Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation

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by  Francesco Bigolin

INTRINSIC LIPSCHITZ GRAPHS IN HEISENBERG GROUPS AND
CONTINUOUS SOLUTIONS OF A BALANCE EQUATION
FRANCESCOBIGOLIN,LAURACARAVENNA,FRANCESCOSERRACASSANO
2
1
0
2 Abstract. In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-
RiemannianHeisenberggroupsintermsoftheirdistributionalgradients. Moreover,weprovethe
b equivalence of different notions of continuous weak solutions to the equation φy+[φ2/2]t = w,
e wherew isaboundedfunctiondependingonφ.
F

4
1

] Contents
G
D
1. Introduction 1
.
h
2. Sub-Riemannian Heisenberg Group 6
t
a
m 3. Different Solutions of the Intrinsic Gradient Differential Equation 9
[
4. Existence of Lagrangian Parameterizations 11

1
5. Equivalence among Distributional PDE and Intrinsic Lipschitz Condition. 14
v
3
5.1. Some properties of distributional solutions 14
8
0
5.2. Proof of the equivalence 17
3
. 6. Further Equivalences 19
2
0
6.1. Lagrangian solutions are distributional solutions 20
2
1
6.2. Distributional solutions are broad solutions 22
:
v
i Appendix A. From partial to full Lagrangian parameterizations 24
X
References 28
r
a
1. Introduction
In the lastyearsit has been largelydeveloped the study ofintrinsic submanifolds inside the Heisen-
berg groups Hn or more general Carnot groups, endowed with their Carnot-Carath´eodory metric
structure,alsonamedsub-Riemannian. Byanintrinsicregular(orintrinsicLipschitz)hypersurfaces
wemeanasubmanifoldwhichhas, intheintrinsicgeometryofHn, thesamerolelikeaC1 (orLips-
chitz)regulargraphhasintheEuclideangeometry. Intrinsicregulargraphshadseveralapplications
Date:February15,2012.
F.B. is supported by PRIN 2008 and University of Trento, Italy. L.C. has been supported by Centro De Giorgi
(SNS di Pisa), by the ERC Starting Grant CONSLAW and the UK EPSRC Science and Innovation award to the
Oxford Centre for Nonlinear PDE (EP/E035027/1). F.S.C. is supported by GNAMPA, PRIN 2008 and University
ofTrento,Italy.
1
2 F.BIGOLIN,L.CARAVENNA,F.SERRACASSANO
within the theory of rectifiable sets and minimal surfaces in CC geometry, in theoretical computer
science, geometry of Banach spaces and mathematical models in neurosciences.
We postpone complete definitions of Hn to Section 2. We only remind that the Heisenberg group
Hn = Cn R R2n+1 is the simplest example of Carnot group, endowed with a left-invariant
× ≡
metric d (equivalent to its Carnot-Carath´eodory metric), not equivalent to the Euclidean metric.
Hn isa(∞connected,simplyconnectedandstratified)Liegroupandhasasufficientlyrichcompatible
underlying structure, due to the existence of intrinsic families of left translations and dilations and
depending to the horizontal fields X ,...,Y . We call intrinsic any notion depending directly by the
1 n
structure and geometry of Hn. For a complete description of Carnot groups [6, 16, 21, 22, 23] are
recommended.
As we said, we will study intrinsic submanifolds in Hn. An intrinsic regular hypersurface S Hn
⊂
is locally defined as the non critical level set of an horizontal differentiable function, more precisely
thereexistslocallyacontinuousfunctionf :Hn Rsuchthatf(P)=0,thereexistsinthesenseof
→
distributions Hf =(X1f,...,Ynf),itiscontinuousandnon-vanishingforP S . Intrinsicregular
hypersurfaces∇canbelocallyberepresentedasX -graphbyafunctionφ:ω ∈W R2n R,where
1
W = x = 0 , through an implicit function theorem (see [12]). In [2, 4, 5⊂] the≡parame→trization φ
1
{ }
has been characterized as weak solution of a system of non linear first order PDEs φφ=w, where
w :ω R2n 1 and φ =(X ,...,X ,∂ +φ∂ ,Y ,...,Y ), seeTheorem2.7. Bya∇nintrinsicpoint
− 2 n y t 2 n
of view→, the operato∇r φφ is the intrinsic gradient of the function φ : W R. In particular, [5]
showsthatφisacontin∇uousdistributionalsolutionoftheproblem φφ=w→withw C0(ω,R2n 1)
−
∇ ∈
if and only if φ induces an intrinsic regular graph.
Let us point out that an intrinsic regular graph can be very irregular from the Euclidean point of
view: indeed,thereareexamplesofintrinsicregulargraphsinH1 whicharefractalintheEuclidean
sense ([19]).
The aim of our work is to characterize Intrinsic Lipschitz graphs in terms of the intrinsic distribu-
tional gradient. In the Euclidean setting, Lipschitz graphs can be either defined
by means of cones: there exists a cone which translated to any point of the surface locally
•
intersects it only in the vertex;
in a metric way: there exists L>0 such that φ(x) φ(y) Ly x for every x,y ω;
• | − |≤ | − | ∈
by the distributional derivatives, which must be absolutely continuous measures with L
∞
•
density.
Intrinsic Lipschitz graphs in Hn have been introduced and studied in [15]. In particular the equiva-
lence of the first two points for intrinsic Lipschitz graphs has been established. A subset S Hn is
⊂
intrinsic Lipschitz if at each point P S there is an intrinsic cone with vertex P and fixed opening,
∈
intersecting S only in P. As consequence, the metric definition (see Definition 2.8) is given with
respect to the the graph quasidistance d , see (2.5), i.e the function φ : (ω,d ) R is meant
φ φ
→
Lipschitz in classical metric sense. This notion turned out to be the right one in the setting of the
intrinsic rectifiability in Hn. Indeed it was proved in [15] that the notion of rectifiable set in terms
of an intrinsic regular hypersurfaces is equivalent to the one in terms of intrinsic Lipschitz graphs.
WewilldenotebyLipW(ω)theclassofallintrinsicLipschitzfunctionφ:ω →RandbyLipW,loc(ω)
the one of locally intrinsic Lipschitz function. Notice that LipW(ω) is not a vector space and that
Lip(ω)(cid:40)LipW,loc(ω)(cid:40)C0lo,1c/2(ω),
where Lip(ω) and C0,1/2(ω) denote respectively the classes of Euclidean Lipschitz and 1/2-H¨older
loc
functions in ω. For a complete presentation of intrinsic Lipschitz graphs and functions [8, 15, 20]
are recommended.
Thefirstmainresultofthispaperisthecharacterizationofaparametrizationφ:ω Rofanintrin-
sic Lipschitz graph as a continuous distributional solution of φφ=w, where w →L (ω,R2n 1).
∞ −
∇ ∈
INTRINSIC LIPSCHITZ GRAPHS IN Hn 3
Theorem 1.1. Let ω W R2n be an open set, φ : ω R be a continuous function and
w ∈ L∞(ω;R). φ ∈ Lip⊂W,loc(ω≡;R) if and only if there exists→w ∈ L∞(ω;R2n−1) such that φ is a
distributional solution of the system φφ=w.
∇
WestressthatthisisindeeddifferentfromprovingaRademachertheorem,whichismorerelatedto
a pointwise rather than distributional characterization for the derivative, see [15]. Nevertheless, we
find that the density of the (intrinsic) distributional derivative is indeed given by the function one
finds by Rademacher theorem. We also stress that there are a priori different notions of continuous
solutions φ:ω R to φφ=w, which express the Lagrangian and Eulerian viewpoints. They will
→ ∇
turn out to be equivalent descriptions of intrinsic Lipschitz graphs, when the source w belongs to
L (ω;R2n 1). This is proved in Section 6 and it is summarized as follows.
∞ −
Theorem 1.2. The following conditions are equivalent
(i) φ is a distributional solution of the system φφ=w with w L (ω;R2n 1);
∞ −
(ii) φ is a broad solution of φφ=w, i.e. there∇exists a Borel fu∈nction wˆ L (ω;R2n 1) s.t.
∞ −
∇ ∈
(B.1): w(A)=wˆ(A) 2n-a.e. A ω;
L ∈
(B.2): for every continuous vector field φ having an integral curve Γ C1(( δ,δ);ω), φ
∇i ∈ −
satisfies
(cid:90) s
φ(Γ(s)) φ(Γ(0))= wˆ(Γ(r)) dr s [ δ,δ].
− ∀ ∈ −
0
In the statement, L (ω;R2n 1) denotes the set of functions from ω to R2n 1, while L (ω;R2n 1)
∞ − − ∞ −
denotes the equivalence classes of Lebesgue measurable functions in L (ω;R2n 1) which are iden-
∞ −
tified when differing on a Lebesgue negligible set. We will keep this notation throughout the paper:
its relevance is remarked by Examples 1.3, 1.4 below.
Outline of the proofs. With the intention of focusing on the nonlinear field, we fix the attention
on the case n=1. The variables will be denoted by t and y, and the subscripts [] , [] will denote
t y
· ·
the distributional derivatives ∂ =∂ , ∂ =∂ in the Euclidean sense w.r.t. this variables.
∂t t ∂y y
Given a continuous distributional solution φ of the PDE
(cid:20)φ2(cid:21)
φφ=φ + =w
y
∇ 2
t
we first prove that it is Lipschitz when restricted along any characteristic curve Γ(y) = (y,γ(y)),
γ˙ =φ Γ, adapting an argument by Dafermos. On the other hand, by a construction based on the
◦
classical existence theory of ODEs with continuous coefficients, we can define a change of variable
(y,χ(y,τ))whichstraightenscharacteristics. ThischangeofvariablesdoesnotenjoyBVorLipschitz
regularity,itfailsinjectivityinanessentialway,thoughitiscontinuousandweimposeanimportant
monotonicityproperty. Thismonotonicity,relyingonthefactthatwebasicallyworkindimension2,
istheregularitypropertywhichallowsusthechangeofvariables. Asweexemplifybelow,weindeed
haveanapproachdifferentfromprovidingaregularLagrangianflowofAmbrosio-DiPerna’stheory,
and it is essentially two dimensional. After the change of variables, the PDE is, roughly, linear, and
weindeedfindafamilyofODEsforφonthefamilyofcharacteristicscomposingχ,withcoefficients
which now are not anymore continuous, but which are however bounded. By generalizing a lemma
on ODEs already present in [4], we prove the 1/2-H¨older continuity on the vertical direction (y
constant), andaposterioriinthewholedomain. Thisarethemainingredientsforestablishingthat
φ defines indeed a Lipschitz graph: given two points, we connect them by a curve made first by
a characteristic curve which joins the two vertical lines through the points, then by the remaining
vertical segment. We manage this way to control the variation of φ between the two points with
their graph distance d , checking therefore the metric definition of intrinsic Lipschitz graphs.
φ
4 F.BIGOLIN,L.CARAVENNA,F.SERRACASSANO
y y y
t t t
φ(y,t) φ(y,t) φ(y,t)
t t t
Figure 1. Illustration of Examples 1.3, 1.4. Characteristics are drown for three
particularcontinuousdistributionalsolutiontotheequationφ +[φ2/2] =sgn(t)/2.
y t
Below the graphs of the corresponding functions φ are depicted.
The other side of Theorem 1.1 is based on the possibility of suitably approximating an intrinsic
Lipschitz graph with intrinsic regular graphs. A geometric approximation is provided by [8]. We
also provide a more analytic, and weaker, approximation as a byproduct of the change of variable
χ which straightens characteristics, by mollification.
WestopnowforawhileinordertoclarifythefeaturesofthestatementinTheorem1.2, andwhyit
is so important to differentiate between L (ω;R2n 1) and L (ω;R2n 1). Lagrangian formulations
∞ − ∞ −
are affected by altering the representative, as the following example stresses.
(cid:112)
Example 1.3 (Figure 1). Consider the continuous function φ(y,t) = t in the domain ω =
| |
(0,1) ( 1,1). For simplicity, it does not depend on y. It is easy to calculate
× −
∂ ∂ φ2 (cid:26) 1/2 ift 0
φφ= φ+ = ≥ =:w(y,t).
∇ ∂y ∂t 2 1/2 ift<0
−
Consider the specific characteristic curve (y,γ(y)) := (y,0). Even if γ˙(y) = φ(y,γ(y)) = 0, the
derivative of φ along this characteristic curve is not the right one:
∂ 1
φ(y,0)=0= =w(y,0).
∂y (cid:54) 2
Equation (3.5) holds however on every characteristic curves provided we choose correctly an L -
∞
representative wˆ L (ω;R) of the source w: it is enough to consider
∞
∈

1/2 ift>0

wˆ(y,t):= 0 ift=0
 1/2 ift<0
−
Notice that w(y,t)=wˆ(y,t) for 2-a.e. (y,t) ω. (cid:3)
L ∈
Before outlining Theorem 1.2 we exemplify other features mentioned above by similar examples.
Example 1.4 (Figure 1). Let ω =(0,1) ( 1,1), and choose
× −
(cid:112)
φ(y,t)= sgnt t.
− | |
Again
∂ ∂ φ2 (cid:26) 1/2 ift>0
φφ= φ+ = =:w(y,t).
∇ ∂y ∂t 2 1/2 ift 0
− ≤
INTRINSIC LIPSCHITZ GRAPHS IN Hn 5
One easily sees that characteristics do collapse in an essential way. Considering instead
(cid:112)
φ(y,t)=sgnt t
| |
characteristics do split in an unavidable way. Therefore, while it is proved in [10] that in Exam-
ple 1.3 one can choose for changing variable a flux which is better than a generic other—theregular
Lagrangian flow—we are not always in this case. This is the main reason why we refer in our first
change of variables to a monotonicity property.
We have now motivated the further study for the stronger statement of Theorem 1.2. In order to
prove it, we consider also the weaker concept of Lagrangian solution: the idea is that the reduction
on characteristics is not required on any characteristic, but on a set of characteristic composing the
changeofvariablesχthatonehaschosen. Indeed,exhibitingasuitablesetofcharacteristicsforthe
change of variables χ is part of the proof. Therefore, a L -representative w for the source of the
∞ χ
ODEs related to χ is provided by taking the y-derivative of φ(y,χ(y,τ)), which by construction is
alsothesecondy-derivativeofχ(y,τ). Certainly,thereisanadditionalfurthertechnicalityincoming
back from (y,τ) to (y,t), which can be overcome. However, if one changes the set of characteristics
in general one arrives to a different function wχ(cid:48) L∞(ω;R).
∈
We have called broad solution a function which satisfies the reduction on every characteristic curve.
In order to have this stronger characterization, we give a different argument borrowed from [1]. We
define a universal source term wˆ in an abstract way, by a selection theorem, at each point where
thereexistsacharacteristiccurvewithsecondderivative. Aftershowingthatthisiswelldefined,we
have provided a universal representative of the intrinsic gradient of φ. In cases as Examples 1.3, 1.4
it extends the one, defined only almost everywhere, provided by Rademacher theorem.
Outline of the paper. The paper is organized as follows. In Section 2 we recall basic notions
about the Heinsenberg groups. In Section 3 we fix instead notations relative to the PDE, mainly
specifying the different notions of solutions we will consider. One of them will involve a change of
variables, for passing to the Lagrangian formulation, which is mainly matter of Section 4 and it is
basically concerned with classical theory on ODEs. Then Appendix A also explains how to extend
a partial change of variables of that kind to become surjective, and provides a counterexample to
its local Lipschitz regularity. In Section 5 we prove the equivalence among the facts that either
a continuous function φ describes a Lipschitz graph or it is a distributional solution to the PDE
φφ=w, w L . The further equivalencies are finally matter of Section 6.
∞
∇ ∈
With some simplification, we can illustrate the main connections by the following papillon. As
mentioned above, there is also a connection with the existence of smooth approximations, which is
not emphasized. Here it is noticed in Corollary 6.4 and applied relatively to the equivalence with
the distributional formulation. See [8] for a different smoothing.
Intrinsic Lips(cid:71)(cid:71)chitz (Def. 2.8) (cid:50)(cid:50)Broad (Def. 3.7)
Cor.4.5 Th.6.7
(cid:37)(cid:37)
L.5.5 L.5.7 Lagr(cid:52)(cid:52)angian (D(cid:99)(cid:99) ef. 3.6) clear
clear
Th.5.3(n=1)
(cid:15)(cid:15) (cid:41)(cid:41) (cid:7)(cid:7)
(cid:114)(cid:114)
Distributional (Def. 3.1) Cor.6.1 L.4.2 Broad* (Def. 3.8)
Acknowledgements. WewarmlythankStefanoBianchiniandGiovanniAlbertiforusefuldiscussions
and important suggestions in particular on the subject of Section 6.
6 F.BIGOLIN,L.CARAVENNA,F.SERRACASSANO
2. Sub-Riemannian Heisenberg Group
Definition: a noncommutative Lie group. We denote the points of Hn Cn R R2n+1 by
≡ × ≡
P =[z,t]=[x+iy,t]=(x,y,t), z Cn, x,y Rn, t R.
∈ ∈ ∈
If P =[z,t], Q=[z ,t] Hn and r >0, the group operation reads as
(cid:48) (cid:48)
∈
(cid:20) (cid:21)
1
(2.1) P Q:= z+z ,t+t m( z,z¯ ) .
· (cid:48) (cid:48)− 2(cid:61) (cid:104) (cid:48)(cid:105)
The group identity is the origin 0 and one has [z,t] 1 = [ z, t]. In Hn there is a natural one
−
− −
parameter group of non isotropic dilations δ (P):=[rz,r2t] , r >0.
r
The group Hn can be endowed with the homogeneous norm
P :=max z , t1/2
(cid:107) (cid:107)∞ {| | | | }
and with the left-invariant and homogeneous distance
d (P,Q):= P 1 Q .
−
∞ (cid:107) · (cid:107)∞
The metric d is equivalent to the standard Carnot-Carath´eodory distance. It follows that the
Hausdorff dim∞ension of (Hn,d ) is 2n+2, whereas its topological dimension is 2n+1.
∞
The Lie algebra h of left invariant vector fields is (linearly) generated by
n
∂ 1 ∂ ∂ 1 ∂ ∂
X = y , Y = + x , j =1,...,n, T =
j j j j
∂x − 2 ∂t ∂y 2 ∂t ∂t
j j
and the only nonvanishing commutators are
[X ,Y ]=T, j =1,...n.
j j
We also use the notation X :=Y for j =n+1,...,2n.
j j n
−
Horizontal fields and differential calculus. We shall identify vector fields and associated first
order differential operators; thus the vector fields X ,...,X , Y ,...,Y generate a vector bundle
1 n 1 n
on Hn, the so called horizontal vector bundle HHn according to the notation of Gromov (see [16]),
that is a vector subbundle of THn, the tangent vector bundle of Hn. Since each fiber of HHn can
be canonically identified with a vector subspace of R2n+1, each section ϕ of HHn can be identified
with a map ϕ:Hn R2n+1. At each point P Hn the horizontal fiber is indicated as HHn and
P
→ ∈
each fiber can be endowed with the scalar product , and the associated norm that make
P P
(cid:104)· ·(cid:105) (cid:107)·(cid:107)
the vector fields X ,...,X , Y ,...,Y orthonormal.
1 n 1 n
Definition 2.1. A real valued function f, defined on an open set Ω Hn, is said to be of class
⊂
C1(Ω) if f C0(Ω) and the distribution
H
∈
Hf :=(X1f,...,Ynf)
∇
is represented by a continuous function.
Definition 2.2. We shall say that S Hn is an H-regular hypersurface if for every P S there
⊂ ∈
exist an open ball U (P,r) and a function f C1(U (P,r)) such that
H
∞ ∈ ∞
i: S U (P,r)= Q U (P,r):f(Q)=0 ;
ii: H∩f(P∞)=0. { ∈ ∞ }
∇ (cid:54)
Hf(P)
The horizontal normal to S at P is ν (P):= ∇ .
S
− Hf(P)
|∇ |
In the spirit of [2, 3, 4, 5] we set W:= (x,y,t) Hn : x =0 R2n, V:= (x,y,t) Hn : x =
1 2
0,...,y = 0,...,y = 0 R. There{fore if A∈ W then A =}≡(0,x ,...,x{,y ,y ,.∈..,y ,t), we
1 n 2 n 1 2 n
} ≡ ∈
will write A=(y ,v,t), where v =(x ,...,x ,y ,...,y ) if n 2 and A=(y ,t)=(y,t) if n=1.
1 2 n 2 n 1
≥
INTRINSIC LIPSCHITZ GRAPHS IN Hn 7
Definition 2.3. A set S Hn is an X -graph if there is a function φ : ω W V such that
1
⊂ ⊂ →
S =G1 (ω):= A φ(A)e :A ω .
H,φ { · 1 ∈ }
Let us recall the following results proved in [12].
Theorem 2.4 (ImplicitFunctionTheorem). Let Ω be an open set in Hn, 0 Ω, and let f C1(Ω)
H
∈ ∈
be such that X f > 0. Let S := (x,y,t) Ω : f(x,y,t) = 0 ; then there exist a connected open
1
neighborhood of 0 and a unique c{ontinuou∈s function φ:ω W} [ h,h] such that S =Φ(ω),
U ⊂ → − ∩U
where h>0 and Φ is the map defined as
ω (y ,v,t) Φ(y ,v,t)=(y ,v,t) φ(y ,v,t)e
1 1 1 1 1
(cid:51) (cid:55)→ ·
and given explicitly by
(cid:16) y (cid:17)
Φ(y ,v,t)= φ(y ,v,t),x ,...,x ,y ,y ,...,y ,t 1φ(y ,v,t) if n 2
1 1 2 n 1 2 n 1
− 2 ≥
(cid:16) y (cid:17)
Φ(y ,τ)= φ(y ,t),y ,t 1φ(y ,t) if n=1.
1 1 1 1
− 2
Let n 2, A =(x0,...,x0,y0,...,y0,t0) R2n and define
≥ 0 2 n 1 n ∈
I (A ) :=(cid:8)(x ,...,x ,y ,...,y ,t) R2n : y y0 <r,
r 0 (cid:80)n [(2x x0n)2+1(y yn0)2]∈<r2, t| 1t−0 <1|r(cid:9).
i=2 i− i i− i | − |
When n=1 and A =(y0,t0) R2 let
0
∈
I (A ):=(cid:8)(y,t) R2 : y y0 <r, t t0 <r(cid:9).
r 0
∈ | − | | − |
Following [2, 3, 25] we define the graph quasidistance d on ω. We set O := (x,y,t) Hn : x =
φ 1 1
0,t=0 , T:= (x,y,t) Hn : x =0,...,y =0 { ∈
1 n
} { ∈ }
Definition 2.5. For A=(y ,v,t), B =(y ,v ,t) ω we define
1 1(cid:48) (cid:48) (cid:48) ∈
(2.2) dφ(A,B):=(cid:107)πO1(Φ(A)−1·Φ(B))(cid:107)∞+(cid:107)πT(Φ(A)−1·Φ(B))(cid:107)∞
If n 2 we have explicitly
≥
(cid:12) (cid:12)1/2
(cid:12) 1 (cid:12)
dφ(A,B)=|(y1(cid:48),v(cid:48))−(y1,v)|+(cid:12)(cid:12)t(cid:48)−t− 2(φ(A)+φ(B))(y1(cid:48) −y1)+σ(v,v(cid:48))(cid:12)(cid:12) ;
where σ(v,v )= 1(cid:80)n (v v v v ). If n=1 and A=(y ,t),B =(y ,t) ω we have
(cid:48) 2 j=2 n+j j(cid:48) − j n(cid:48)+j 1 1(cid:48) (cid:48) ∈
(cid:12) (cid:12)1/2
(cid:12) 1 (cid:12)
dφ(A,B)=|y1(cid:48) −y1|+(cid:12)(cid:12)t(cid:48)−t− 2(φ(A)+φ(B))(y1(cid:48) −y1)(cid:12)(cid:12) .
An intrinsic differentiable structure can be induced on W by means of d , see [2, 3, 25]. We remind
φ
that a map L:W R is W-linear if it is a group homeomorphism and L(ry ,rv,r2t)=rL(y ,v,t)
1 1
for all r >0 and (y→,v,t) W. We remind then the notion of φ-differentiablility.
1
∈ ∇
Definition 2.6. Let φ:ω W R be a fixed continuous function, and let A ω and ψ :ω R
0
⊂ → ∈ →
be given.
We say that ψ is φ-differentiable at A if there is an W-linear functional L:W R such
0
• ∇ →
that
(2.3) lim ψ(A)−ψ(A0)−L(A−01·A) =0.
A→A0 dφ(A0,A)
We say that ψ is uniformly φ-differentiable at A if there is an H-linear functional L :
0
• W R such that ∇
→ (cid:26) ψ(B) ψ(A) L(B 1 A) (cid:27)
−
(2.4) lim sup | − − · | =0
r→0A,B∈Ir(A0) dφ(A,B)
A(cid:54)=B
8 F.BIGOLIN,L.CARAVENNA,F.SERRACASSANO
If φ is uniformly φ-differentiable at A , then φ is φ-differentiable at A .
0 0
∇ ∇
In [2] it has been proved that each H- regular graph Φ(ω) admits an intrinsic gradient φφ
C0(ω;R2n), in sense of distributions, which shares a lot of properties with the Euclidean gr∇adien∈t.
Indeed, since W = exp(span X ,...,X ,Y ,...,Y ,T ), it is possible to define the differential
2 n 1 n
{ }
operators given, in sense of distributions, by
1 ∂ ∂ 1 ∂ ∂ 1 ∂
Wφφ:=Y φ+ T(φ2)= φ+x φ+ (φ2)= φ+ (φ2),
1 1
2 ∂y ∂t 2∂t ∂y 2∂t
(2.5) (cid:26) 1 1
(X φ,...,X φ,Wφφ,Y φ,...,Y φ) if n 2
φφ:= 2 n 2 n ≥ .
∇ Wφφ if n=1
Wealsodenoteby φ :=( φ,..., φ )thefamilyofvectorfieldsonR2n, φ :=X forj =2,...,n,
∇ ∇2 ∇2n ∇j j
φ =Wφ :=Y + φT and φ :=Y for j =n+2,...,2n.
∇n+1 1 ∇j j−n
Thefollowingcharacterizationswereprovedin[2,3,5]. Thedefinitionsofbroad*anddistributional
solution of the system φφ=w are recalled in Section 3.
∇
Theorem 2.7. Let ω W R2n be an open set and let φ:ω R be a continuous function. Then
⊂ ≡ →
(i) The set S := Φ(ω) is an H-regular surface and ν1(P) < 0 for all P S, where ν (P) =
S ∈ S
(ν1(P),...,ν2n(P)) is the horizontal normal to S at P.
S S
is equivalent to each one of the following conditions:
(ii) There exists w C0(ω;R2n 1) and a family (φ ) C1(ω) such that, as (cid:15) 0+,
− (cid:15) (cid:15)>0
∈ ⊂ →
φ(cid:15) →φ and ∇φ(cid:15)φ(cid:15) →w in L∞loc(ω),
and φφ=w in ω, in sense of distributions.
(i) Ther∇e exists w C0(ω;R2n 1) such that φ is a broad* solution of the system φφ=w.
−
(ii) There exists w ∈C0(ω;R2n 1) such that φ is a distributional solution of φφ∇=w.
−
∈ ∇
(iii) φ is uniformly φ-differentiable at A for all A ω.
∇ ∈
Introduction to the concern of this paper.
Let us now introduce the concept of intrinsic Lipschitz function and intrinsic Lipschitz graph.
Definition 2.8. Let φ : ω W R. We say that φ is an intrinsic Lipschitz continuous function
⊂ →
in ω and write φ LipW(ω), if there is a constant L>0 such that
∈
(2.6) φ(A) φ(B) Ld (A,B) A,B ω
φ
| − |≤ ∀ ∈
MifoφreovLeirpWw(eωs)afyortheavterφyiωs(cid:48)a(cid:98)loωca.lly intrinsic Lipschitz function in ω and we write φ ∈ LipW,loc(ω)
∈
We remark that when φ is intrinsic Lipschitz, then there exists C >0 such that
1
d (A,B) d (Φ(A),Φ(B)) Cd (A,B) A,B ω.
φ φ
C ≤ ∞ ≤ ∀ ∈
Inparticular,thegraphdistanced isalsoequivalenttotheCarnot-Carath´eodorydistancerestricted
φ
to the corresponding points on the graph of the Lipschitz intrinsic hypersurface. This means that φ
is Lipschitz continuous also in the classical sense when evaluated on any fixed integral curve of the
vector field Wφ, while it is 1/2-H¨older on the lines where t is fixed.
In [8] is proved the following characterization for intrinsic Lipschitz functions.
Theorem 2.9. Let ω W be open and bounded, let φ:ω R. Then the following are equivalent:
⊂ →
(i) φ∈LipW,loc(ω)
INTRINSIC LIPSCHITZ GRAPHS IN Hn 9
(ii) tCh(eωre)e>xis0ts{uφckh}tkh∈aNt⊂ C∞(ω) and w ∈ (L∞loc(ω))2n−1 such that ∀ω(cid:48) (cid:98) ω there exists C =
(cid:48)
(ii1) φk k N uniformly converges to φ on the compact sets of ω;
(ii2) { φ}kφ∈k(A) C 2n-a.e. x ω(cid:48), k N;
|∇ |≤ L ∈ ∈
(ii3) φkφk(A) w(A) 2n-a.e. A ω.
∇ → L ∈
Moreover if (ii) holds, then φφ(A)=w(A) 2n-a.e. A ω.
∇ L ∈
Let us finally recall the following Rademacher type Theorem, proved in [15].
Theorem 2.10. If φ LipW(ω) then φ is φ-differentiable for 2n-a.e A ω.
∈ ∇ L ∈
3. Different Solutions of the Intrinsic Gradient Differential Equation
Even when w C0(ω), where ω is an open subset of W R2, the equation
∈ ≡
(cid:20)φ2(y,t)(cid:21)
(3.1) φ (y,t)+ =w(y,t) in ω
y
2
t
allows in general for discontinuous solution. However, it is the case n=1 of the system (2.5)

X φ=w
 j j
(3.2) φφ=w Wφ(φ)=w j =2,...,n
∇ ⇔ Y φ=w n+1
j n+j
wherew C0(ω,R2n 1)andthissystem,byTheorem2.7,describesanH-regularsurfaceS :=Φ(ω)
−
∈
which is an X -graph. Since we want to study in the present paper intrinsic Lipschitz graphs, then
1
we do not require anymore the continuity of w but we allow w L (ω;R2n 1). Notwithstanding
∞ −
∈
that, the continuity of φ remains natural.
There are a priori different notions of continuous solutions φ : ω R. We recall some of them in
→
this section: distributional, Lagrangian, broad, broad*. All of them will finally coincide.
After giving in the present sections the definitions for all n, we will focus in the next one the
analysis on the non-linear equation in the case n = 1, which conveys the attention on the planar
case (3.1). We will remind this reduction by adopting often the variables (s,τ) instead of (y,t).
The generalization to other cases n 2 of most of the lemmas is straightforward, because the fields
≥
X and Y are linear. It is not basically in Lemma 4.4, where we prefer taking advantage of the
j j
continuity of χ; however, we have no reason to prove it in full generality.
We recall that in general solutions are not smooth, even if we assume the continuity—see e.g. Ex-
ample A.2 below. The equation is then interpreted in a distributional way.
Definition 3.1 (Distributional solution). A continuous function φ : ω R is a distributional
→
solution to (3.2) if for each ϕ C (ω)
∈ ∞c
(cid:90) (cid:90) (cid:90) (cid:90)
(3.3) φX ϕd 2n = w ϕd 2n, φY ϕd 2n = w ϕd 2n j =2,...,n
j j j j+n
L − L L − L
ω ω ω ω
and
(cid:90) (cid:18) (cid:19) (cid:90)
∂ 1 ∂
φ ϕ+ φ2 ϕ d 2n = w ϕd 2n.
n+1
∂y 2 ∂t L − L
ω ω
WeconsidernowdifferentversionsfortheLagrangianformulationofthePDE.Thefirstonesomehow
englobes a choice of trajectories for passing from Lagrangian to Eulerian variables, and imposes the
evolution equation on these trajectories. If B is a subset R2n, we will denote
B := y R: (y ,v,t) B , B := t R: (y ,v,t) B .
v,t { 1 ∈ 1 ∈ } y1,v { ∈ 1 ∈ }
10 F.BIGOLIN,L.CARAVENNA,F.SERRACASSANO
Definition 3.2 (Lagrangian parameterization). A partial Lagrangian parameterization associated
to a continuous function φ:ω R and to the balance law (3.2) is any couple (ω˜,χ) with ω˜ R2n,
usually open, and χ:ω˜ (y ,v→,τ) R Borel, such that ⊂
1
(cid:51) (cid:55)→
(L.1): the function Υ(y ,v,τ)=(y ,v,χ(y ,v,τ)) is valued in ω:
1 1 1
(L.2): χ is nondecreasing in the τ variable for each fixed y ,v;
1
(L.3): for each v,τ fixed, χ(y ,v,τ) is absolutely continuous in y and almost everywhere
1 1
∂
(3.4) χ(y ,v,τ)=φ(Υ(y ,v,τ)).
1 1
∂y
1
We call it (full) Lagrangian parameterization if χ(y ,v,τ) is onto the section ω for all y ,v.
1 y1,v 1
We remark again that we emphasized in this definition the nonlinear PDE of the system: a La-
grangianparameterizaitonprovidesacoveringofω bycharacteristiclinesforthatequation. Indeed,
a covering by characteristic lines of the other equations is immediately given by an expression like
(cid:40)
t yix i=2,...,n
χ (x ,...,x ,y ,...,y ,t)= − 2 i , Ψ (y ,v,t)=(y ,v,χ (y ,v,t)).
i 2 n 1 n t+ xiy i=n+2,...,2n i 1 1 i 1
2 i
Moreover, the reduction along characteristics for the linear equations, and thus the equivalence
between Lagrangian and distributional solution, holds with less technicality.
Definition 3.3. A (partial) parameterization (ω˜,χ) extends the (partial) parameterization (ω˜ ,χ˜),
(cid:48) (cid:48)
we denote (ω˜ ,χ˜) (ω˜,χ), if there exists a Borel injective map
(cid:48) (cid:48)
(cid:22)
J :ω˜ (y ,v,τ) (y ,v,j(y ,v,τ)) ω˜ such that χ J =χ˜.
(cid:48) 1 1 1 (cid:48)
(cid:51) (cid:55)→ ∈ ◦
When (ω˜ ,χ˜) (ω˜,χ) and (ω˜,χ) (ω˜ ,χ˜) they are called equivalent.
(cid:48) (cid:48) (cid:48) (cid:48)
(cid:22) (cid:22)
Remark 3.4. ThenotionofLagrangianparameterizationgivenabovedoesnotconsistinadifferent
formulation for the notion of regular Lagrangian flow in the sense by Ambrosio-Di Perna (see [9]
for an effective presentation). Particles are really allowed both to split and to join, therefore in
particular the compressibility condition here is not required, while instead we have a monotonicity
property.
Notation 3.5. When we need to distinguish letters, we denote with¯functions defined on ω but
·
possibly related to a parameterization, with˜functions defined on ω˜, and withˆfunctions defined
· ·
on ω not related to specific parameterizations.
Notice that a full Lagrangian parameterization is continuous in the two variables for free: indeed,
e.g. for n=1, by monotonicity one has that for each s
sup χ(s,τ ) χ(s,τ) inf χ(s,τ ).
(cid:48) (cid:48)
τ(cid:48)<τ ≤ ≤τ(cid:48)>τ
By the surjectivity then equality must hold. Considering the Lipschitz continuity in the other
variables one gains the joint continuity in (s,τ). The same holds for n > 1 provided that χ is
continuous on the hyperplane y = 0, since the argument above gives the continuity only on the
1
planes where v is constant. We do not mind about continuity in v.
BeforegivingthenotionofLagrangiansolution,werecallthatasetA Rnisuniversallymeasurable
⊂
if it is measurable w.r.t. every Borel measure. Universally measurable sets constitute a σ-algebra,
whichincludesanalyticsets. Afunctionf :Rn Rissaiduniversallymeasurableifitismeasurable
→
w.r.t. this σ-algebra. In particular, it will be measurable w.r.t. any Borel measure.
Notice that Borel counterimages of universally measurable sets are universally measurable. Then
thecompositionϕ ψ ofanyuniversallymeasurablefunctionϕwithaBorelfunctionψ isuniversally
◦
measurable. This composition would be nasty if ϕ were just Lebesgue measurable.
Since restrictions of Borel functions on Borel sets are Borel, all the terms in the following definition
are thus meaningful.