SCUOLA NORMALE SUPERIORE DI PISA
Perfezionamento in Matematica
XVIII ciclo, anni 2003-2005
PhD Thesis
Federica Dragoni
CARNOT-CARATHEODORY
METRICS AND VISCOSITY
SOLUTIONS
Advisor: Prof. Italo Capuzzo Dolcetta
Contents
Introduction. 5
1 Sub-Riemannian geometries. 11
1.1 Basic definitions and main properties. . . . . . . . . . . . . . . 11
1.1.1 Historical introduction and Didos problem. . . . . . . 11
1.1.2 Some new development: the visual cortex. . . . . . . . 15
1.1.3 Riemannian metrics. . . . . . . . . . . . . . . . . . . . 17
1.1.4 Carnot-Caratheodory metrics and the Hormander con-
dition. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.1.5 Notions equivalent to the Carnot-Caratheodory distance. 27
1.1.6 Chows Theorem. . . . . . . . . . . . . . . . . . . . . . 33
1.1.7 Relationship between the Carnot-Caratheodory dis-
tance and the Euclidean distance. . . . . . . . . . . . . 36
1.1.8 Sub-Riemannian geodesics. . . . . . . . . . . . . . . . . 39
1.1.9 The Grusin plane. . . . . . . . . . . . . . . . . . . . . . 44
1.2 Carnot groups. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.2.1 Nilpotent Lie groups. . . . . . . . . . . . . . . . . . . . 49
1.2.2 Calculus on Carnot groups. . . . . . . . . . . . . . . . 53
1.3 The Heisenberg group. . . . . . . . . . . . . . . . . . . . . . . 56
1.3.1 The polarized Heisenberg group. . . . . . . . . . . . . . 57
1.3.2 The canonical Heisenberg group. . . . . . . . . . . . . 60
1.3.3 Equivalence of the two definitions. . . . . . . . . . . . . 64
2 Viscosity solutions and metric Hopf-Lax formula. 71
2.1 An introduction to the theory of viscosity solutions. . . . . . . 71
2.1.1 Viscosity solutions for continuous functions. . . . . . . 73
3
4 Contents.
2.1.2 Discontinuous viscosity solutions. . . . . . . . . . . . . 79
2.2 The generalized eikonal equation. . . . . . . . . . . . . . . . . 87
2.3 The Hopf-Lax function. . . . . . . . . . . . . . . . . . . . . . . 97
2.3.1 Optimal control theory and Hopf-Lax formula. . . . . . 97
2.3.2 Some properties of the Euclidean Hopf-Lax function. . 102
2.4 The metric Hopf-Lax function. . . . . . . . . . . . . . . . . . . 105
2.4.1 Properties of the metric Hopf-Lax function. . . . . . . 106
2.4.2 The Hopf-Lax solution for the Cauchy problem. . . . . 117
2.4.3 Examples and applications. . . . . . . . . . . . . . . . 122
3 Carnot-Caratheodory inf-convolutions. 127
3.1 Definition and basic properties. . . . . . . . . . . . . . . . . . 127
3.2 Inf-convolutions and logarithms of heat kernels. . . . . . . . . 131
3.2.1 The Euclidean approximation and the Large Deviation
Principle. . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.2.2 Applicability of the Large Deviation Principle: the
proof of Varadhan for the Riemannian case. . . . . . . 136
3.3 Carnot-Caratheodory inf-convolutions and ultraparabolic
equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.3.1 Heat kernels for hypoelliptic operators. . . . . . . . . . 141
3.3.2 Limiting behavior of solutions of subelliptic heat equa-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
A The Legendre-Fenchel transform. 157
Bibliography. 165
Introduction.
Sub-Riemannian geometries have many interesting applications in very dif-
ferent settings, as optimal control theory ([17, 26, 27]), calculus of variations
([2, 4]) or stochastic differential equations ([16]). Many physical phenomena
seem to induce in a natural way an associated sub-Riemannian structure, for
example, one can think of Berrys phase problem, a swimming microorgan-
ism (studied in [75]), the optimal control in laser-induced population transfer
(see [20]) and the perceptual completion in the visual cortex ([35]). Sub-
Riemannian geometries (known also as Carnot-Caratheodory spaces) arise
whenever there are privileged and prohibited paths. In fact, the main differ-
ence between these geometries and the Riemannian geometries is the need to
move along some prescribed vector fields. A Sub-Riemannian metric is in-
deed a Riemannian metric defined only on a subbundle of the tangent bundle
to the manifold. More precisely, let X = {X1, ..., Xm} be a family of vectorfields, defined on a n-dimensional manifold M (with in general m n), asub-Riemannian metric is a Riemannian metric
,
defined on the fibers of
H := Span(X ) TM . A subbundle H is usually called distribution.For sake of simplicity, from now on we will always assume thatM = Rn. This
in particular implies that the tanget space at any point x is equal to Rn and
the tangent bundle is isomorphic to R2 n. An admissible (or X -horizontal)curve is any absolutely continuous curve : [0, T ] Rn, such that
(t) Span(X1((t)), ..., Xm((t))), a.e. t [0, T ].
Since the Riemannian metric,
is defined along the fibers of H, for thehorizontal curves and, only for the horizontal curves, we can introduce a
6 Introduction.
length-functional as follows:
l() =
T
0
(t), (t)
12dt.
A sub-Riemannian (or Carnot-Caratheodory) distance on Rn can be de-
fined, for any x, y Rn, as
d(x, y) = inf{l() | admissible curve joiningx to y}. (1)
It is obvious that, whenever there are not admissible curves joining x to y,
then d(x, y) = +. For this purpose the Hormander condition is introduced.In fact, in Carnot-Caratheodory spaces satisfying the Hormander condition,
it is always possible to join two given points by an admissible curve (Chows
Theorem, [17, 75]). Then the associated Carnot-Caratheodory distance is
finite. The Hormander condition (known also as bracket generating condi-
tion) is satisfied if the Lie algebra associated to the distribution H = Span(X )spans the whole tangent space at any point of the manifold (that in this par-
ticular case is at any point equal to Rn). We recall that the bracket between
two vector fields X and Y is the vector field defined as [X, Y ] = XY Y X,acting by derivation on smooth real functions. The Lie algebra L(X ) associ-ated to X = {X1, ..., Xm} is the set of all the brackets between elements ofX , so the Hormander condition holds, if and only if,
Span(L(X1(x), ..., Xm(x))
)= TxM = R
n, for any x Rn.
The first chapter is dedicated to the study of sub-Riemannian geometries
and topological and metric implications of the Hormander condition.
In the second chapter, we are interested in solving some first-order
nonlinear partial differential equations (PDEs) related to the Hormander
condition. Therefore we will introduce the theory of viscosity solutions
for continuous and discontinuous functions. Later we concentrate on the
two particular nonlinear PDEs: The eikonal equation and an evolution
Hamilton-Jacobi equation.
We first solve the generalized eikonal equation:
H0(x,Du) = 1, (2)
Introduction. 7
with vanishing condition at some fixed point y Rn and where H0(x, p) isthe geometrical Hamiltonian defined by
H0(x, p) = |(x)p|, (3)
with (x) Hormander-matrix (i.e. a m n real-valued matrix with smoothcoefficiente and such that its rows satisfy the Hormander condition).
Later we consider an evolution Hamilton-Jacobi equation of the form:
ut + (H0(x,Du)) = 0 (4)
where is a suitable positive and convex function and H0(x, p) satisfies the
structural assumpiton (2). The main model for the PDE (4) is
ut +1
|(x)Du| = 0, with > 1. (5)
We solve both the previous PDEs by using the Hormander condition and
suitable representative formulas. In order to solve the eikonal equation, fixed
the point y Rn, we define the minimal-time function
d(x, y) = infX()Fx,y
T (X()), (6)
where Fx,y is the set of all the trajectories joining x to y in a time T (X())which are solutions of the differential inclusion
X(t) H0(X(t), 0) = T (X(t))B1(0),
with B1(0) unit Euclidean ball in Rm centered at the origin.
It is possible to show under very weak assumptions that a minimal-time
distance is a generalized distance (i.e. a distance not always symmetric)
solving a Dynamical Programming Principle:
d(x, y) = infX()Fx,y
[t+ d(X(t), y)], for any 0 < t < d(x, y).
For our particular Hamiltonian in (3), the minimal-time distance turns out to
be equivalent to the Carnot-Caratheodory distance associated to the matrix
(x). By using the Dynamical Programming Principle, we prove that u(x) =
d(x, y) solves in the viscosity sense the corresponding eikonal equations on
8 Introduction.
Rn\{y}. By a sub-Riemanninan generalization of the Rademachers Theorem(see [75, 77, 80]), the viscosity result implies that the Carnot-Caratheodory
distance is a almost everywhere solution, too (at least in Carnot groups).
To solve the eikonal equation is the key-point in order to solve the Cauchy
problem for Eq. (4) with lower semicontinuous initial data g(x). To get
the existence of a viscosity solution for this class of PDEs, we use a suitable
representative formula: the metric Hopf-Lax formula given by
u(t, x) = infyRn
[g
