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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