _Big quantum cohomology of Fano complete intersections_ , 2022, arXiv 1501.03683.
  * [Hu22b] by same author, _On Genus 1 Gromov-Witten invariants of Fano complete intersections_ , 2022, arXiv 2203.08091.
  * [Kon95] Maxim Kontsevich, _Enumeration of rational curves via torus actions_ , The moduli space of curves (Texel Island, 1994), Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 335–368. MR 1363062
  * [Liu13] Chiu-Chu Melissa Liu, _Localization in Gromov-Witten theory and orbifold Gromov-Witten theory_ , Handbook of moduli. Vol. II, Adv. Lect. Math. (ALM), vol. 25, Int. Press, Somerville, MA, 2013, pp. 353–425. MR 3184181
  * [MS21] Dusa McDuff and Kyler Siegel, _Counting curves with local tangency constraints_ , J. Topol. 14 (2021), no. 4, 1176–1242. MR 4332489
  * [MS22] Giosuè Muratore and Csaba Schneider, _Effective computations of the Atiyah-Bott formula_ , J. Symbolic Comput. 112 (2022), 164–181. MR 4383164
  * [Mur21] Giosuè Muratore, _A recursive formula for osculating curves_ , Ark. Mat. 59 (2021), no. 1, 195–211. MR 4256011
  * [Mur23] Giosuè Muratore, _Enumeration of Rational Contact Curves via Torus Actions_ , Michigan Math. J. 73 (2023), no. 4, 875–894. MR 4634985
  * [Oda78] Tadao Oda, _Lectures on torus embeddings and applications. (Based on joint work with Katsuya Miyake.)_ , Lect. Math. Phys., Math., Tata Inst. Fundam. Res., vol. 58, Springer, Berlin; Tata Inst. of Fundamental Research, Bombay, 1978 (English).
  * [OSC23] _OSCAR – Open Source Computer Algebra Research system, Version 0.13.0_ , 2023\.
  * [Sat00] Hiroshi Sato, _Toward the classification of higher-dimensional toric Fano varieties_ , Tohoku Math. J. (2) 52 (2000), no. 3, 383–413. MR 1772804
  * [Spi99] Holger Spielberg, _A formula for the Gromov-Witten invariants of toric varieties_ , Prépublication de l’Institut de Recherche Mathématique Avancée [Prepublication of the Institute of Advanced Mathematical Research], vol. 1999/11, Université Louis Pasteur, Département de Mathématique, Institut de Recherche Mathématique Avancée, Strasbourg, 1999, Thèse, Université Louis Pasteur (Strasbourg I), Strasbourg, 1999. MR 1692603
  * [Spi02] by same author, _Multiple quantum products in toric varieties_ , Int. J. Math. Math. Sci. 31 (2002), no. 11, 675–686. MR 1928909
  * [WROM86] Robert Alan Wright, Bruce Richmond, Andrew Odlyzko, and Brendan D. McKay, _Constant time generation of free trees_ , SIAM J. Comput. 15 (1986), no. 2, 540–548. MR 837603


# Hölder regularity of continuous solutions to balance laws and applications
in the Heisenberg group

Laura Caravenna  Laura Caravenna, Universita‘ degli Studi di Padova,
Dipartimento di Matematica Tullio Levi-Civita, Via Trieste 63, 35121 Padova,
Italy, Email<EMAIL_ADDRESS>,  Elio Marconi  Elio Marconi,
Universita‘ degli Studi di Padova, Dipartimento di Matematica Tullio Levi-
Civita, Via Trieste 63, 35121 Padova, Italy, Email<EMAIL_ADDRESS>and
Andrea Pinamonti  Andrea Pinamonti, Department of Mathematics, University of
Trento, Trento 38123, Italy, Email<EMAIL_ADDRESS>
###### Abstract.

We prove Hölder regularity of any continuous solution $u$ to a $1$-D scalar
balance law $u_{t}+[f(u)]_{x}=g$, when the source term $g$ is bounded and the
flux $f$ is nonlinear of order $\ell\in\mathbb{N}$ with $\ell\geq 2$. For
example, $\ell=3$ if $f(u)=u^{3}$. Moreover, we prove that at almost every
point $(t,x)$, it holds $u(t,x+h)-u(t,x)=o(|h|^{\frac{1}{\ell}})$ as $h\to 0$.
Due to Lipschitz regularity along characteristics, this implies that at almost
every point $(t,x)$, it holds
$u(t+k,x+h)-u(t,x)=o((|h|+|k|)^{\frac{1}{\ell}})$ as $|(h,k)|\to 0$. We apply
the results to provide a new proof of the Rademacher theorem for intrinsic
Lipschitz functions in the first Heisenberg group.

##  1\. Introduction

The effect of non-linearity in conservation laws is two-folds: on the one hand
it prevents the existence of smooth solutions due to shocks formation. On the
other hand it has some regularizing effects. The most striking example is the
estimate by Oleinik [34], which proves that $L^{\infty}$ initial data are
immediately regularized to $BV_{\mathrm{loc}}$ at positive times, for entropy
solutions of the classical Burgers’ equation

$u_{t}+[u^{2}/2]_{x}=0.$

Furthermore, [4] proved that $BV_{\mathrm{loc}}$ regularity improves to
$SBV_{\mathrm{loc}}$, then generalized to more general $1d$ scalar balance
laws in [35] and to systems of conservation and balance laws in [9]. Similar
results have been proven for Hamilton-Jacobi equations in several space
dimensions [10].

A possible approach to prove the $BV_{\mathrm{loc}}$ estimate in the scalar
case relies on the existence of a family non-crossing (backward)
characteristics [21, Chapter 10] along which the solution is constant. The
non-crossing condition is a geometric constraint which forces
$BV_{\mathrm{loc}}$ regularity. This approach has been extended to the more
general conservation law of the form:

$\displaystyle u_{t}+[f(u)]_{x}=0,$ (1.1)

see [13, 1] for the case of convex fluxes $f$ and [19, 17, 31], for the case
of general smooth fluxes. The final result is that a fractional regularity of
entropy solutions can be obtained quantifying the non-linearity of the flux
$f$.

Motivated by some applications to the theory of rectifiable sets in the
Heisenberg group in the present paper we take into account the presence of the
source $g\in L^{\infty}$ in (1.1), i.e.

$\displaystyle u_{t}+[f(u)]_{x}=g,$ (1.2)

We prove a regularity result for the continuous solutions of (1.2) where $f\in
C^{\ell}(\mathbb{R})$ is nonlinear of order $\ell\geq 2$ (see Definition A).
More precisely, we first prove that every continuous solution to (1.2) is
indeed locally $1/\ell-$Hölder continuous and, even better,

$u(t,x+h)-u(t,x)=o\left(\sqrt[\ell]{|h|}\right)\quad\mbox{as}\quad h\to 0$
(1.3)

at Lebesgue almost every point $(t,x)$. It is known that, even with analytic
fluxes $f$ and continuous sources terms $g$, continuous solutions to (1.2)
improve to Hölder continuous regularity but it does not go beyond. Indeed,
even with quadratic flux continuous solutions to (2.1) might exhibit nasty
fractal behaviour and in general they are neither Sobolev nor BV [28, 3].
Moreover, if the flux $f$ is strictly convex but not analytic the result
generally fails [3, § 5], which motivates the finite order nonlinearity
Assumption A on the flux. Similar results have been obtained also for
conservation laws in several space dimensions by means of the kinetic
formulation (see for example [30, 26]), nevertheless the conjectured optimal
regularity has not been proven yet. We mention that also in the case of
Hamilton-Jacobi equations, optimal Hölder regularity has been recently proved
(see [18] and references therein).

In order to describe the application of our result to the theory of
rectifiable sets in the Heisenberg group we recall that the notion of
Lipschitz submanifolds in sub-Riemannian geometry was introduced, at least in
the setting of Carnot groups, by B. Franchi, R. Serapioni and F. Serra Cassano
in a series of papers [23, 24, 25] through the theory of intrinsic Lipschitz
graphs (see also [15, 16]). Roughly speaking, a subset $S\subset\mathbb{G}$ of
a Carnot group $\mathbb{G}$ is intrinsic Lipschitz if at each point $P\in S$
there is an intrinsic cone with vertex P and fixed opening, intersecting $S$
only in $P$. Remarkably, this notion turned out to be the right one in the
setting of the intrinsic rectifiability in the simplest Carnot group, namely
the Heisenberg group $\mathbb{H}^{n}$. Indeed, it was proved in [25] that the
notion of rectifiable set in terms of an intrinsic regular hypersurfaces is
equivalent to the one in terms of intrinsic Lipschitz graphs. We also remark
that intrinsic Lipschitz functions played a crucial role in the recent paper
[32], where the longstanding question of determining the approximation ratio
of the Goemans-Linial algorithm for the Sparsest Cut Problem was settled, see
also [33] for different applications. We address the interested reader to [24,
37] for a complete introduction to the theory of intrinsic Lipschitz
functions. One of the main open questions in this area of research is whether
a Rademacher type theorem holds. Namely, assume that a splitting
$\mathbb{G}=\mathbb{W}\mathbb{V}$ of a Carnot group $\mathbb{G}$ is fixed, is
it true that every intrinsically Lipschitz function
$u:\mathbb{W}\to\mathbb{V}$ is intrinsically differentiable almost everywhere?
In [22, 24], it is proved that the answer is _yes_ if $\mathbb{G}$ is of step
two or a group of type $\star$ and $\mathbb{V}=\mathbb{R}$. More recently, D.
Vittone in [37] proved that the answer is also _yes_ in the case of the
Heisenberg group without any a priori assumption on the splitting. We also
address the interested reader to [7, 8, 29] for further partial results.
Remarkably, in [27] the authors constructed intrinsic Lipschitz graphs of
codimension $2$ in Carnot groups which are not intrinsically differentiable
almost everywhere. Insipired by the results contained in [6], in [11], the
first named author together with Bigolin and Serra Cassano provided a
characterization of intrinsic Lipschitz graphs in the Heisenberg group in
terms of a system of non linear first order PDEs. Moreover, they proved the
equivalence of different notions of continuous weak solutions to the equation

$\displaystyle u_{t}+[u^{2}/2]_{x}=g,$ (1.4)

where $g$ is a bounded function. It turns out that the question whether an
intrinsically Lipschitz function $u$ is intrinsically differentiable or is
equivalent to (1.3). This observation allows us to provide a completely PDEs
based proof of the Rademacher theorem in $\mathbb{H}^{1}$. We point out that
the proof provided in [24] requires some deep and nontrivial results in
geometric measure theory, namely the fact that the subgraph of an intrinsic
Lipschitz function $u$ is a set with locally finite $\mathbb{H}-$perimeter and
that at almost every point of the graph of $u$ there is an approximate tangent
plane. Another interesting feature of our approach is its potential
applicability to those Carnot groups where the aforementioned geometric tools
are not yet available e.g. the Engel group.

Structure of the paper: In § 2 we revise an elementary estimate on continuous
solutions to balance laws when the flux is convex. When the source is bounded,
such estimate is the key to Hölder continuity and Lipschitz continuity along
characteristics. We prove in § 3 Hölder regularity of any continuous solution
$u$ to a $1d$ scalar balance law, when the source term is bounded and the flux
is nonlinear of order $\ell$ in the sense of § A. The Hölder regularity is
then refined in § 4, where we show that at Lebesgue almost every point $(t,x)$
it holds $u(t,x+h)-u(t,x)=o(|h|^{\frac{1}{\ell}})$ as $h\to 0$. In § 5 we
finally prove the Rademacher theorem for intrinsic Lipschitz functions in
$\mathbb{H}^{1}$.

##  2\. Preliminary results

We recall the definition of characteristics of the equation

$\partial_{t}u+[f(u)]_{x}=g\,,\qquad f\in
C^{\ell}(\mathbb{R}),\ell\in\mathbb{N},\ g\ \text{bounded}.$ (2.1)

######  Definition 2.1.

A characteristic associated to a continuous solution $u$ to (2.1) is any
function $\gamma\in C^{1}(I)$, defined on a real interval $I\subset\mathbb{R}$
and satisfying the ordinary differential equation

$\dot{\gamma}(t)=f^{\prime}(u(s,\gamma(t)))\qquad\text{for }t\in I.$

The following well-known result will be helpful later on (see [20, 2]):

######  Proposition 2.2.

Let $g\in L^{1}_{\mathrm{loc}}(\Omega)$, $\Omega\subset\mathbb{R}^{2}$ be open
and assume that $u\in C^{0}(\Omega)$ is a distributional solution of (2.1).
Given a characteristic $\gamma:[t_{1},t_{2}]\to\mathbb{R}$ and
${{\mathfrak{h}}}>0$, assume that the following sets

$\begin{split}S^{+}(\gamma,{{\mathfrak{h}}})\doteq&\leavevmode\nobreak\
\\{(t,x)\in[t_{1},t_{2}]\times\mathbb{R}:x\in[\gamma(t),\gamma(t)+{{\mathfrak{h}}}]\\},\\\
S^{-}(\gamma,{{\mathfrak{h}}})\doteq&\leavevmode\nobreak\
\\{(t,x)\in[t_{1},t_{2}]\times\mathbb{R}:x\in[\gamma(t)-{{\mathfrak{h}}},\gamma(t)]\\}\end{split}$

are contained in $\Omega$. If $f$ is convex on the interval
$u(S^{+}(\gamma,{{\mathfrak{h}}})\cup S^{-}(\gamma,{{\mathfrak{h}}}))$, then

$\displaystyle\int_{\gamma(t_{2})}^{\gamma(t_{2})+{{\mathfrak{h}}}}u(t_{2},x)dx-\int_{\gamma(t_{1})}^{\gamma(t_{1})+{{\mathfrak{h}}}}u(t_{1},x)dx\leq\int_{S^{+}(\gamma,{{\mathfrak{h}}})}g\,dxdt$
(2.2a)
$\displaystyle\int_{\gamma(t_{2})-{{\mathfrak{h}}}}^{\gamma(t_{2})}u(t_{2},x)dx-\int_{\gamma(t_{1})-{{\mathfrak{h}}}}^{\gamma(t_{1})}u(t_{1},x)dx\geq\int_{S^{-}(\gamma,{{\mathfrak{h}}})}g\,dxdt.$
(2.2b)

The analogous statement holds for concave fluxes $f$ reversing the
inequalities (2.2a) and (2.2b).

$x$$t$$\sigma$$t_{2}=\sigma+\varepsilon$$t_{1}=\sigma-\varepsilon$$S^{-}$$S^{+}$$\gamma(t)$$\gamma(t)+\varrho\varepsilon^{\ell}$$\gamma(t)-\varrho\varepsilon^{\ell}$$\mathbf{S_{\varrho}}(\gamma,\varepsilon,\sigma)$$\mathbf{{{\mathfrak{h}}}=\varrho\varepsilon^{\ell}}$$\mathbf{\varepsilon=\varrho^{-\frac{1}{\ell}}\sqrt[\ell]{{{{\mathfrak{h}}}}{}}}$
Figure 1. A region $S_{\varrho}$ of the Vitali covering (4.2) and
$S^{\pm}(\gamma|_{[t_{1},t_{2}]},{{\mathfrak{h}}})$ in Proposition 2.

######  Remark 2.3.

Dividing estimates (2.2a) and (2.2b) by ${{\mathfrak{h}}}$ and letting
${{\mathfrak{h}}}\to 0$ we get that $t\mapsto u(t,\gamma(t))$ is Lipschitz
continuous with constant bounded by $\|g\|_{L^{\infty}}$. The same result was
proven in [2, Theorem 30] under the assumption that
$\mathscr{L}^{1}(\overline{\mathrm{Infl}(f)})=0$, where $\mathrm{Infl}(f)$
denotes the set of inflections points of $f$.

##  3\. Hölder regularity

The Hölder regularity of continuous solutions to (1.4) in [12, 11, Corollary
5.5] has been generalized in [14] to the case of $\ell$-nonlinear convex
fluxes. We describe in § A what we mean by $\ell$-nonlinear function: here we
only mention that, in particular, are $\ell$-nonlinear the functions for which
at any point there is a derivative of order between $2$ and $\ell$ which does
not vanish. In this section we extend the local $\frac{1}{\ell}$-Hölder
regularity of continuous solutions without convexity hypothesis, so that it
applies, for example, to $f(u)=u^{3}$.

######  Proposition 3.1.

Let $f\in C^{2}(\mathbb{R})$ be nonlinear of order $\ell$ according to
Definition A. Consider an open connected set $\Omega\subset\mathbb{R}^{2}$ and
let $u\in C^{0}(\Omega)$ and $g\in L^{\infty}(\Omega)$ be such that

$\partial_{t}u+[f(u)]_{x}=g\quad\mbox{in }\mathcal{D}^{\prime}(\Omega).$

Then $u\in C^{0,\alpha}_{\mathrm{loc}}(\Omega)$ with $\alpha=\frac{1}{\ell}$.

We first prove the following claim.

######  Lemma 3.2.

In the hypothesis of Proposition 3, consider $(t,x_{1}),(t,x_{2})\in\Omega$
such that $f^{\prime\prime}$ does not vanish in the closed interval $I$ with
endpoints $u(t,x_{1})=u_{1}$ and $u(t,x_{2})=u_{2}$: set

$\displaystyle q=\frac{\min_{I}f^{\prime\prime}}{\max_{J}f^{\prime\prime}}\,,$
$\displaystyle G=\|g\|_{L^{\infty}}\,,$
$\displaystyle\tau=\frac{q}{2G}|u_{2}-u_{1}|\,,$ $\displaystyle
C=\|f^{\prime\prime}\|_{L^{\infty}(B_{G\tau}(I))}\,,$ $\displaystyle
L=\|f^{\prime}\|_{L^{\infty}(B_{G\tau}(I))}\,,$

given any $J\supset B_{\frac{1}{2}|u_{2}-u_{1}|}(I)$. If
$\|g\|_{L^{\infty}}=0$ rather fix above any $G>0$. Assume that

$B_{|x_{1}-x_{2}|+L\tau+CG\tau^{2}}(t,x_{1})\subset\Omega\,.$ (3.1)

Then, denoting by $c_{\ell}$ the constant of nonlinearity in Definition A, we
get

$|u_{2}-u_{1}|\leq\mathtt{c}|x_{2}-x_{1}|^{\frac{1}{\ell}},\qquad\mbox{with}\
\mathtt{c}=\sqrt[\ell]{\frac{4G}{qc_{\ell}}}.$

###### Proof of Lemma 3.

We consider two characteristics $\gamma_{1},\gamma_{2}$ passing through
$(t,x_{1}),(t,x_{2})$. Without loss of generality we assume $x_{1}<x_{2}$. We
can also assume $f^{\prime}(u_{1})>f^{\prime}(u_{2})$, the argument in the
opposite case is similar considering evolution in the past.

Since $\gamma_{1}$ is a characteristic, then

$\gamma_{1}(t+\tau)=x_{1}+\int_{t}^{t+\tau}f^{\prime}(u(s,\gamma_{1}(s)))ds.$
(3.2)

By Remark 2 the Lipschitz function $u(\cdot,\gamma_{1}(\cdot))$ restricted to
$[t,t+\tau]$ has image in $[u_{1}-G\tau,u_{1}+G\tau]$. By the chain rule on
$\sigma\mapsto f^{\prime}(u(\sigma,\gamma_{1}(\sigma)))$ we estimate

$\begin{split}\left|f^{\prime}(u(s,\gamma_{1}(s)))-f^{\prime}(u_{1})\right|=&\left|\int_{t}^{s}\partial_{\sigma}f^{\prime}(u(\sigma,\gamma_{1}(\sigma)))d\sigma\right|\leq
CG(s-t)\qquad\forall s\in[t,t+\tau].\end{split}$ (3.3)

Estimating the integral in (3.2) by (3.3), and doing similarly with
$\gamma_{2}$, we see that

$\displaystyle\gamma_{1}(t+\tau)\geq
x_{1}+f^{\prime}(u_{1})\tau-\frac{CG}{2}\tau^{2}\,.$ (3.4a)
$\displaystyle\gamma_{2}(t+\tau)\leq
x_{2}+f^{\prime}(u_{2})\tau+\frac{CG}{2}\tau^{2}.$ (3.4b)

Notice that by the choice of $\tau$ and (3.1) the set
$\\{(s,\gamma_{1}(s)),(s,\gamma_{2}(s)):s\in[t,t+\tau]\\}\subset\Omega$ and
since $2G\tau\leq{|u_{2}-u_{1}|}$ we have

$(u_{2}-G\tau,u_{2}+G\tau)\cap(u_{1}-G\tau,u_{1}+G\tau)=\emptyset\,.$

In particular there is no $s\in[t,t+\tau)$ for which
$\gamma_{1}(s)=\gamma_{2}(s)$, and therefore $\gamma_{1}(s)<\gamma_{2}(s)$ for
every $s\in[t,t+\tau)$. Concatenating the previous inequalities (3.4), as
$\gamma_{1}(\tau)-\gamma_{2}(\tau)\leq 0$ we have

$f^{\prime}(u_{1})-f^{\prime}(u_{2})\leq\frac{x_{2}-x_{1}}{\tau}+CG\tau.$

Lemma A allows to absorb
$CG\tau=\frac{q}{2}\|f^{\prime\prime}\|_{L^{\infty}(B_{\tau
G}(I))}|u_{2}-u_{1}|$ in the l.h.s., so that

$\left(f^{\prime}(u_{1})-f^{\prime}(u_{2})\right)|u_{1}-u_{2}|q\leq{4G}(x_{2}-x_{1}).$

By Lemma A we have $f^{\prime}(u_{1})-f^{\prime}(u_{2})\geq
c_{\ell}|u_{1}-u_{2}|^{\ell-1}$, thus we arrive to the claim

$|u_{1}-u_{2}|\leq\left(\frac{{4G}}{qc_{\ell}}\right)^{\frac{1}{\ell}}\sqrt[\ell]{x_{2}-x_{1}}\,.\qed$

######  Example 3.3.

The solution $u(t,x)=\operatorname{sign}x\cdot\sqrt{|x|}$ to
$\partial_{t}u+\partial_{x}u^{2}=\operatorname{sign}x$ has Hölder constant
$\sqrt{2}=\left(\frac{2\|g\|_{L^{\infty}}}{qc_{\ell}}\right)^{\frac{1}{\ell}}$.

###### Proof of Proposition 3.

Given any compact set $K\subset\Omega$ whose $t$-sections are intervals, set
$J=u(K)$. By Proposition A the set $Z=\\{v\in J\ :\ f^{\prime\prime}(v)=0\\}$
is finite: set

$\overline{\mathtt{c}}=\frac{\max\\{|w-v|\ :\ w,v\in J\\}}{\max\\{|w-v|\ :\
B_{\delta}(Z)\cap[v,w]=\emptyset\,,\ w,v\in
J\\}}\qquad\delta=\frac{1}{3}\min\\{|v-w|\ :\ v,w\in Z\\}\,.$

Observe that for every $x<y$ it is possible to find $x_{1},x_{2}\in[x,y]$ such
that $|u(t,x)-u(t,y)|\leq\bar{c}|u(t,x_{1})-u(t,x_{2})|$ and
$B_{\delta}(Z)\cap[x_{1},x_{2}]=\emptyset$, so that for some $c>0$111Doing a
rough estimate, there is not point in looking for constants better than
$\mathtt{c}=\sqrt[\ell]{\frac{\min_{J\setminus
B_{\delta}(Z)}|f^{\prime\prime}|}{\max_{J}|f^{\prime\prime}|}\frac{4G}{c_{\ell}}}$.
by Lemma 3 we have

$|u(t,x)-u(t,y)|\leq\overline{\mathtt{c}}|u(t,x_{1})-u(t,x_{2})|\leq\overline{\mathtt{c}}\mathtt{c}|x_{1}-x_{2}|^{\frac{1}{\ell}}\leq\overline{\mathtt{c}}\mathtt{c}|y-x|^{\frac{1}{\ell}}\,.$

We thus proved that $u$ is $\frac{1}{\ell}-$Hölder in space on compact sets
$K$ whose $t$-sections are intervals. In order to recover the regularity in
time, recall that $u$ is $\|g\|_{L^{\infty}}-$Lipschitz along characteristics
by Remark 2: as in at Step 2 in the proof of [14, Corollary 2.2] we get
$\frac{1}{\ell}-$Hölder continuity on subsets of $\Omega$ of the form
$\\{(t,x)\in[t_{1},t_{2}]\times\mathbb{R}\ :\ \gamma_{1}(t)\leq
x\leq\gamma_{2}(t)\\}$, with $\gamma_{1}$, $\gamma_{2}$ characteristics. Since
the interior of such characteristic regions cover any compact subset of
$\Omega$ we get the thesis. ∎

######  Remark 3.4.

By looking at the proof of Proposition 3 we have that the $\ell$-Hölder
constant of $u$ on a rectangle $K$ depends only on $f$ restricted to $u(K)$.
More precisely it depends on $c_{\ell}$ and on the minimum distance between
zeros of $f^{\prime\prime}$.

##  4\. Finer Hölder regularity

The main result of this section is the following:

######  Proposition 4.1.

Let $f\in C^{2}$ be nonlinear of order $\ell$ as in Definition A. Let
$\Omega\subset\mathbb{R}^{2}$ and $u\in C^{0}(\Omega)$, $g\in
L^{\infty}(\Omega)$ be such that

$\partial_{t}u+[f(u)]_{x}=g\qquad\mbox{in }\mathcal{D}^{\prime}(\Omega).$

Then for $\mathscr{L}^{2}-$a.e. $(t,x)\in\Omega$ it holds

$A(t,x)\doteq\limsup_{y\to x}\frac{|u(t,y)-u(t,x)|}{\sqrt[\ell]{|y-x|}}=0.$
(4.1)

By Remark 3, the function $A$ is locally essentially bounded.

We will prove (4.1) for points $(t,x)$ which are Lebesgue points of $g$ with
respect to the following suitable coverings of $\Omega$. Such covering was
first introduced in [14] for proving that for $\mathscr{L}^{2}-$a.e. $(t,x)$
the derivative of $\sigma\mapsto u(\sigma,\gamma(\sigma))$ along any
characteristic $\gamma$ through $(t,x)$ is the Lebesgue value of $g$: for
being self-contained, we prove this again in § 5 (Case 2) for the quadratic
flux; the same argument works for $\ell$-nonlinear fluxes by the estimates in
this section.

Given $\varrho>0$, define regions translating a characteristic $\gamma$ of
${{\mathfrak{h}}}=\varrho\varepsilon^{\ell}$, between times
$\sigma\pm\varepsilon$:

$\begin{split}\mathcal{V}_{\varrho}\doteq&\leavevmode\nobreak\
\left\\{S_{\varrho}(\gamma,\varepsilon,\sigma)\
\big{|}\,\sigma\in\mathbb{R},\,\varepsilon>0,\,\gamma\in
C^{1}((\sigma-\varepsilon,\sigma+\varepsilon)\,:\,\dot{\gamma}(t)=f^{\prime}(u(t,\gamma(t)))\right\\}\,,\\\
S_{\varrho}(\gamma,\varepsilon,\sigma)\doteq&\leavevmode\nobreak\
\left\\{(t,x)\in[\sigma-\varepsilon,\sigma+\varepsilon]\times\mathbb{R}\ :\
\gamma(t)-\varrho\varepsilon^{\ell}\leq
x\leq\gamma(t)+\varrho\varepsilon^{\ell}\right\\}.\end{split}$ (4.2)

######  Proposition 4.2.

Consider any sequence $\\{\varrho_{j}\\}_{j\in\mathbb{N}}$ and
$\mathcal{V}_{\varrho_{j}}$ as in (4.2). For every $q\in L^{\infty}(\Omega)$
a.e. $(t,x)\in\Omega$ is a Lebesgue point of $q$ with respect to the covering
$\mathcal{V}_{\varrho_{j}}$, namely there is $O\subset\Omega$ with
$\mathscr{L}^{2}(\Omega\setminus O)=0$ such that for every $(t,x)\in O$ and
every $j\in\mathbb{N}$ it holds

$\lim_{\begin{subarray}{c}(t,x)\in S\in\mathcal{V}_{\varrho_{j}}\\\
\operatorname{diam}(S)\downarrow
0\end{subarray}}\frac{1}{\mathscr{L}^{2}(S)}\int_{S}|q-q(t,x)|dxdt=0.$

The proof of the above proposition can be found in [14, § 3.1]: the key point
is that one can define through the sets
$S_{\varrho}(\gamma,\varepsilon,\sigma)$ a quasi-distance for which the
measure $\mathscr{L}^{2}$ is doubling so that Lebesgue differentiation theorem
applies.

We are now in position to prove Proposition 4.

###### Proof of Proposition 4.

Since $u\in C^{0,\alpha}_{\mathrm{loc}}(\Omega)$ by Proposition 3, for any
$c\in\mathbb{R}$ it holds $A(t,x)=0$ at every $(t,x)$ which is a point of
density one of the set $u^{-1}(\\{c\\})$. Since $Z=\\{v\ :\
f^{\prime\prime}(v)=0\\}$ is discrete by Lemma A, the statement thus holds for
$\mathscr{L}^{2}$-a.e. $(t,x)\in u^{-1}(Z)$.

We now prove the claim for $\mathscr{L}^{2}$-a.e. $(t,x)\in\Omega\setminus
u^{-1}(Z)$, namely such that $f^{\prime\prime}(u(t,x))\neq 0$. Given two
sequences $\varrho_{j},\delta_{j}\downarrow 0$, $\varrho_{j}+\delta_{j}<1$, we
consider the coverings $\mathcal{V}_{\varrho_{j}}$ in (4.2) and we set

$r_{j}=\frac{\varrho_{j}^{\ell}}{1+\delta_{j}}\quad\text{and denote\
}\varepsilon=\varepsilon_{j}({{\mathfrak{h}}})=\frac{\sqrt[\ell]{(1+\delta_{j}){{\mathfrak{h}}}}}{\varrho_{j}}\text{
for }{{\mathfrak{h}}}>0\,,$

so that $r_{j}\varepsilon^{\ell}={{\mathfrak{h}}}$ and
$\varrho_{j}\varepsilon^{\ell}=\varrho_{j}^{-(\ell-1)}(1+\delta_{j}){{\mathfrak{h}}}$.
We prove that $A(t,x)=0$ for every $(t,x)\in\Omega\setminus u^{-1}(Z)$ which
is a Lebesgue point of $g$ with respect to each covering
$\mathcal{V}_{\varrho_{j}}$ for $j\in\mathbb{N}$. Proposition 4 grants that
such points have full Lebesgue measure in $\Omega\setminus u^{-1}(Z)$.

Up to a translation we may assume that $(t,x)=(0,0)$: let $x_{k}\to 0$ be a
sequence such that
$A(0,0)=\lim_{k\to\infty}\frac{|u(0,x_{k})-u(0,0)|}{\sqrt[\ell]{|x_{k}|}}$. Up
to a symmetry with respect to the axis $x=0$, and up to repeating the
following argument for negative times, is not restrictive to consider the case
for which $x_{k}\downarrow 0$, $u(0,x_{k})<u(0,0)$ and
$f^{\prime\prime}(u(0,0))>0$. Denote by $\gamma_{0}$ and $\gamma_{k}$ two
characteristics such that $\gamma_{0}(0)=0$ and $\gamma_{k}(0)=x_{k}$. Set
$u_{0}(t)=u(t,\gamma_{0}(t))$ and $u_{k}(t)=u(t,\gamma_{k}(t))$. Let us fix
$j\in\mathbb{N}$ and denote by $S_{k}$ the set
$S_{\varrho_{j}}(\gamma_{0},\varepsilon_{j}(x_{k}),0)$. With the standard
convention $\inf\emptyset=+\infty$, set

$t_{k}^{*}=\varepsilon_{j}(x_{k})\wedge\inf\left\\{t>0\ \ :\ u_{0}(t)\leq
u_{k}(t)\right\\}\,.$

Notice that $t^{*}_{k}>0$, being $u$ continuous and $u_{k}(0)<u_{0}(0)$. Since
$x_{k},t^{*}_{k}\to 0$ as $k\to\infty$ and $u$ is continuous, there is
$\bar{k}$ such that for every $k>\bar{k}$ the function $u$ restricted to
$S_{k}$ takes values in $u^{-1}(\\{f^{\prime\prime}>0\\})$. We will assume
$k\geq\bar{k}$ in the following of the proof: thus

$\dot{\gamma}_{0}(t)=f^{\prime}(u_{0}(t))\geq
f^{\prime}(u_{k}(t))=\dot{\gamma}_{k}(t)\qquad\text{in $[0,t_{k}^{*}]$}\,$

and $\gamma_{k}\leq\gamma_{0}+x_{k}$ by the initial condition, so that by the
choice $\varrho_{j}\varepsilon^{\ell}>(1+\delta_{j}){{\mathfrak{h}}}$ we have

$\\{(t,x)\in[0,t^{*}_{k}]\times\mathbb{R}:\gamma_{0}(t)-\delta_{j}x_{k}\leq
x\leq\gamma_{k}(t)+\delta_{j}x_{k}\\}\subset S_{k}\,.$ (4.3)

If $t\in[0,t_{k}^{*}]$, Proposition 2 gives the one sided estimates

$\displaystyle\int_{\gamma_{k}(t)}^{\gamma_{k}(t)+\delta_{j}x_{k}}u(t,x)dx-\int_{x_{k}}^{x_{k}+\delta_{j}x_{k}}u(0,x)dx\leq$
$\displaystyle\int_{0}^{t}\int_{\gamma_{k}(t)}^{\gamma_{k}(t)+\delta_{j}x_{k}}g(s,x)dxds\,,$
(4.4a)
$\displaystyle\int^{\gamma_{0}(t)}_{\gamma_{0}(t)-\delta_{j}x_{k}}u(t,x)dx-\int^{0}_{-\delta_{j}x_{k}}u(0,x)dx\geq$
$\displaystyle\int_{0}^{t}\int_{\gamma_{0}(t)-\delta_{j}x_{k}}^{\gamma_{0}(t)}g(s,x)dxds\,.$
(4.4b)

Denote by $\mathtt{c}$ the $\frac{1}{\ell}$-Hölder constant of $u$ restricted
to $S_{\bar{k}}$. Then when $\\{t\\}\times[x-r,x+r]\subset S_{k}$

$\int_{x}^{x+r}\left|u(t,q)-u(t,x)\right|\,dq\leq\mathtt{c}r^{\frac{\ell+1}{\ell}},\qquad\int_{x-r}^{x}\left|u(t,q)-u(t,x)\right|\,dq\leq\mathtt{c}r^{\frac{\ell+1}{\ell}}.$
(4.5)

By (4.3) and by Proposition 4 in particular for every $t\in[0,t^{*}_{k}]$
there is $\eta_{k}\downarrow 0$ such that

$\left|\int_{0}^{t}\int_{\gamma_{k}(t)}^{\gamma_{k}(t)+\delta_{j}x_{k}}g(s,x)dxds-t\delta_{j}x_{k}g(0,0)\right|\leq\eta_{k}\mathscr{L}^{2}(S_{k})\,.$
(4.6)

Subtracting equations (4.4) and dividing by $\delta_{j}x_{k}$, from (4.3),
(4.5) and (4.6) we deduce that

$\begin{split}u_{0}(t)-u_{k}(t)&\geq
u_{0}(0)-u_{k}(0)-2\frac{\eta_{k}\mathscr{L}^{2}(S_{k})}{\delta_{j}x_{k}}-2\mathtt{c}\sqrt[\ell]{\delta_{j}x_{k}}\qquad\text{for
every $t\in[0,t^{*}_{k}]$.}\end{split}$ (4.7)

### Case 1: there is $t\in[0,t^{*}_{k}]$ such that
$u_{0}(t)-u_{k}(t)\leq\frac{1}{2}(u_{0}(0)-u_{k}(0))$.

From (4.7) it holds

$\begin{split}\frac{u(0,0)-u(0,x_{k})}{\sqrt[\ell]{x_{k}}}\leq&\leavevmode\nobreak\
4\frac{\mathscr{L}^{2}(S_{k})}{\delta_{j}x_{k}\sqrt[\ell]{x_{k}}}\eta_{k}+4\mathtt{c}\frac{\sqrt[\ell]{\delta_{j}x_{k}}}{\sqrt[\ell]{x_{k}}}\\\
=&\leavevmode\nobreak\
\frac{16(1+\delta_{j})^{\frac{\ell+1}{\ell}}}{\delta_{j}\varrho_{j}^{\ell}}\eta_{k}+4\mathtt{c}\sqrt[\ell]{\delta_{j}}.\end{split}$
(4.8)

### Case 2: for every $t\in[0,t^{*}_{k}]$ it holds
$u_{0}(t)-u_{k}(t)>\frac{1}{2}(u_{0}(0)-u_{k}(0))$.

In particular we have
$t^{*}_{k}=\frac{\sqrt[\ell]{(1+\delta_{j})x_{k}}}{\varrho_{j}}$ and
$\gamma_{k}$, $\gamma_{0}$ do not intersect in $[0,t^{*}_{k}]$. By Lemma A,
for every $t\in[0,t^{*}_{k}]$ it holds

$\begin{split}\gamma_{0}^{\prime}(t)-\gamma_{k}^{\prime}(t)=&\leavevmode\nobreak\
f^{\prime}(u_{0}(t)-f^{\prime}(u_{k}(t))\\\ \geq&\leavevmode\nobreak\
2c_{\ell}\left(u_{0}(t)-u_{k}(t)\right)^{\ell-1}\geq\frac{c_{\ell}}{2^{\ell-2}}(u_{0}(0)-u_{k}(0))^{\ell-1}\end{split}$

Integrating with respect to time in $[0,t^{*}_{k}]$ and imposing that
$\gamma_{0}(t^{*}_{k})\leq\gamma_{k}(t^{*}_{k})$ we obtain

$0\geq\gamma_{0}(t^{*}_{k})-\gamma_{k}(t^{*}_{k})\geq\gamma_{0}(0)-\gamma_{k}(0)+\frac{c_{\ell}}{2^{\ell-1}}(u_{0}(0)-u_{k}(0))^{\ell-1}t^{*}_{k}.$

Recalling that $\gamma_{0}(0)=0$ and $\gamma_{k}(0)=x_{k}$ this implies

$\frac{u(0,0)-u(0,x_{k})}{\sqrt[\ell]{x_{k}}}\leq
2\left(\frac{x_{k}}{t^{*}_{k}c_{\ell}}\right)^{\frac{1}{\ell-1}}\frac{1}{\sqrt[\ell]{x_{k}}}=2\left(\frac{\varrho_{j}}{c_{\ell}\sqrt[\ell]{1+\delta_{j}}}\right)^{\frac{1}{\ell-1}}$
(4.9)

Taking first the limit as $k\to\infty$, where $\eta_{k}$ vanishes, (4.8) and
(4.9) show that $A(0,0)=0$ because $\delta_{j}$ and $\varrho_{j}$ can be taken
arbitrarily small. ∎

##  5\. A new proof of the Rademacher theorem for intrinsic Lipschitz
functions

Aim of this section is to provide a new proof of the Rademacher theorem for
intrinsic Lipschitz functions in the first Heisenberg group $\mathbb{H}^{1}$.
We start by recalling the main definitions, we refer the interested reader to
[36] for a complete introduction to the subject.

We denote the points of
$\mathbb{H}^{1}\equiv\mathbb{C}\times\mathbb{R}\equiv\mathbb{R}^{3}$ by

$P=[z,t]=[x+iy,t]=(x,y,t),\qquad z\in\mathbb{C},\ x,y\in\mathbb{R},\
t\in\mathbb{R}.$

If $P=[z,t]$, $Q=[z^{\prime},t^{\prime}]\in\mathbb{H}^{1}$ and $r>0$, the
group operation reads as

$P\cdot
Q:=\left[z+z^{\prime},t+t^{\prime}-\frac{1}{2}\operatorname{Im}(\langle
z,\bar{z^{\prime}}\rangle)\right].$ (5.1)

The group identity is the origin $0$ and one has $[z,t]^{-1}=[-z,-t]$. In
$\mathbb{H}^{1}$ there is a natural one parameter group of non isotropic
dilations $\delta_{r}(P):=[rz,r^{2}t]$ , $r>0$. The group $\mathbb{H}^{1}$ can
be endowed with the homogeneous norm

$\|P\|_{\infty}:=\max\\{|z|,|t|^{1/2}\\}$

and with the left-invariant and homogeneous distance

$d_{\infty}(P,Q):=\|P^{-1}\cdot Q\|_{\infty}.$

The metric $d_{\infty}$ is equivalent to the standard Carnot-Carathéodory
distance. It follows that the Hausdorff dimension of
$(\mathbb{H}^{1},d_{\infty})$ is $4$, whereas its topological dimension is
clearly $3$. The Lie algebra $\mathfrak{h}_{1}$ of left invariant vector
fields is (linearly) generated by

$X=\partial_{x}-\frac{1}{2}y\partial_{t},\quad
Y=\partial_{y}+\frac{1}{2}x\partial_{t},\quad T=\partial_{t}$

and the only nonvanishing commutators are

$[X,Y]=T.$

In the spirit of [6] we set $\mathbb{W}:=\\{(x,y,t)\in\mathbb{H}^{1}:\
x=0\\}\equiv\mathbb{R}^{2}$. Therefore, if $A\in\mathbb{W}$, we write
$A=(y,t)$.

Following [6] we define the graph quasidistance as:

######  Definition 5.1.

For $A=(y,t),\,B=(y^{\prime},t^{\prime})\in\omega$ we define

$d_{\varphi}(A,B)=|y-y^{\prime}|+\left|t^{\prime}-t-\frac{1}{2}(\varphi(A)+\varphi(B))(y^{\prime}-y)\right|^{1/2}.$

An intrinsic differentiable structure can be induced on $\mathbb{W}$ by means
of $d_{\varphi}$, see [6]. We remind that a map $L:\mathbb{W}\to\mathbb{R}$ is
$\mathbb{W}$-linear if it is a group homeomorphism and $L(ry,r^{2}t)=rL(y,t)$
for all $r>0$ and $(y,t)\in\mathbb{W}$. We recall then the notion of
$\varphi$-differentiablility.

######  Definition 5.2.

Let $\varphi:\omega\subset\mathbb{W}\rightarrow\mathbb{R}$ be a fixed
continuous function, and let $A_{0}\in\omega$ and
$\psi:\omega\rightarrow\mathbb{R}$ be given. We say that $\psi$ is
$\varphi$-differentiable at $A_{0}$ if there is an $\mathbb{W}$-linear
functional $L:\mathbb{W}\rightarrow\mathbb{R}$ such that

$\lim_{A\rightarrow A_{0}}\frac{\psi(A)-\psi(A_{0})-L(A_{0}^{-1}\cdot
A)}{\mathop{}\\!\mathrm{d}_{\varphi}(A_{0},A)}=0.$ (5.2)

It is well known [6] that given a $\mathbb{W}$-linear functional
$L:\mathbb{W}\rightarrow\mathbb{R}$ there exists a unique
$\hat{w}\in\mathbb{R}$ such that $L(A)=L((y,t))=\hat{w}y$. Let us now
introduce the concept of intrinsic Lipschitz function.

######  Definition 5.3.

Let $\varphi:\omega\subset\mathbb{W}\to\mathbb{R}$. We say that $\varphi$ is
an intrinsic Lipschitz continuous function and we write $\varphi\in
Lip_{\mathbb{W}}(\omega)$ if there exists a constant $L>0$ such that

$|\varphi(A)-\varphi(B)|\leq Ld_{\varphi}(A,B)\quad\forall A,B\in\omega.$

We say that $\varphi$ is locally intrinsic Lipschitz and we write $\varphi\in
Lip_{\mathbb{W},loc}(\omega)$ if $\varphi\in
Lip_{\mathbb{W},loc}(\omega^{\prime})$ for any $\omega^{\prime}\Subset\omega$.

The following characterization is proved in [11].

######  Theorem 5.4.

Let $\omega\subset\mathbb{R}^{2}$ be an open set and let
$\varphi:\omega\to\mathbb{R}$ be a continuous. Then $\varphi\in
Lip_{\mathbb{W},loc}(\omega)$ if and only if there exists $g\in
L^{\infty}(\omega)$ such that

$\varphi_{y}+[\varphi^{2}/2]_{t}=g\quad\mbox{in}\
\mathcal{D}^{\prime}(\omega).$

Let us finally recall the following Rademacher type Theorem, proved in [24]
(see also [37]). As recalled in the Introduction the original proof requires
deep results in geometric measure theory, namely the fact that the subgraph of
an intrinsic Lipschitz function $\varphi$ is a set with locally finite
$\mathbb{H}$-perimeter and that at almost every point of the graph of
$\varphi$ there is an approximate. Here instead we propose a completely PDEs
based proof which uses the results of Section 4.

######  Theorem 5.5.

If $\varphi\in Lip_{\mathbb{W}}(\omega)$ then $\varphi$ is
$\varphi$-differentiable $\mathcal{L}^{2}-$a.e. in $\omega$.

More precisely, we prove $\varphi$-differentiability at those points which are
Lebesgue points of $g$ for all the coverings (4.2), for any fixed sequence
$\rho_{i}$ decreasing to 0.

###### Proof.

Let $A=(t,x)$ be a Lebesgue point of $g$ with respect to the coverings
$\mathcal{V}_{\varrho_{j}}$ of Lemma 4. We show that any sequence
$A_{n}=(t_{n},x_{n})\to A_{0}$ has a (not relabeled) subsequence for which

$\lim_{n\to\infty}\frac{|\varphi(A_{n})-\varphi(A_{0})-g(0,0)t_{n}|}{d_{\varphi}(A_{n},A_{0})}=0.$

We recall the explicit expression
$d_{\varphi}(A_{n},A_{0})=|t_{n}-t|+\sqrt{\left|x_{n}-x-\frac{\varphi(A_{n})+\varphi(A_{0})}{2}\cdot(t_{n}-t)\right|}$.

For simplicity of notation, assume $A_{0}=(0,0)$,
$t_{n},x^{\prime}_{n}\in(0,\frac{1}{1+\|g\|_{L^{\infty}}})$. Let
$\gamma_{n}\in C^{1}([0,t_{n}])$ be a characteristic of $\varphi$ through
$(t_{n},x_{n})$ and set $A^{\prime}_{n}=(0,\gamma_{n}(0))=(0,x^{\prime}_{n})$:
by Remark (2)

$\left|x_{n}-\frac{\varphi(A_{n})+\varphi(A_{0})}{2}\cdot
t_{n}-x^{\prime}_{n}\right|=\left|\int_{0}^{t_{n}}\varphi(s,\gamma(s))\,ds-\frac{\varphi(A_{n})+\varphi(A_{0})}{2}\cdot
t_{n}\right|\leq\frac{\|g\|_{L^{\infty}}}{4}t_{n}^{2}\,.$ (5.3)

Up to a (not relabeled) subsequence one of the following two cases is
satisfied.

###  Case 1: it holds
$\lim_{n\to\infty}\frac{|t_{n}|}{\sqrt{|x^{\prime}_{n}|}}=0$.

In this case it follows from Proposition 4 and Remark 2 that:

$\begin{split}|\varphi(A_{n})-\varphi(A_{0})-g(0,0)t_{n}|\leq&\leavevmode\nobreak\
\|g\|_{L^{\infty}}|t_{n}|+|\varphi(A_{n})-\varphi(A^{\prime}_{n})|+|\varphi(A^{\prime}_{n})-\varphi(A_{0})|\\\
\leq&\leavevmode\nobreak\
2\|g\|_{L^{\infty}}t_{n}+o\left(\sqrt{x^{\prime}_{n}}\right)=o\left(\sqrt{x^{\prime}_{n}}\right)\,.\end{split}$

We conclude since (5.3) and $t_{n}=o(\sqrt{x^{\prime}_{n}})$ imply
$d_{\varphi}(A_{n},A_{0})=\sqrt{x^{\prime}_{n}}\cdot(1+o(\sqrt{x^{\prime}_{n}}))$.

###  Case 2: there is $\delta\in(0,1)$ such that
$\frac{|t_{n}|}{\sqrt{|x^{\prime}_{n}|}}\geq\delta>0$.

Let $\varepsilon\in(0,1)$ and consider
$S_{n}^{\varepsilon}=\\{(s,x)\in[0,t_{n}]\times\mathbb{R}:\gamma(s)-\varepsilon
t_{n}^{2}\leq x\leq\gamma(s)-\varepsilon t_{n}^{2}\\}$. Denote by $\mathtt{c}$
the $\frac{1}{2}-$Hölder constant of $\varphi$ in a ball $B$ centered at
$A_{0}$. For $n$ sufficiently large $A_{n},A^{\prime}_{n}\in B$: by estimates
like (4.4)-(4.5) we get

$|\varphi(A_{n})-\varphi(A_{0})-g(0,0)t_{n}|\leq
4\mathtt{c}\sqrt{\varepsilon}t_{n}+\frac{1}{2\varepsilon
t_{n}^{2}}\int_{S_{n}^{\varepsilon}}|g-g(0,0)|.$ (5.4)

Just for notational convenience, suppose $\varrho_{0}=1$. Since
$S_{n}^{\varepsilon}\subset
S_{1}\left(\gamma_{n},t_{n}\sqrt{\frac{1}{\delta^{2}}+\varepsilon},0\right)\in\mathcal{V}_{1}$
and $(0,0)$ is a Lebesgue point of $g$ with respect to the covering
$\mathcal{V}_{1}$, then

$\eta_{n}\doteq\frac{1}{\mathscr{L}^{2}\left(S_{1}\left(\gamma_{n},t_{n}\sqrt{\frac{1}{\delta^{2}}+\varepsilon},0\right)\right)}\int_{S_{n}^{\varepsilon}}|g-g(0,0)|$

vanishes as $n\to\infty$. The measure of the set is
$4\left({\frac{1}{\delta^{2}}+\varepsilon}\right)^{\frac{3}{2}}t_{n}^{3}$. It
follows from (5.4) that

$\begin{split}\frac{|\varphi(A_{n})-\varphi(A_{0})-g(0,0)t_{n}|}{t_{n}}\leq&\leavevmode\nobreak\
4\mathtt{c}\sqrt{\varepsilon}+\frac{\mathscr{L}^{2}\left(S_{1}\left(\gamma_{n},t_{n}\sqrt{\frac{1}{\delta^{2}}+\varepsilon},0\right)\right)}{2\varepsilon
t_{n}^{3}}\eta_{n}\\\ =&\leavevmode\nobreak\
4\mathtt{c}\sqrt{\varepsilon}+\frac{2\left(\frac{1}{\delta^{2}}+\varepsilon\right)^{\frac{3}{2}}}{\varepsilon}\eta_{n}.\end{split}$

Recalling that $|t_{n}|\leq d_{\varphi}(A_{n},A_{0})$, the claim follows by
letting $n\to\infty$ since $\varepsilon$ is arbitrarily small. ∎

##  Appendix A Nonlinear functions of a given order

We specify the assumption of finite order nonlinearity, for brevity also
called $\ell$-nonlinearity.

######  Definition A.1.

Let $I\subset\mathbb{R}$ be an interval. A function $f$ differentiable at $v$
is “nonlinear of order $\ell>1$ with constant $c>0$ at $v$” if there exists
$\delta>0$ such that for every $v+h\in I\cap B_{\delta}(v)$ one has

$\left|f(v+h)-f(v)-f^{\prime}(v)h\right|\geq c\left|h\right|^{\ell}\,.$

If the inequality holds for all $v,v+h\in I$ we call $f$ “nonlinear in $I$ of
order $\ell$ with constant $c$”.

Of course, if $f$ is nonlinear of order $\ell>1$ with constant $c>0$ at $v$
then it is also nonlinear at $v$ of every order $\ell^{\prime}\geq\ell$ with
any constant $0<c^{\prime}\leq c$ just because $|h|^{\ell^{\prime}-\ell}\leq
1$ when $0<|h|\leq\delta<1$.

######  Remark A.2.

The best order of nonlinearity at a point $v$ is the order $\ell$ of the first
non-vanishing derivative at the point, higher than the first. It can be proved
just applying the definition of differentiability $\ell$-times recursively,
and the constant is arbitrarily close to $\frac{1}{\ell!}|f^{(\ell)}(v)|$.

######  Lemma A.3.

Suppose $f\in C^{\ell}(I)$. If
$\sum_{k=2}^{\ell}\frac{1}{k!}\left|f^{(k)}\right|\geq c$ in the interval $I$
and $I$ has length $d$ then $f$ is nonlinear in $I$ of order $\ell$ with
constant $c\cdot\min\\{d^{2-\ell},1\\}$. Moreover, if $k\in\mathbb{N}$ is the
minimum exponent of nonlinearity of $f$ at $v$ with $c_{0}$ the infimum of the
relative constants, then $f^{(k)}(v)=k!c_{0}$ and $f^{(j)}(v)=0$ for
$j=2,\dots,k-1$.

###### Proof.

Let $v,v+h\in I$. Just by Taylor’s expansion, if $f^{(k)}(v)=0$ for $2\leq
k<j\leq\ell$ one has

$\left|f(v+h)-f(v)-f^{\prime}(v)h\right|=\left|\sum_{k=2}^{j-1}\frac{1}{k!}f^{(k)}(v)h^{k}+\frac{1}{j!}f^{(j)}(\xi)h^{j}\right|\geq
c|h|^{j}\geq\frac{c}{d^{\ell-j}}|h|^{\ell}\,,$

thanks to the assumtion $\frac{1}{j!}\left|f^{(j)}\right|\geq c$ and that $K$
has diameter $d$, so that $|h|^{\ell-j}\leq d^{\ell-j}$.

Suppose now that $k$ is the minimum exponent of nonlinearity of $f$ at $v$: by
the first part of the statement in particular $f^{(j)}(v)=0$ for
$j=2,\dots,k-1$ and that $|f^{(k)}(v)|\leq c_{0}k!$. If

$\left|f(v+h)-f(v)-f^{\prime}(v)h\right|\geq(c_{0}-\varepsilon)|h|^{\ell}\qquad\forall\varepsilon>0\,,\
|h|\leq\delta$

then is $\xi$ between $v$ and $v+h$ such that
$f(v+h)-f(v)-f^{\prime}(v)h=\frac{f^{(k)}}{k!}(\xi)$, thus $|f^{(k)}(v)|\geq
c_{0}k!$. ∎

Nonlinearity provides a lower bound on the increments also of the first
derivative of $f$:

######  Lemma A.4.

Suppose $f$ is nonlinear in an interval $I$ of order $\ell$ with constant $c$.
Then

$\left|f^{\prime}(v)-f^{\prime}(w)\right|\geq 2c|v-w|^{\ell-1}\qquad\forall
v,w\in I\,.$ (A.1)

###### Proof.

Just by Definition (A) of nonlinearity when $h=w-v$ we have

$\left|f^{\prime}(w)h-f^{\prime}(v)h\right|=\left|f(v+h)-f(v)-f^{\prime}(v)h+f(w-h)-f(w)+f^{\prime}(w)h\right|\geq
2c|h|^{\ell}\,.\qed$

######  Lemma A.5.

If $f\in C^{2}(J)$ is nonlinear of order $\ell$ and constant $c$ in a compact
interval $J$, then:

  1. (1)

When $\ell=2$ the second derivative of $f$ has lower bound $2c$ in $J$ and it
holds

${|f^{\prime}(v)-f^{\prime}(w)|}{}\geq
q_{w,v}\cdot\|f^{\prime\prime}\|_{L^{\infty}(J)}\cdot|v-w|\,,$ (A.2)

for all $v,w\in J$, with
$q_{w,v}\doteq\frac{2c}{\|f^{\prime\prime}\|_{L^{\infty}(J)}}$ belonging in
$(0,1]$ and independent on $w,v$.

  2. (2)

When $\ell>2$ the set $Z=\\{v\in J:f^{\prime\prime}(v)=0\\}$ is finite and
inequality (A.2) holds whenever $[w,v]\cap Z=\emptyset$ with
$q_{w,v}\doteq\frac{\min_{[w,v]}|f^{\prime\prime}|}{\max_{J}|f^{\prime\prime}|}$
belonging in $(0,1]$.

###### Proof.

  1. (1)

When $\ell=2$ the statement follows immediately by (A.1).

  2. (2)

At any accumulation point $\overline{v}$ of $D$ necessarily
$f^{(j)}(\bar{v})=0$ for every $j\geq 2$, thus by Taylor’s expansion we would
have
$f(\overline{v}+h)-f(\overline{v})-f^{\prime}(\overline{v})h=o(h^{\ell})$:
this would contradict the finite order of nonlinearity of $f$. In particular,
also when $\ell>2$ the set $D$ must be discrete.

If $[u,v]\cap K=\emptyset$ then $f$ is nonlinear of order $2$ in $[u,v]$ and
${|f^{\prime}(v)-f^{\prime}(w)|}=|f^{\prime\prime}(\xi)||v-w|$ with
$\xi\in(v,w)$. Writing
$|f^{\prime\prime}(\xi)|\geq\min_{[u,v]}|f^{\prime\prime}|\cdot\frac{\max_{J}|f^{\prime\prime}|}{\max_{J}|f^{\prime\prime}|}$
one has the claim. ∎

## References

  * [1] S. Adimurthi, S. S. Ghoshal, G. D. Veerappa Gowda.  Finer regularity of an entropy solution for 1-d scalar conservation laws with non uniform convex flux.  Rend. Semin. Mat. Univ. Padova, 132, 1–24, 2014.
  * [2] G. Alberti, S. Bianchini, L. Caravenna.  Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux I.  Journal of Differential Equations, 261(8): 4298–4337, 2016.
  * [3] G. Alberti, S. Bianchini, L. Caravenna. Eulerian, Lagrangian and Broad continuous solutions to a balance law with non convex flux II. http://preprints.sissa.it/xmlui/handle/1963/35197
  * [4] L. Ambrosio, C. De Lellis.  A note on admissible solutions of 1d scalar conservation laws and 2d Hamilton–Jacobi equations.  J. Hyperbolic Differ. Equ., 1(4):813–826, 2004.
  * [5] L. Ambrosio, B. Kleiner, and E. Le Donne. Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane. J. Geom. Anal., 19(3):509–540, 2009.
  * [6] L. Ambrosio, F. Serra Cassano, and D. Vittone. Intrinsic regular hypersurfaces in Heisenberg groups. J. Geom. Anal. 16(2): 187–232, 2006.
  * [7] G. Antonelli and A. Merlo. Intrinsically Lipschitz functions with normal target in Carnot groups. Annales Fennici Mathematici, 46 (1) 571–579, 2021
  * [8] G. Antonelli and A. Merlo. On rectifiable measures in Carnot groups: structure theory. arXiv e-prints, page arXiv:2009.13941, 2020.
  * [9] S. Bianchini, L. Caravenna.  SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one Space Dimension.  Commun. Math. Phys., 313:1–33, 2012.
  * [10] S. Bianchini, C. De Lellis, R. Robyr.  SBV Regularity for Hamilton–Jacobi Equations in $\mathbb{R}^{n}$.  Arch Rational Mech Anal, 200, 1003–1021, 2011.
  * [11] F. Bigolin, L. Caravenna, and F. Serra Cassano. newblock Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation.  Ann. Inst. H. Poincaré C Anal. Non Linéaire 32, no. 5, 925–963, 2015.
  * [12] F. Bigolin, F. Serra Cassano.  Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs,  Adv. Calc. Var. 3:69–97, 2010.
  * [13] C. Bourdarias, M. Gisclon, and S. Junca.  Fractional $BV$ spaces and applications to scalar conservation laws.  J. Hyperbolic Differ. Equ., 11(4):655–677, 2014.
  * [14] L. Caravenna.  Regularity estimates for continuous solutions of $\alpha$-convex balance laws  Communications on Pure and Applied Analysis, 16(2): 629–644, 2017.
  * [15] G. Citti, M. Manfredini, A. Pinamonti and F. Serra Cassano.  Smooth approximation for intrinsic Lipschitz functions in the Heisenberg group  Calc. Var. Partial Differential Equations, no. 3-4, 1279–1308, 2014
  * [16] G. Citti, M. Manfredini, A. Pinamonti and F. Serra Cassano.  Poincaré-type inequality for Lipschitz continuous vector fields  J. Math. Pures Appl., (9) 105, no. 3, 265–292, 2016.
  * [17] K. S. Cheng.  A regularity theorem for a nonconvex scalar conservation law.  J. Differential Equations, 61(1):79–127, 1986.
  * [18] M. Cirant, G. Verzini.  Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms.  Advances in Mathematics, 409, 2022.
  * [19] C. M. Dafermos.  Regularity and large time behaviour of solutions of a conservation law without convexity.  Proc. Roy. Soc. Edinburgh Sect. A, 99(3-4):201–239, 1985.
  * [20] C. M. Dafermos.  Continuous solutions for balance laws.  Ric. Mat., 55(1):79–91, 2006.
  * [21] C. M. Dafermos.  Hyperbolic conservation laws in continuum physics, volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, fourth edition, 2016.
  * [22] B. Franchi, M. Marchi, and R. P. Serapioni. Differentiability and approximate differentiability for intrinsic Lipschitz functions in Carnot groups and a Rademacher theorem. Anal. Geom. Metr. Spaces, 2:258–281, 2014.
  * [23] B. Franchi, R. Serapioni, and F. Serra Cassano. Intrinsic Lipschitz graphs in Heisenberg groups. J. Nonlinear Convex Anal., 7(3):423–441, 2006.
  * [24] B. Franchi, R. Serapioni, and F. Serra Cassano. Differentiability of intrinsic Lipschitz func- tions within Heisenberg groups. J. Geom. Anal., 21(4):1044–1084, 2011.
  * [25] B. Franchi and R. P. Serapioni. Intrinsic Lipschitz graphs within Carnot groups. J. Geom. Anal., 26(3):1946–1994, 2016.
  * [26] B. Gess and X. Lamy.  Regularity of solutions to scalar conservation laws with a force.  Ann. Inst. H. Poincaré Anal. Non Linéaire, 36(2):505–521, 2019.
  * [27] A. Julia, S. Nicolussi Golo, and D. Vittone. Area of intrinsic graphs and coarea formula in Carnot Groups. Bull. Lond. Math. Soc. 53, no. 6, 1766–1775, 2021.
  * [28] B. Kirchheim and F. Serra Cassano. Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), III(4):871–-896, 2004.
  * [29] E. Le Donne and T. Moisala. Semigenerated step-3 Carnot algebras and applications to sub- Riemannian perimeter, Math. Z. 299, no. 3-4, 2257–2285, 2021.
  * [30] P.-L. Lions, B. Perthame, and E. Tadmor.  A kinetic formulation of multidimensional scalar conservation laws and related equations.  J. Amer. Math. Soc., 7(1):169–191, 1994.
  * [31] E. Marconi.  Regularity estimates for scalar conservation laws in one space dimension.  J. Hyperbolic Differ. Equ., 15(4):623–691, 2018.
  * [32] A. Naor and R. Young. Vertical perimeter versus horizontal perimeter, Annals of Mathematics 188, no. 1, 171–279, 2018.
  * [33] A. Naor and R. Young. Foliated corona decompositions, Acta Mathematica, Vol. 229, No. 1, pp. 55-200, 2022.
  * [34] O. A. Oleĭnik.  Discontinuous solutions of non-linear differential equations.  Amer. Math. Soc. Transl. (2), 26:95–172, 1963.
  * [35] R. Robyr.  SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function,  J. Hyperbolic Differ. Equ., 5(2):449–475, 2008.
  * [36] F. Serra Cassano.  Some topics of geometric measure theory in Carnot groups,  Geometry, analysis and dynamics on sub-Riemannian manifolds. Vol. 1, EMS Ser. Lect. Math., Eur. Math. Soc., Zürich, 2016, pp. 1–121.
  * [37] D. Vittone. Lipschitz graphs and currents in Heisenberg groups. Forum Math. Sigma 10 (2022), Paper No. e6, 104 pp.