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NOTES FOR THE COURSE:

NONLINEAR SUBELLIPTIC EQUATIONS

ON CARNOT GROUPS

ANALYSIS AND GEOMETRY IN METRIC SPACES

JUAN J. MANFREDI

Abstract. The main objective of this course is to present an exten-sion of Jensens uniqueness theorem for viscosity solutions of secondorder uniformly elliptic equations to Carnot groups. This is done viaan extension of the comparison principle for semicontinuous functionsof Crandall-Ishii-Lions. We first present the details for the Heisenberggroup, where the ideas and the calculations are easier to appreciate. Wethen consider the general case of Carnot groups and present applicationsto the theory of convex functions and to minimal Lipschitz extensions.The last lecture is devoted to Cordes estimates in the Heisenberg group.

Contents

Special Note: 21. Lecture I: The Euclidean case 31.1. Calculus 31.2. Jets 51.3. Viscosity Solutions 61.4. The Maximum Principle for Semicontinuous Functions 71.5. Extensions and generalizations 82. Lecture II: The Heisenberg group 82.1. Calculus in H 92.2. Taylor Formula 92.3. Subelliptic Jets 10

2.4. Fully Non-Linear Equations 112.5. The Subelliptic Maximum Principle 132.6. Examples of Comparison Principles 162.7. Degenerate Elliptic Equations, Constant Coefficients 162.8. Uniformly Elliptic Equations with no First Derivatives

Dependence. 17

Date: May 26, 2003.The author thanks the Centro Internazionale per la Ricerca Matematica (CIRM) of

Trento, the MIUR-PIN Project Calcolo delle Variazion GNAMPA-CNR for their gra-cious invitation to speak at the third School on Analysis and Geometry in Metric Spaces.Partially supported by NSF award DMS-0100107.

1


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2 JUAN J. MANFREDI

3. Lecture III: The Maximum Principle for Riemannian VectorFields 18

3.1. Jets 214. Lecture IV: The Maximum Principle in Carnot Groups 254.1. Calculus in G 264.2. Taylor Formula 264.3. Subelliptic Jets 284.4. Fully Non-Linear Equations 294.5. Absolutely Minimizing Lipschitz Extensions 325. L ecture V: Convex functions on Carnot groups 33

5.1. Convexity in the Viscosity Sense inRn

345.2. Convexity in Carnot Groups 355.3. Regularity of Convex Functions 366. Lecture VI: Subelliptic Cordes Estimates 376.1. Cordes conditions 386.2. HW2,2-interior regularity for p-harmonic functions in H 39References 41

Special Note:

The analysis of non-linear subelliptic equations is going through a burstof activity! During the preparation of these notes at least three preprintshave appeared that contain substantial improvements in the field.

Changyou Wang [W1] has found a new approach to Jensens theo-rem on Carnot groups that is more powerful than ours. He extendsalso Jensens uniqueness theorem for infinite harmonic functions togeneral Carnot groups, previously known only for the Heisenberggroup [Bi].

Cristian Gutierrez and Anna Maria Montanari [GM1], [GM2] haverecently identified the Monge-Ampere measure of a convex functionon the Heisenberg group and proved the twice pointwise differentia-

bility a. e. of convex functions (subelliptic Aleksandrov theorem).Another presentation of similar results has been given by NicolaGarofalo and Federico Tournier [GTo].

It is a pleasure to thank Thomas Bieske and Andr as Domokos who reada preliminary version of these notes and made valuable suggestions. Ihave benefited from many discussions with Zoltan Balogh, Matthew Rickley,Roberto Monti, Jeremy Tyson and Martin Riemann while visiting the Uni-versity of Bern and revising these notes. While at Trento, Bianca Stroffoliniand Petri Juutinen made numerous suggestions. Finally, I would like tothank Andrea Caruso and Francesco Borrell form the University of Cataniafor their invaluable help in the preparation of the final version of these notes.


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NONLINEAR SUBELLIPTIC EQUATIONS 3

Nevertheless, the responsibility for any mistakes and omissions is solely theauthors.

1. Lecture I: The Euclidean case

We first present an elementary discussion in Rn starting at the maximumprinciple for smooth functions and arriving to the notion of viscosity solu-tions. This notion is based on using jets or pointwise generalized derivativesobtained from considering expansions for semi-continuous functions similarto Taylor expansions for smooth functions.

1.1. Calculus. Let be a (bounded, smooth) domain inRn

. Let f: Rbe a C2-function that has a local maximum at a point x0 . It followsfrom Taylors theorem that:

Df(x0) = 0

and the second derivative is negative semi-definite

D2f(x0) 0.This last inequality must be interpreted in the matrix sense: for every Rnwe have

D2f(x0), 0.Our first version of the maximum principle is just a restatement of this result

for the difference of two functions.

Theorem 1. Maximum Principle, First version: If the differenceuvhas a local maximum at a point x0, then

(1.1) Du(x0) = Dv(x0)

and

(1.2) D2u(x0) D2v(x0).Let us see how to use this theorem to prove a uniqueness principle for

smooth solutions of elliptic partial differential equations. Consider the equa-tion

(1.3) u + f(Du) + u = 0,where f is a smooth function. Suppose that u and v are smooth solutions of(1.3) on having identical boundary values. Consider the function (uv)+.If it is positive at a point, the uv has a positive interior local maximum, sayat the point x0. From Theorem 1 and (1.3) we get the following relations:

Du(x0) = Dv(x0),(1.4)

D2u(x0) D2v(x0),(1.5)0 = u(x0) + f(Du(x0)) + u(x0)(1.6)0 = v(x0) + f(Dv(x0)) + v(x0)(1.7)


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4 JUAN J. MANFREDI

Subtracting (1.7) from (1.6) we get

0 = u(x0) + v(x0) + u(x0) v(x0).From (1.5) if follows that u(x0) + v(x0) 0. Since we clearly haveu(x0) v(x0) > 0 we get a contradiction. Therefore (u v)+ 0, orequivalently u v. A symmetric argument gives v u.

Therefore, we have proved uniqueness for the Dirichlet problem for theequation (1.3).

Note however that the above argument does not apply directly to theLaplace equation u = 0. A careful examination of the previous argumentshows that we can, in fact, prove the following:

Theorem 2. Comparison Principle, Early Version: Assume thatu v on , that u is a subsolution of (1.3)

u + f(Du) + u 0and that v is a supersolution of (1.3)

v + f(Dv) + v 0,then we have

u vin .

Note that the first order terms play no essential role in the maximum or

comparison principles in this section.The more flexible Theorem 2 allows us to expand the class of equations

for which we can prove uniqueness for the Dirichlet problem. In the case ofthe Laplace equation, we can proceed as follows. Let u and v be harmonicfunctions (u = 0, v = 0) on a bounded domain , continuous on, and with the same boundary values on . For > 0 small defineu = u + x

21 and v = v x21. Here is the basis of the argument. For > 0

we can always find > 0 small enough so that u is a subsolution and v is asupersolution of the equation u + u = 0. Therefore, by the comparisonprinciple Theorem (2), the difference u v attains its maximum in at apoint in . Next, compute the maximum

max (u v) max (2x2

1) Constant and observe that

u v Constant in . Letting 0 we obtain u v.

Modifications of these simple ideas can be used to prove comparison prin-ciples for many elliptic equations of the form

F(x,u,Du,D2u) = 0

as long as F is increasing in u and decreasing in D2u, but the hypothesis ofC2 regularity seems quite necessary. The main purpose of this introductionis to present a version of theorems 1 and 2 for non-smooth functions in Rn.


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We need a notion of generalized derivative in order to find the analogue of(1.1) and (1.2) for non-smooth functions. Of course, there is already anextensive theory of generalized derivatives or distributions that has provenextremely useful for understanding linear and quasilinear equations. How-ever, distributions are not always useful in nonlinear problems. Take as anexample the -Laplace equation

ni,j=1

2u

xixj

u

xi

u

xj= 0.

It is not clear at all what a distributional solution should be.

1.2. Jets. The notion of generalized derivative that we shall use is basedon the Taylor theorem. Suppose that u : Rn R is a C2-function. Taylorat 0 says:

u(x) = u(0) + Du(0), x + 12D2u(0)x, x + o(|x|2)

as x 0. Ifu is not necessarily smooth, we could define a generalized firstderivative as the vector and a generalized second derivative as an n nsymmetric matrix X if

(1.8) u(x) = u(0) + , x +1

2X x, x + o(|x|2

)

as x 0. This turns out to be very strict for non-smooth functions.The key idea is to split (1.8) in two parts. Since we still need to evalu-ate functions pointwise, the class of semicontinuous functions (upper andlower) is the natural class of non-smooth functions to be consider. Recallthat upper-semicontinuous functions are locally bounded above and lower-semicontinuous functions are locally bounded below.

Definition 1. A pair (, X), where Rn and X is an n n symmetricmatrix, belongs to the second order superjet of an upper semicontinuousfunction u at the point x0 if

u(x0 + h) u(x0) + , h + 12X h, h + o |h|2

as h 0. The collection of all of these pairs, is denoted by J2,+u(x0).(Later on, when we need to emphasize the Euclidean structure we will write

J2,+eucl

u(x0).)

There is no apriori reason why the set J2,+u(x0) should be non-emptyfor an arbitrary upper-semicontinuous function. However, a simple argu-ment translating paraboloids shows that indeed J2,+u(x0) is non-empty fora dense set of points x0. For lower semi-continuous functions, we considersubjets:


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6 JUAN J. MANFREDI

Definition 2. A pair (, X), where Rn and X is an n n symmetricmatrix, belongs to the second order subjet of a lower semicontinuous functionv at the point x0 if

v(x0 + h) v(x0) + , h + 12X h, h + o |h|2

as h 0. The collection of all of these pairs, is denoted by J2,v(x0).(Later on, when we need to emphasize the Euclidean structure we will write

J2,

euclv(x0).)

If u is continuous and J2,+(u, x0) J2,(u, x0) = , then it contains onlyone pair (, X). Moreover, the function u is differentiable at x0, the vector = Du(x0) and we say that u is twice pointwise differentiable at x0 andwrite D2u(x0) = X.

A special class of jets is determined by smooth functions that touch afunction u from above or below at a point x0. Denote by K

2,(u, x0) thecollection of pairs (x0), D2(x0)where C2() touches u from below at x0; that is, (x0) = u(x0) and(x) < u(x) for x = x0. Similarly, we define K2,+(u, x0) using smooth testfunctions that touch a function u from above. In fact, all jets come fromtest functions.

Theorem 3. (Crandall, Ishii, [C]) We always haveK2,+(u, x0) = J

2,+(u, x0)

andK2,(u, x0) = J2,(u, x0).

1.3. Viscosity Solutions. Consider equations of the form

F(x,u,Du,D2u) = 0

for a continuous F increasing in u and decreasing in D2u. We say that anupper-semicontinuous function u is a viscosity subsolution of the aboveequation, if whenever x0 and (, X) J2,+u(x0) we have

F(x0, u(x0), , X ) 0.Similarly, we define viscosity supersolutions by using second order subjetsof lower-semicontinuos functions. A viscosity solution is both a viscositysubsolution and a viscosity supersolution. With this notion of weak solution,we have increased enormously the class of functions possibly qualifying assolutions. However, a remarkable result of Jensen [J1] states that for secondorder uniformly elliptic equations there is at most one viscosity solution.

The proof of this fundamental uniqueness result is based on a powerfulnon-linear regularization technique from Control Theory involving first orderHamilton-Jacobi equations - see [E] for an enlightening discussion. Once youhave the comparison principle, one can also adapt techniques from classicalpotential theory to prove existence (Perrons method).


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1.4. The Maximum Principle for Semicontinuous Functions.

Theorem 4. (Crandall-Ishii-Jensen-Lions, 1992, [CIL]): Let the func-tion u be upper semicontinuous in a bounded domain . Let the function vbe lower semi-continuous in . Suppose that for x we have

limsupyx

u(y) lim infyx v(y),

where both sides are not+ or simultaneously. Ifuv has an interiorlocal maximum

sup

(u v) > 0then we have:

For > 0 we can find points p, q such thati)

lim (p q) = 0,

where (p) = |p|2,ii) there exists a point p such that p p (and so does q by (i))

andsup

(u v) = u(p) v(p) > 0,

iii) there exist n n symmetric matrices X, Y and vectors so that

(, X) J2,+

(u, p),iv)

(, Y) J2,(u, q),and

v)X Y.

The last statement means that if Rn we haveX, Y, .

In fact, a stronger statement than (v) holds, which we shall need later on.Set

A = D2p,q((p q))and

C = (A2 + A).

Then for all , Rn we have(1.9) X, Y, C , .Statement (v) follows from 1.9 by setting = and noting that A, = 0for R2n.

A minor technical detail: we have used the closures of the second ordersuper and subjets, J2,+(u, p) and J

2,(v, p). These are defined by takingpointwise limits as follows: A pair (, X) J2,+(u, p) if there exist sequences


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8 JUAN J. MANFREDI

of points pm p, vectors m and matrices Xm X as m suchthat u(pm) u(p) and (m, Xm) J2,+(u, pm).1.5. Extensions and generalizations. The notion of viscosity solution isso robust that it makes sense on Lie groups, Riemannian and sub-Riemannianmanifolds and even more general structures generated by vector fields. Theextension of the Crandall-Ishii-Lions-Jensen machinery to these more gen-eral structures is the objective of this course.

2. Lecture II: The Heisenberg group

In this section we consider the simplest Carnot group, the Heisenberggroup H = (R3, ), where is the group operation given by

(x,y ,z) (x, y, z) = (x + x, y + y, z + z + 12

(xy yx)).H is a Lie group with Lie algebra h generated by the left-invariant vectorfields

X1 =

x y

2

z

X2 =

y+

x

2

z

X3 =

z

The only non-trivial commuting relation is X3 = [X1, X2]. Thus H is anilpotent Lie group of step 2. The horizontal tangent space at a pointp = (x,y,z) is Th(p) = linear span{X1(p), X2(p)}. A piecewise smoothcurve t (t) H is horizontal if (t) Th((t)) whenever (t) exists.Given two points p, q H denote by

(p, q) = {horizontal curves joining p and q}.There are plenty of horizontal curves. We have the classical:

Theorem 5. (Chow) (p, q) = .For convenience, fix an ambient Riemannian metric in R3 so that

{X1, X2, X3}is an orthonormal frame and

Riemannian vol. element = Haar measure of H = Lebesgue meas. in R3.The Carnot-Caratheodory metric is then defined by

dcc(p, q) = inf{length() : (p, q)}.It depends only on the restriction of the ambient Riemannian metric to thehorizontal distribution generated by the horizontal tangent spaces.


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Since horizontal curves are preserved by left translations dcc is left-invariant.The exponential mapping exp : h H is a global diffeomorphism. We useexponential coordinates of the first kind:

p = (x,y,z) = exp(xX1 + yX2 + zX3).

There is a family of non-isotropic dilations t, parametrized by t > 0

t(x,y ,z) = (tx,ty,t2z)

that are group homomorphisms.References for Carnot groups and Carnot-Caratheodory spaces include:

[B], [FS], [H], [G], [GN], and [Lu].

2.1. Calculus in H. Given a function u : H R we considerDu = (X1u, X2u, X3u) R3,

the (full) gradient of u. As a vector field, this is written

Du = (X1u) X1 + (X2u) X2 + (X3u) X3.

The horizontal gradient of u is

D0u = (X1u, X2u) R2,or as a vector field D0u = (X1u) X1 + (X2u) X2.

Theorem 6. Ball-Box Theorem (simple version):(See[B], [G], [NSW])

Set |p| = dcc(p, 0), the Carnot-Caratheodory gauge and|p|H = ((x2 + y2)2 + z2)1/4

the Heisenberg gauge. We have

dcc(p, 0) |p|H |x| + |y| + |z|1/2and

Vol(B(0, r)) r4,where B(0, r) is the Carnot-Carathedory ball centered at 0 of radius r.

Remark 1. With our choice of vector fields we have

|(0, 0, z)|cc = 4 |z|1/22.2. Taylor Formula. Suppose that u : H R is a smooth function.Euclidean Taylor at 0 says:

u(x,y ,z) = u(0, 0, 0)

+ ux x + uy y + uz z+

1

2

uxx x2 + uyy y2 + uzz z2

+2uxyxy + 2uxzxz + 2uyzyz}+ o(x2 + y2 + z2)


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10 JUAN J. MANFREDI

Using the fact that |p|2 x2 + y2 + |z|, we obtain the horizontal Taylorexpansion

u(x,y ,z) = u(0, 0, 0)

+ ux x + uy y + uz z+

1

2

uxxx

2 + uyy y2 + 2uxyxy

+ o(|p|2).

At another point p0 = (x0, y0, z0), we get the horizontal Taylor formula by

left-translation:

u(p) = u(p0) + Du(p0), p10 p +1

2(D20u(p0))h, h + o(|p10 p|2),

where Du(p0) is the full gradient of u at p0, the matrix

(D20u) =

X21 u

12 (X1X2u + X2X1u)

12 (X1X2u + X2X1u) X

22 u

is the horizontal symmetrized second derivative and the vector and h =(x x0, y y0) is the horizontal projection of p10 p. We could also haveused the non-symmetrized second derivative

D20u =

X21 u X1X2u

X2X1u X22 u

.

We prefer to use the symmetrized version since the quadratic form associatedto a matrix A is determined by its symmetric part 12

A + At

. Here At

denotes the transpose of A.

2.3. Subelliptic Jets. Let u be an upper-semicontinuous real function inH. The second order superjet of u at p0 is defined as

J2,+(u, p0) =

(, X) R3 S2(R) such that

u(p) u(p0) + , p10 p +1

2Xh, h + o(|p10 p|2)

Similarly, for lower-semincontinuous u, we define the second order subjet

J2,(u, p0) =

(, Y) R3 S2(R) such that

u(p) u(p0) + , p10 p +1

2Yh, h + o(|p10 p|2)

.


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One easy way to get jets is by using smooth functions that touch u fromabove or below.

K2,+(u, p0) =

(D(p0),(D

2(p0))) : C2 in X1, X2, C1 in X3,

(p0) = u(p0)

(p) u(p), p = p0 in a neighborhood ofp0

As in the Euclidean case every jet can be obtained by this method.

Lemma 1. ([C] for the Euclidean case and [Bi] for the subelliptic case) Wealways have

K2,+(u, p0) = J2,+(u, p0)

and

K2,(u, p0) = J2,(u, p0)

We also define the closure of the second order superjet of an upper-semicontinuous function u at p0, denoted by J

2,+(u, p0), as the set of pairs(, X) R3 S2(R) such that there exist sequences of points pm and pairs(m, Xm) J2,+(u, pm) such that

(pm, u(pm), m,Xm)

(p0, u(p0), ,

X)

as m . The closure of the second order subjet of a lower-semicontinuousfunction v at p0, denoted by J

2,(u, p0) is defined in an analogous manner.

2.4. Fully Non-Linear Equations. Consider a continuous function

F :H R h S(R2) R(p, u, , X) F(p, u, , X).

We will assume that F is proper; that is, F is increasing in u and F isdecreasing in X.Definition 3. A lower semicontinuous function u is a viscosity supersolu-tion of the equation

F(p, u(p), Du(p), (D2u(p))) = 0

if whenever (, X) J2,(u, p0) we have

F(p0, u(p0), , X) 0.Equivalently, if touches u from below, is C2 in X1, X2 and C

1 in X3, thenwe must have

F(p0, u(p0), D(p0), (D2(p0))

) 0.


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12 JUAN J. MANFREDI

Definition 4. An upper semicontinuous functionu is a viscosity subsolutionof the equation

F(p, u(p), Du(p), (D2u(p))) = 0if whenever (, X) J2,+(u, p0) we have

F(p0, u(p0), , X) 0.Equivalently, if touches u from above, is C2 in X1, X2 and C

1 inX3, thenwe must have

F(p0, u(p0), D(p0), (D2(p0))

) 0.

Note that if u is a viscosity subsolution and (, X) J2,+(u, p0) then, bythe continuity of F, we still have

F(p0, u(p0), , X) 0.A similar remark applies to viscosity supersolutions and the closure of secondorder subjets.

A viscosity solution is defined as being both a viscosity subsolution anda viscosity supersolution. Observe that since F is proper, it follows easilythat if u is a smooth classical solution then u is a viscosity solution.

Examples of F:

Subelliptic Laplace equation (the Hormander-Kohn operator):

0u = X21 u + X22 u = 0 Subelliptic p-Laplace equation, 2 p < :

pu = |D0u|p20u + (p 2)|D0u|p40,u

= div (|D0u|p2D0u) = 0Strictly speaking we need p 2 for the continuity of the correspondingF. In the Euclidean case it is possible to extend the definition to the fullrange p > 1. This is a non-trivial matter not yet studied in the case ofthe Heisenberg group (to the best of my knowledge.) See [JLM] for theEuclidean case.

Subelliptic -Laplace equation ([Bi]):

0,u = 2i,j

(Xiu)(Xju)XiXju

= (D20u)D0u, D0u Naive subelliptic Monge-Ampere

det(D20u) = fHere the corresponding F(X) = det Xis only proper in the cone of positivesemidefinite matrices. As mentioned in the introduction, Gutierrez andMontanari have considered the Monge-Ampere operator

det(D20u) +

3

4(T u)2

.


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They show, among other things, that it can always be defined in the senseof measures for a convex function. See [GM1] and [GM2].

In order to have a reasonable theory we need a comparison principle,since it implies uniqueness. It also implies existence (under some additionalhypothesis) via Perrons method (See [C] and [IL]).

The comparison principle for viscosity solutions is based on the max-imum principle for semicontinuous functions(Crandall-Ishii-Lions, [CIL],in Rn). This principle gives a substitute for the maximum principle forsmooth functions easily obtained from the subelliptic Taylor formula.

If u, v C2() and u v has a local maximum at p , we haveDu(p) = Dv(p)

and

(D20u(p)) (D20v(p))

2.5. The Subelliptic Maximum Principle. The subelliptic version ofthe maximum principle for semicontinuous functions was proved by Bieske[Bi] for the case of the Heisenberg group.

Theorem 7. Maximum principle for semicontinuous functions: Letu be upper semi-continuous in a bounded domain H. Letv be lowersemi-continuous in . Suppose that for x we have

limsupyx u(y) lim infyx v(y),where both sides are not+ or simultaneously. Ifuv has an interiorlocal maximum

sup


For > 0 we can find points p, q H such thati)

lim (p q

1 ) = 0,

where

(x,y ,z) = x4

+ y4

+ z2

,ii) There exists a point p such that p p (and so does q by (i))

and

sup

(u v) = u(p) v(p) > 0,iii) there exist symmetric matrices

X, Y S(R2)and vectors

R3so that


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14 JUAN J. MANFREDI

iv)

(, X) J2,+(u, p),v)

(, Y) J2,(v, q),and

vi)X Y + o(1)

as .

The last statement means that if R

2

we haveX, Y , , a()||2,where a() 0 as .

Proof. For > 0 set

M = sup

{u(p) v(q) (p q1)}.

By semi-continuity M < and M is attained at a point (p, q). Notethat M is decreasing in and uniform bounded. We have

M/2 u(p) v(q)

2(p q1 )

M/2 2 (p q1 ) M

M/2 M 1

2 (p q1 )

Conclude first that

lim (p

q1 ) = 0.Next we observe,

sup

(u(p) v(p)) M = u(p) v(q) (p q1 )

Since p p as well as q p (for a subsequence of s tending to ) weget

sup

(u(p) v(p)) = limM = u(p) v(p).

We apply now the Euclidean maximum principle for semicontinuousfunctions of Crandall-Ishii-Lions [CIL]. There exist 33 symmetric matricesX, Y so that

Dp((p q1)), X J2,+eucl. (u, p)

and Dq((p q1)), Y J2,eucl. (v, q)with the property

X, Y, C ,


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where the vectors , R3, andC = (A2 + A)

and

A = D2p,q((p q1))are 6 6 matrices.

We need now a way to get subelliptic jets from euclidean jets.

Lemma 2. Subelliptic jets from Euclidean jets: Let(, X) J2,+eucl.

(u, p)be a second order Euclidean superjet. Then, we have

(DLp , (DLp X (DLp)t)22) J2,+(u, p)In the lemma the subindex 2 2 indicates the principal 2 2 minor of

a 3 3 matrix. The mapping Lp is just left multiplication by p in H. Onecan easily see that its differential is given by

DLp =

1 0 y/20 1 x/2

0 0 1

.

Proof. Easy when p = 0. Left translate for general p.

Using this lemma we conclude that DLp Dp(p q1), (DLp X (DLp)t)22

J2,+(u, p)

and DLq Dq(p q1), (DLq Y (DLq)t)22

J2,(v, q).

Our choice of (p q1) implies thatDLp Dp(p q1) = DLq Dq(p q1).

We call this common value . Let = (1, 2) R2. Write

p = (1, 2, 12(2x1 1y1 )),

where we have set p = (x1, y

1 , z

1 ).

The vector p is chosen so that

((DLp)t)1 p = (1, 2, 0)

Similarly set

q = (1, 2,1

2(2x

2 1y2 )),

where we have set q = (x2, y

2 , z

2 ). It satisfies

((DLq)t)1 q = (1, 2, 0).


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16 JUAN J. MANFREDI

Set

X = (DLp X (DLp)t)22and

Y = (DLq Y (DLq)t)22.With these choices we have

X, Y , == Xp, p Yq, q C(p q), p q =

= ||2

x2y1 x1y22

+ z1 z2

2

o(1) ||2. by (i).

Remark 2. Bieske [Bi] has considered more general functions of the form

(x,y,z) = x2n + y2n + z2m

Estimate (vi) in this case is more complicated. It takes the form

X, Y, 2m2||2 (z1 z2 +1

2(z2 y

1 z1 y1 ))2+4m

2m2||2 2 1m (p q1 ) 2m2||2 ( (p q1 )) 1

1

m (p q1 ).This improvement is necessary to prove uniqueness for the subelliptic -Laplacian, [Bi].

2.6. Examples of Comparison Principles. We have included a coupleof examples where the comparison principle holds. As explained in [CIL],once we have the subelliptic version of the maximum principle, many morecases can be analyzed. See also [IL] for more examples.

2.7. Degenerate Elliptic Equations, Constant Coefficients.

Theorem 8. Let F be continuous and independent of p. Suppose that Fsatisfies

F

u > 0,

F is decreasing in X, and|F(u,, X) F(u,, Y)| w(X Y),

where w 0 as X Y 0. Let u be an upper semicontinuous subsolutionand v a lower semicontinuous supersolution of

F(u,Du, (D20u)) = 0


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in a domain such that

lim supyx

u(y) liminfyx v(y), x ,

where both sides are not + or simultaneously. Thenu(x) v(x) for all x .

Proof. Let us proceed by contradiction. Suppose that sup(u v) > 0 andapply the subelliptic maximum principle from 2.5. For large enough wehave:

0 < (u(p) v(q)) F(u(p), , X) F(v(q), , X)+ F(v(q), , Y) F(v(q), , Y)

F(v(q), , Y) F(v(q), , X) F(v(q), , Y) F(v(q), , Y + R) w(R) 0 as ,

where we have used the facts that u is a subsolution, that v is a supersolution,the inequality

X Y + R,the fact that F is proper and the continuity of F.

2.8. Uniformly Elliptic Equations with no First Derivatives De-pendence.

Theorem 9. LetF be continuous, proper, independent of and satisfying

|F(p, u, X) F(q,u, X)| w(p, q),where w(p, q) 0 as q1 p 0. Suppose thatF uniformly elliptic in thefollowing sense. There exists a constant > 0 such that

F(p, u, X) F(p, u, Y) trace(X Y)whenever X Y. Let u be an upper semicontinuous subsolution of

F(p, u, (D20u)) = 0

in a domain , v a lower semicontinuous supersolution, u v on as in2.7. Then

u(x) v(x) for all x .Proof. Suppose this is not the case. Then sup (u v) > 0. Let be anon-negative smooth function such that

(D20) < 0

everywhere in the bounded domain ( could be a quadratic polynomial).Note that for > 0 small enough, the function u v has an interior maxi-mum, where v = v + .


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18 JUAN J. MANFREDI

Claim. The function v is a strict supersolution of

F(p, u, (D20u)) = 0.

Proof. Let be a smooth test function touching v from below at p0

(p0) = v(p0)

(p) < v(p), p = p0.Then (p) (p) is a smooth test function that touches v from below atp0. Thus, we have

F(p0, (p0) (p0), (D2

0) (D2

0)) 0,which implies

F(p0, (p0), (D20)

(D20)) 0.Using the uniform ellipticity we get

F(p0, (p0), (D20)

) F(p0, (p0), (D20)(D20)) trace ((D20)(p0)).

We conclude that

F(p0, (p0), (D20)

) trace ((D20))(p0) > 0thereby proving the claim.

Apply the subelliptic maximum principle to u v. Since F is increasingin u:

0 < F(p, u(p), X) F(p, v(q), X)+F(q, v(q), X) F(q, v(q), X) 0 + w(p, q))

getting a contradiction for large enough.

3. Lecture III: The Maximum Principle for Riemannian VectorFields

Consider a frame X = {X1, X2, . . . , X n} in Rn consisting of n linearlyindependent smooth vector fields. Write

Xi(x) =

ni,j=1

aij(x)

xj

for some smooth functions aij(x). Denote by A(x) the matrix whose (i, j)-entry is aij(x). We always assume that det(A(x)) = 0 in Rn.

We need to write down an appropriate Taylor theorem. We will useexponential coordinates as done in [NSW]. Fix a point p Rn and lett = (t1, t2, . . . , tn) denote a vector close to zero. We define the exponential


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based at p of t, denoted by p(t), as follows: Let be the unique solutionto the system of ordinary differential equations

(s) =ni=1

tiXi((s))

satisfying the initial condition (0) = p. We set p(t) = (1) and note thisis defined in a a neighborhood of zero. In fact we have the following Calculuslemma.

Lemma 3. Writep(t) = 1p(t),

2p(t), . . . ,

np (t). Note that we can think

of Xi(x) as the i-th row ofA(x). Similarly Dkp(0) is the k-th column ofA(p) so that

Dp(0) = A(p).

For the second derivative we get

D2kp(0)h, h = At(p)h, D(At(p)h)kfor all vectors h Rn.

In particular, the mapping t p(t) is a diffeomorphism taking a aneighborhood of 0 into a neighborhood of p.

The gradient of a function u relative to the frame X is

DXu = (X1(u), X2(u), . . . , X n(u)).The second derivative matrix D2Xu is an n n - not necessarily symmetricmatrix with entries Xi(Xj(u)). We shall be interested in the quadraticform determined by this matrix, which is the same as the quadratic formdetermined by the symmetrized second derivative

(D2Xu) =

1

2

D2Xu + (D

2Xu)

t

.

The Taylor expansion from [NSW] can be stated as follows:

Lemma 4. Letu be a smooth function in a neighborhood of p. We have:

u (p(t)) = u(p) + DXu(p), t +1

2D2Xu(p) t, t + o(|t|2)as t 0.

A natural question is how DXu and (D2Xu)

change if we change frames.We could write down a long (and uninteresting) formula for the changebetween two general frames. Here is how to go from the canonical frame{ x1 , x2 , . . . xn } to the frame X. The gradient relative to the canonicalframe is denoted .Lemma 5. For smooth functions u we have

DXu = A u


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20 JUAN J. MANFREDI

and for all t Rn

D2Xu t, t = A D2u At t, t + nk=1

At t, At tk u

xk.

The Taylor series gives the following counterpart of theorem 1.

Lemma 6. Letu and v be smooth functions such that u v has an interiorlocal maximum at p. Then we have

(3.1) DXu(p) = DXv(p)

and

(3.2)

D2Xu(p) D2Xv(p) .

For the purposes of illustrations let us consider some examples:

Example 1. The canonical frame

This is just { x1 , x2 , . . . xn }. The first and second derivatives are just theusual ones and the exponential mapping is just addition

p(t) = p + t.

Example 2. The Heisenberg groupThe frame is given by the left invariant vector fields {X1, X2, X3}. Forp = (x,y ,z) the matrix A is just

A(p) =

1 0 y/20 1 x/2

0 0 1

.

A simple calculation shows that

At t, D At tk = 0

not only for k = 1 and k = 2 but also for k = 3. That is, although A is not

constant, we have that Lemma 5 simplifies to

(3.3) D2Xu t, t = A D2u At t, t.The exponential mapping is just the group multiplication

p(t) = p 0(t) = (x + t1, y + t2, z + t3 + (1/2)(xt2 yt1)).From Lemma 5 we see that the additional simplification of (3.3) occurswhenever D2kp(0) = 0. This is true for all step 2 groups as it can be seenfrom the Campbell-Hausdorff formula.

Example 3. The Engel group


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This is a step 3 group for which the analogue of (3.3) does not work. Denoteby p = (x,y ,z ,w) a point in R4. The frame is given by the vector fields:

X1 =

x y2 z +

xy12 z2

w

X2 =y +

x2z +

x2

12w

X3 =z +

x2w

X4 =w

So that the matrix A is

A =

1 0 y2 xy12 z20 1 x

2

x2

120 0 1 x20 0 0 1

Let t = (t1, t2, t3, t4). Direct calculations shows that

At t, D At t1 = 0

At t, D At t2 = 0

At t, D At t3 = 0

but for k = 4 we get

At t, D At tk = x

3t1t2 y

3t21 +

1

2t1t3.

Therefore, for a smooth function u we have

DXu = A Duand for all t Rn

D2Xu t, t = A D2u At t, t +

x

3t1t2 y

3t21 +

1

2t1t3

u

w.

3.1. Jets. To define second order superjets of an upper-semicontinuous func-tion u, let us consider smooth functions touching u from above a a pointp.

K2,+(u, p) = C2in a neighborhood of p, (p) = u(p),

(q) u(q), q= p in a neighborhood of p

Each function K2,+(u, p) determines a pair (, A) by

(3.4) =

X1(p), X2(p), . . . , X n(p)

Aij =

12

Xi(Xj())(p) + Xj(Xi())(p)

.

This representation clearly depends on the frame X. Using the Taylor the-orem for and the fact that touches u from above at p we get

(3.5) u (p(t)) u(p) + , t + 12At,t + o(|t|2)


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22 JUAN J. MANFREDI

We may also consider J2,+X (u, p) defined as the collections of pairs (, X)such that (3.5) holds. Using the identification given by (3.4) it is clear that

K2,+(u, p) J2,+X (u, p).In fact, we have equality. This is the analogue of the Crandall-Ishii Lemmaof [C].

Lemma 7.K2,+(u, p) = J2,+X (u, p).

Proof. Given a pair (, X) J2,+X (u, p) we must find a C2 function so that(3.4) holds. Given any pair (, Y) the version of the lemma for the canonicalframe in [C] gives a C2 function touching u from above at p such thatD(p) = and D2(p) = Y. Using Lemma 5 we get

DX(p) = A(p) and

D2X t, t = A Y At t, t +nk=1

At t, D At tkk.

Thus, it suffices to solve for (, Y) the equations

= A(p) and

X t, t = A Y At t, t + nk=1

At t, D At tkk.

Theorem 10. [BBM] The maximum principle for semicontinuousfunctions: Letu be upper semi-continuous in a bounded domain Rn.Letv be lower semi-continuous in . Suppose that for x we have

limsupyx


where both sides are not + or simultaneously. If u v has a positiveinterior local maximum

sup (u v) > 0then we have:

For > 0 we can find points p, q Rn such thati)

lim |p q|

2 = 0,

ii) there exists a point p such that p p (and so does q by (i))and

sup

(u v) = u(p) v(p) > 0,iii) there exist symmetric matricesX, Y and vectors + , so that


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iv)

(+ , X) J2,+X (u, p),v)

( , Y) J2,X (v, q),vi)

+ = o(1)and

vi)

X

Y + o(1)

as .Proof. The idea of the proof as in the case of theorem 7 is to use the Eu-clidean theorem to get the jets and then twist them into position. As in theproof of theorem 7, for > 0 we get points p and q so that (i) and (ii)hold. We apply now the Euclidean maximum principle for semicontin-uous functions of Crandall-Ishii-Lions [CIL]. There exist n n symmetricmatrices X, Y so that

( Dp((p, q)), X) J2,+eucl. (u, p)and

( Dq((p, q)), Y) J2,eucl. (v, q)with the property

(3.6) X, Y, C , where the vectors , Rn, and

C = (A2 + A)

andA = D2p,q((p, q))

are 2n 2n matrices.Let us now twist the jets according to Lemma 5. Call + = Dp((p, q))

and = Dq((p, q)). By our choice of we get + = . Set

+

=A

(p) +

and

= A(q) .We see that

|+ | = |A(p) A(q)||+ | C|p q||Dp((p, q))| C (p, q)= o(1),

where we have used the fact that |p q||Dp(p, q)| C(p, q), property (i)and the smoothness, in the form of a Lispchitz condition, ofA(p).


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24 JUAN J. MANFREDI

The second order parts of the jets are given by

X t, t = A(p)XAt(p) t, t +nk=1

At(p) t, D(At(p) t)k[p](+ )k

and

Y t, t = A(q)YAt(q) t, t +nk=1

At(q) t, D(At(p) t)k[q]( )k.

In order to estimate their difference we write

X

t, t

Y

t, t

=

XA

t(p)

t,At(p)

t

YA

t(q)

t,At(q)

t

+nk=1At(p) t, D(At(p) t)k[p](+ )knk=1At(q) t, D(At(p) t)k[q]( )k.

Using inequality 3.6, we get

X t, t Y t, t C(A(p) t A(q) t) ,A(p) t A(q) t+

nk=1At(p) t, D(At(p) t)k[p] pk (p, q)

nk=1At(q) t, D(At(p) t)k[q] pk (p, q)

To estimate the first term in the right hand side we note that symmetriesof give a block structure to D2p,q so that we have

C(

), (

)

C

|

|2.

Replacing by A(p) t and by A(q) t, using the smoothness of A,and property (i) we get that this first term is o(1). The second and thirdterm together are also o(1) since their difference is estimated by |p q||Dp(p, q)|.

Once we have the maximum principle (Theorem 10) we get easily com-parison theorems for viscosity solutions for fully nonlinear equations of thegeneral form

F(x, u(x), DXu(x), (D2Xu(x))

) = 0where F is continuous and proper (increasing in u and decreasing in (D2Xu(x))

).)as it is done in [CIL]. Here is an example:

Corollary 1. Suppose F(p, z, , X) satisfies

(r s) F(p, r, , X) F(p, s, , X),|F(p, r, , X) F(q,r,,X)| 1(|p q|),|F(p, r, , X) F(p, r, , Y)| 2(|X Y|) an|F(p, r, , X) F(p, r, , X)| 3(| |),

where the constant > 0 and the functions i : [0, ) [0, ) satisfyi(0

+) = 0 for i = 1, 2, 3. Let u be an upper-continuous viscosity solutionand v a lower semi-continuous viscosity supersolution to

F(x, u(x), DXu(x), (D2Xu(x))

) = 0


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in a domain so that for all p we havelimsupq,qp

u(q) lim infq,qp

v(q)

and both sides are not or simultaneously. Thenu(p) v(p)

for all p .

4. Lecture IV: The Maximum Principle in Carnot Groups

A Carnot group

Gof step r

1 is a simply connected nilpotent Lie group

whose Lie algebra g is stratified. This means that g admits a decompositionas a vector space sum

g = g1 g2 grsuch that

[g1, gj ] = gj+1

for j = 1, . . . , r with gk = {0} for k > j. Note that g is generated as a Liealgebra by g1.

Let mj = dim(gj) and choose a basis ofgj formed by left-invariant vectorfields Xi,j , i = 1, . . . , mj. The dimension ofG as a manifold is m = m1 +m2 + . . . + mr. The horizontal tangent space at a point p G is

Th(p) = linear span{

X1,1(p), X2,1(p), . . . , X m1,1(p)}

.

As in the Heisenberg group case, we say that a piecewise smooth curvet (t) is horizontal if (t) Th((t)) whenever (t) exists. Given twopoints p, q G denote by

(p, q) = {horizontal curves joining p and q}.Chows theorem states that (p, q) = .

For convenience, fix an ambient Riemannian metric in G so that X ={Xi,j}1mj ,j=1r is a left invariant orthonormal frame andRiemannian vol. element = Haar measure of G = Lebesgue meas. in Rm.

The Carnot-Caratheodory metric is then defined by

dcc(p, q) = inf{length() : (p, q)}.It depends only on the restriction of the ambient Riemannian metric tothe horizontal distribution generated by the horizontal tangent spaces. Theexponential mapping exp: g G is a global diffeomorphism.

A point p G has exponential coordinates (pi,j)1imj ,1jr if

p = exp

rj=1

mji=1

pi,jXi,j

.

Denoting by the group operation in G, the mapping (p, q) p q haspolynomial entries when written in exponential coordinates.


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26 JUAN J. MANFREDI

The non-isotropic dilations are the group homorphisms given by

t

rj=1

mji=1

pi,jXi,j

= r

j=1

mji=1

tjpi,jXi,j,

where t > 0.Recall that references for Carnot groups and Carnot-Caratheodory spaces

include: [B], [FS], [H], [G], [GN], and [Lu].

4.1. Calculus in G. Given a function u : G R we considerDXu = (Xi,ju)1imj ,1jr Rm,

the (full) gradient of u. As a vector field, this is written

DXu =r

j=1

mji=1

(Xi,ju)Xi,j

The horizontal gradient of u is

D0u = (Xi,1u)1im1

Rm1,

or as a vector field D0u =m1i=1(Xi,1u)Xi,1.

Theorem 11. Ball-Box Theorem:(See[B], [G], [NSW]) Set|p| = dcc(p, 0),the Carnot-Caratheodory gauge and let

|p|G = rj=1

mji=1

|pi,j |2 r!

j

1

2r!

be a smooth gauge. Then:

dcc(p, 0) |p|G r

j=1

mji=1

|pi,j |1

j

and

Vol(B(0, r)) rQ,

where B(0, r) is the Carnot-Caratheodory ball centered at 0 of radius r andQ =

rj=1jmj is the homogeneous dimension of G.

4.2. Taylor Formula. Suppose that u : G R is a smooth function. Letus write down the Taylor expansion with respect to the frame X at p = 0.


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Write P =rj=1

mji=1pi,jXi,j so that p = (pi,j) = exp(P).

u(exp(P)) = u(0)

+r

j=1

mji=1

(Xi,ju)(0)pi,j

+1

2

i,j,k,l

(Xk,lXi,ju)(0)pk,lpi,j

+ o rj=1

mji=1

|pi,j |2as p 0. We want to replace the quadratic norm in the error term by

|p|2G r

j=1

mji=1

|pi,j |2

j .

For j 3 it is clear that pi,j = o(|pi,j |2

j ) and for j, k 2 it is also clear thatpi,jpl,k = o(|pi,j |

2

j ). Throwing all these terms into the reminder we obtainthe subriemannian Taylor formula

u(exp(P)) = u(0)

+2

j=1

mji=1

(Xi,ju)(0)pi,j

+1

2

m1k,i=1

(Xk,1Xi,1u)(0)pk,1pi,1

+ o|p|2GWe already had defined the horizontal gradient of u as

D0u = (Xi,1u)1im1.

This is the part of the gradient corresponding to g1. It is clear from thesecond order Taylor formula that the part corresponding to g2, which wecall second order horizontal gradient for lack of a better name,

D1u = (Xi,2u)1im2

will also play a role in theory of second order subelliptic equations. We alsowrite

D20u = (Xi,1Xj,1u)m1i,j=1

for the second order derivatives corresponding to g1 and (D20u)

for its sym-metric part 12 (D

20u + (D

20u)

t). With these notations the Taylor formula at 0


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28 JUAN J. MANFREDI

reads

u(p) = u(0)

+ D0u(0), p1 + D1u(0), p2+

1

2(D20u(0)) p1, p1

+ o|p|2G ,where we have written p1 = (pi,1)1im1 and p2 = (pi,2)1im2 .

At another pointp0, we get the horizontal Taylor formula by left-translation.

Lemma 8. If u : G R is a smooth function near p0 we haveu(p) = u(p0)+D0u(p0), (p10 p)1 + D1u(p0), (p10 p)2

+1

2(D20u(p0))(p10 p)1, (p10 p)1 + o(|p10 p|2G)

as p p0.We continue the development of the theory as we did in the Heisenberg

group case.

4.3. Subelliptic Jets. Let u be an upper-semicontinuous real function inG. The second order superjet of u at p0 is defined as

J2,+(u, p0

) = (,, X) Rm1 Rm2 Sm1(R) such thatu(p) u(p0) + , (p10 p)1 + , (p10 p)2

+1

2X(p10 p)1, (p10 p)1 + o(|p10 p|2G)

Similarly, for lower-semincontinuous v, we define the second order subjet

J2,(v, p0) =

(,, Y) Rm1 Rm2 Sm1(R) such that

v(p) v(p0) + , (p10 p)1 + , (p10 p)2

+1

2Y(p10 p)1, (p10 p)1 + o(|p10 p|2G)

As before, one way to get jets is by using smooth functions that touch u fromabove or below. Let 2 denote the class of function such that D0, D1and D20 are continuous. We define

K2,+(u, p0) =

(D0(p0), D1(p0),(D

2(p0))) : 2

(p0) = u(p0)

(p) u(p), p = p0 in a neighborhood of p0

.

The set K2,+(u, p0) is defined analogously.


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Lemma 9. (See [C] for the Euclidean case, [Bi] for the Heisenberg groupcase, and [BM] for the Carnot case) We always have

K2,+(u, p0) = J2,+(u, p0)

and

K2,(u, p0) = J2,(u, p0)

We also define the closure of the second order superjet of an upper-semicontinuous function u at p0, denoted by J

2,+(u, p0), as the set of triples(,, X) Rm1 Rm2 S2(R) such that there exist sequences of points pmand triples (m, mXm) J

2,+

(u, pm) such that(pm, u(pm), m, m, Xm) (p0, u(p0), , , X)

as m . The closure of the second order subjet of a lower-semicontinuousfunction v at p0, denoted by J

2,(u, p0) is defined in an analogous manner.

4.4. Fully Non-Linear Equations. Consider a continuous function

F :G RRm1 Rm2 S(Rm1) R(p, u, , , X) F(p, u, , , X).

We will always assume that F is proper; that is, F is increasing in u and F

is decreasing in X.Definition 5. A lower semicontinuous function v is a viscosity supersolu-tion of the equation

F(p, v(p), D0v(p), D1v(p), (D2v(p))) = 0

if whenever (,, Y) J2,(v, p0) we have

F(p0, v(p0), , , Y) 0.Equivalently, if touches v from below and is in 2, then we must have

F(p0, v(p0), D0(p0), D1(p0), (D2(p0))

) 0.Definition 6. An upper semicontinuous functionu is a viscosity subsolutionof the equation

F(p, v(p), D0v(p), D0v(p), (D2v(p))) = 0

if whenever (,, X) J2,+(u, p0) we have

F(p0, u(p0), , ,X) 0.Equivalently, if touches u from above, is in 2, then we must have

F(p0, u(p0), D0(p0), D0(p0), (D2(p0))

) 0.


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30 JUAN J. MANFREDI

Note that if u is a viscosity subsolution and (,, X) J2,+(u, p0) then, bythe continuity of F, we still have

F(p0, u(p0), , ,X) 0.A similar remark applies to viscosity supersolutions and the closure of secondorder subjets.

A viscosity solution is defined as being both a viscosity subsolution anda viscosity supersolution. Observe that since F is proper, it follows easilythat if u is a smooth classical solution then u is a viscosity solution.

Examples of F:

Subelliptic Laplace equation (the Hormander-Kohn operator):

0u =

1jm1Xj,1Xj,1u

= 0

Subelliptic -Laplace equation:

0,u = m1i,j=1

(X1,iu)(X1,ju)X1,iX1,ju

= (D20u)D0u, D0u

Subelliptic p-Laplace equation, 2

p 1. This is a non-trivial matter not yet studied in the case ofthe Heisenberg group (to the best of my knowledge.) See [JLM] for theEuclidean case.

Naive subelliptic Monge-Ampere det(D20u) = f

Here the corresponding F(X) = det Xis only proper in the cone of positivesemidefinite matrices.

The next step is to generalize to semi-continuous functions the maximumprinciple for smooth functions easily obtained from the subelliptic Taylorformula. If u, v 2() and u v has a local maximum at p , we have

D0u(p) = D0v(p),

D1u(p) = D1v(p),

and

(D20u(p)) (D20v(p))


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Theorem 12. The maximum principle for semicontinuous func-tions: Let u be upper semi-continuous in a bounded domain G. Let vbe lower semi-continuous in . Suppose that for x we have

limsupyx


where both sides are not + or simultaneously. If u v has a positiveinterior local maximum

sup


For > 0 we can find points p, q G such thati)

lim

|p q|2 = 0,ii) there exists a point p such that p p (and so does q by (i))

andsup

(u v) = u(p) v(p) > 0,

iii) there exist m1 m1 symmetric matrices X, Y and vectors + , + , and

so that

iv)

(+ , + , X) J2,+(u, p),

v)( ,

, Y) J2,(v, q),

vi)+ = o(1),+ = o(1)

andvi)

X Y + o(1)as .

Proof. Let us apply Theorem 10 to the Carnot group G endowed with theleft-invariant frame X. We get riemannian jets

(+ , X) J2,+X (u, p)and

( , Y) J2,X (v, q),satisfying

+ = o(1)and

X Y + o(1).All we need to check is that by keeping the parts of + and

in g1 g2

and restricting X and Y to g1 we get subelliptic jets.


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32 JUAN J. MANFREDI

The following lemma follows from the Taylor theorem arguing as in section4.2 regarding higher order derivatives as part of the error term.

Lemma 10. Subelliptic jets from Riemannian jets: Let (, X) J2,+X (u, p) be a second order superjet. Then, we have

(1, 2, Xm1m1) J2,+(u, p)

4.5. Absolutely Minimizing Lipschitz Extensions. Let Rn be adomain. Consider m linearly independent vector fields

{X1, X2, . . . , X m},where m n. If there is an integer r 1 such that at any point x , thelinear span of{X1, X2, . . . , X m} and all their commutators up to order r hasdimension n, the systems of vector fields {X1, X2, . . . , X m} is said to satisfyHormanders condition. In [Ho] Hormander proved that if this condition issatisfied the second order operator X21 + X

22 + . . . + X

2m is hypoelliptic.

The control distance associated to {X1, X2, . . . , X m} is defined using hor-izontal curves as we did in Section 4. This control distance is a genuinemetric by the corresponding version of Chows theorem, see [NSW] and [B],and it satisfies

1

cK|x y| dcc(x, y) cK|x y|1

r

for some constant cK > 0 for x, y K compact subset of .For a function u : R the horizontal gradient of u is

Xu = (X1u, X2u , . . . , X mu).

The Sobolev space HW1,p(), 1 p , consists of functions u Lp()whose distributional horizontal gradient is also in Lp(). Endowed with thenorm

up + Xupthe space HW1,p() is a Banach space. As it is the case in the Euclidean

case, Lipschitz functions in a bounded domain with respect to the Carnot-Caratheodory metric dcc are precisely functions is HW1,(). See [FSS] and

[GN].

Definition 7. A function u HW1,() is an absolute minimizing Lips-chitz extension (AMLE) if whenever D is open, v HW1,(D) andu = v on D we have

XuL(D) XvL(D).Given any Lipschitz function f : R it can always be extended to

an AMLE in . This existence result holds in very general metric spaces asshown by Juutinen [Ju2].


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Jensen proved in [J2] that AMLE in Euclidean space are viscosity solu-tions of the -Laplace equation

ni,j=1

2u

xixj

u

xi

u

xj= 0,

and also proved a uniqueness theorem for solutions of the Dirichlet problem.As a corollary, he therefore obtained that AMLEs are determined by theirboundary values.

For general Carnot groups Bieske and Capogna [BC] proved that AMLEsare viscosity solutions of the corresponding

-Laplace equation

(4.1) ,0u =m

i,j=1

XiuXiuXiXju = 0.

Changyou Wang has recently extended Bieske and Capogna theorem togeneral Hormander vector fields:

Theorem 13. [W1] Let{X1, X2, . . . , X m} be system of Hormander vectorfields and let u be an AMLE. Then u is a viscosity solution of the equation4.1.

The natural question to consider is now the uniqueness of viscosity solu-tions of 4.1. This was proven by Bieske [Bi] in the case of the Heisenberg

group by using the extension of the maximum principle mentioned in Re-mark 2 of Section 2. Bieske has also considered the Grusin plane case in[Bi2], [Bi3]. The case of general Carnot group has recently being settled byChangyou Wang [W1], who introduced subelliptic sup-convolutions. To thebest of my knowledge, the case of general Hormander vector fields is open.

5. Lecture V: Convex functions on Carnot groups

Let us recall the definition of convexity in the viscosity sense:

Definition 8. Let G be an open set and u : R be an upper-semicontinuous function. We say that u is convex in if

(D20u)

0

in the viscosity sense. That is, if p and C2 touches u from aboveat p ((p) = u(p) and (q) u(q) for q near p) we have (D20)(p) 0.

This definition is compatible with the stratified group structure sinceconvexity is preserved by left-translations and by dilations. Also, uniformlimits of convex functions are convex and the supremum of a family of convexfunctions is convex, since these results hold for viscosity subsolutions ingeneral.

Note that we require the a-priori assumption of upper-semicontinuity asit is done in the definition of sub-harmonic functions. This is not neededwhen horizontally convex functions are considered. These are defined by


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34 JUAN J. MANFREDI

requiring that whenever p and the horizontal vector h G1 are suchthat {p t(h) : t (1, 1)} the function of one real variable(5.1) t u(p t(h))is convex for 1 < t < 1. A development of the theory based on the notionof horizontal convexity can be found in [DGN].

It was established in [LMS] that upper-semicontinuous horizontally con-vex functions are indeed convex in the sense of Definition 8. In this sectionwe will show that the reciprocal is also true (see Theorem 15 below.) Thisequivalence has also been established independently by Wang [W2] and Mag-

nani [M].In the Heisenberg group Balogh and Rickley [BR] proved that condition(5.1) by itself, without requiring upper-semicontinuity, suffices to guaranteethat u is continuous - and therefore Lipschitz continuous - and also showedthat horizontally convex functions are convex in the sense of Definition 8.

5.1. Convexity in the Viscosity Sense in Rn. In order to illustrate ourapproach in the case of general Carnot groups, we present here the Euclideanversion of Theorem 15 below.

Theorem 14. Let Rn be an open set and u : R be an upper-semicontinuous function. The following statements are equivalent:

i) whenever x, y

and the segment joining x and y is also in we

have

(5.2) u(x + (1 )y) u(x) + (1 )u(y)for all 0 1.

ii) u is a viscosity subsolution of all equations

F(x, u(x), Du(x), D2u(x)) = 0,

where F(x,z,p,M) is a continuous function in R Rn Snsatisfying the conditions in Section 4.4 and homogeneous; that isF(x,z,p, 0) = 0.

iii) u is a viscosity subsolution of all linear equations with constant co-efficients

F(x,u,Du,D2u) = trace A D2u = 0,where A Sn is positive definite.

iv) x u(Ax) is subharmonic for all A Sn positive definite;v) u satisfies the inequality

D2u 0in the viscosity sense;

vi) u satisfies trace(A D2u) 0 in the sense of distributions for allA Sn positive definite.

A function u is convex if one of the above equivalent statements holds.


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Proof. For the equivalence between i), ii), iii), and v) we refer to [LMS]. Theequivalence between vi) and iii) for any given matrix A is part of viscosityfolklore. This is just the simplest case of the theory of Hessian measuresof Trudinger and Wang [TW1]. To prove the equivalence between iv) andiii) observe that x u(Ax) is subharmonic for all A > 0 if and only iftrace(A D2u(x)At) 0 in the sense of distributions for all A > 0. Thisoccurs precisely when trace(At A D2u(x)) 0 in the sense of distributionfor all A > 0. Since every positive definite matrix B has a positive definitesquare root B = A2 = At A, we see that x u(Ax) is subharmonic for allA > 0 if and only if trace(B D2u(x)) 0 in the sense of distribution for allB > 0. 5.2. Convexity in Carnot Groups. A key observation is that the notionof convexity depends only on the horizontal distribution and not on theparticular choice of a basis ofg1. More precisely, let us consider two linearlyindependent horizontal frames

Xh = {X1, . . . , X m1}, Yh = {Y1, . . . , Y m1}and write Xi =

m1j=1 aijYj, for some constants aij. Let A be the matrix

with entries aij. The matrix A is not singular and the following formulaholds for any smooth function

(D20,X(p)) = A(D20,Y(p))

At.

Thus the matrix (D2h,X(p)) is positive definite if and only if (D2h,Y(p))is positive definite.

Given a frame X we denote by

Xu =

m1i=1

X2i u

the corresponding Hormander-Kohn Laplacian.The main result of this lecture is the analogue to Theorem 14.

Theorem 15. Let G be an open set and u : R be an upper-semicontinuous function. The following statements are equivalent:

i) whenever p

and h

G1 are such that

{p

t(h) : t

(

1, 1)

}

the function of one real variable

t u(p t(h))is convex for 1 < t < 1.

ii) u is a viscosity subsolution of all equations

F(p, u(p), D0u(p), (D20u(p))

) = 0,

where F(x,z,p,M) is proper and homogeneous.iii) u is a viscosity subsolution of all linear equations with constant co-

efficients

F(p, u, D0u, (D20u)

) = trace

A (D20u

) = 0,


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36 JUAN J. MANFREDI

where A Sm1 is positive definite.iv) u satisfies the inequality Yu 0 in the viscosity sense for all

framesY such thatYh = AXh, where A Sm1 is positive definite.v) u satisfies the inequality

(D20u) 0

in the viscosity sense.vi) u satisfies trace(A (D20u)) 0 in the sense of distributions for

all A Sm1 positive definite.A few remarks are in order. Condition i) is called horizontal convexity in

[LMS] and H-convexity in [DGN]. Note that iv) is indeed the analogue ofiv) in Theorem 14. Condition v) is called v-convexity in [LMS] and in [W2].The equivalence of the four viscosity related conditions ii), iii), iv), and v)

follows easily from elementary linear algebra facts as in theorem 14. More-over if one of these conditions holds, then u is locally bounded. This is thecase because u is always a subsolution of the corresponding -Laplacian (see5.3 below). The details in the case of the Heisenberg group are contained inthe proof of Lemma 3.1 in [LMS].

To show that iv) implies vi) we may do it one matrix A Sm1 at atime. Thus, we may assume that A is the identity matrix. Ifu is a boundedviscosity subsolution of the Hormander-Kohn Laplacian, it follows using thesame proofs as in Lemma 2.2 and Lemma 2.3 from [LMS] that u is weak-

subsolution with first horizontal derivatives locally square integrable. If uis not bounded below, we use the truncation uM(x) = max{M, u(x)} andstandard limit theorems (see [TW2].)

To prove the equivalence of (i) with the other conditions we need to estab-lish that convex functions can be approximated by smooth convex functions.Note that this is relatively easy to do for horizontally convex functions sincethe inequality (5.1) is preserved by convolution with a smooth mollifier (seethe proof of Theorem 4.2 in [LMS].) Fortunately, Bonfiglioli and Lanconelli[BL] have characterized subharmonic functions by a sub-mean value prop-erty and proved that subharmonic functions can be approximated by smoothsubharmonic functions. Moreover these approximations are frame indepen-dent. If follows from these results that vi) implies iv) since the implication

holds for smooth functions, and viscosity subsolutions are preserved by lo-cally uniform limits. The complete details are in [JLMS].

5.3. Regularity of Convex Functions. Convex functions are subsolu-tions of all homogeneous ellliptic equations. In particular we consider theHormander-Kohn Laplace equation

hu = (X21 u + X2mu) = 0,and the subelliptic -Laplace equation

(5.3) ,hu = m

i,j=1

(Xiu)(Xju)(XiXju) = 0.


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These equations can certainly be written in the form

F(p, u(p), D0u(p), (D20u)

(p)) = 0.

The proof that a convex function u is bounded is done in two steps. Tocontrol u+ we use the subelliptic version of the De Giorgi-Moser estimatefor subsolutions of linear elliptic equations

supBR

(u+) CB4R

u+ dx.

To control u we use comparison with cones defined using

-harmonic

functions. The details are in [LMS].To prove that a convex funtion is Lipschitz, the key ingredient is that sub-

solutions of (5.3) are Lipschtiz continuous. This was established by Jensen[J2] in the Euclidean case and Bieske [B] for the Heisenberg group. The caseof general Carnot groups follows from Wang [W1].

We collect these results in the following:

Theorem 16. Let G be an open set andu : R be a convex function.LetBR be a ball such that B4R . Thenu is locally bounded and we have.

(5.4) uL(BR) CB4R

|u| dx.

Moreover, u is locally Lipschitz and we have the bound

(5.5) DhuL(BR) C

RuL(B2R).

Here C is a constant independent of u and R. If, in addition, u is C2, thenthe symmetrized horizontal second derivatives are nonnegative

(5.6) (D2hu) 0.

A different proof of this theorem for horizontally convex functions is in[DGN].

6. Lecture VI: Subelliptic Cordes Estimates

The goal of this lecture is to present some estimates of Cordes type forlinear subelliptic partial differential operators in non-divergence form withmeasurable coefficients in the Heisenberg group, including the linearized p-Laplacian. The following regularity theorems follow from these techniques.

Theorem 17. Let

1712 p < 5+

5

2 . Then any p-harmonic function in

the Heisenberg group H initially in HW1,ploc is in HW2,2loc .Theorem 18. Given 0 < < 1 there exists = () such for |p 2| < ,p-harmonic functions in the Heisenberg group H have horizontal derivativesthat are Holder continuous with exponent .


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38 JUAN J. MANFREDI

The results from this section are basend on the work of Marchi [Ma, Ma2]extended by Domokos [D1], which give non-uniform bounds of the HW2,2

(or HW2,p) norm of approximate p-harmonic functions. Using the Cordescondition [Co, T] and Strichartzs spectral analysis [S] we establish HW2,2

estimates for linear subelliptic partial differential operators with measurablecoefficients.

Let H be a domain in the Heisenberg group. Consider the followingSobolev space with respect to the horizontal vector fields Xi as

HW2,2() = {u L2() : XiXju L2() , for all i, j {1, 2}}

endowed with the inner-product

(u, v)HW2,2() =

u(x)v(x) +

2i,j=1

XiXju(x) XiXjv(x)

dx .

HW2,2() is a Hilbert space and let HW2,20 () be the closure of C0 () in

this Hilbert space.Recall that X2u is the matrix of second order horizontal derivatives whose

entries are (X2u)ij = Xj(Xiu), and 0u =2i=1 XiXiu is the subelliptic

Laplacian associated to the horizontal vector fields Xi. The first step inproving Cordes estimates is to control the L2-norm of all the second deriva-

tives by the L2

-norm of the sublaplacian. For the symmetric part of thesecond derivative, this follows easily by integration by parts but to con-trol T u we need a more refined argument based on Strichartz [S] spectralanalysis.

Lemma 11. For all u HW2,20 () we have

X2uL2()

3 0uL2().

The constant

3 is sharp when = H.Proof. Since H and iT commute they share the same system of eigenvec-tors. Strichartz [S] computed them explicitly as well as the corresponding

eigenvalues. Computing L2-norms is now a matter of adding eigenvalues.Details can be found in [DM].

6.1. Cordes conditions. Let us consider now

Au =2

i,j=1

aij(x)XiXju

where the functions aij L(). Let us denote by A = (aij) the 2 2matrix of coefficients.


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Definition 9. [Co, T] We say that A satisfies the Cordes condition K, ifthere exists (0, 1] and > 0 such that

(6.1) 0 0 such that = 31 < 1 and Asatisfies the Cordes condition K,. Then for all u HW2,20 () we have

(6.2) X2uL2

31

1

LAuL2 ,where

(x) =A(x), IA(x)2 .

Proof. This is just Linear Algebra. See [DM] for the details.

6.2. HW2,2-interior regularity for p-harmonic functions in H. Let H be a domain, h HW1,p() and p > 1. Consider the problem ofminimizing the functional

(u) =

|Xu(x)|p dx

over all u

HW1,p() such that u

h

HW1,p

0

(). The Euler equationfor this problem is the p-Laplace equation

(6.3)2i=1

Xi|Xu|p2 Xiu = 0 , in .

A function u HW1,p() is called a weak solution of (6.3) if

(6.4)2i=1

|Xu(x)|p2Xiu(x) Xi(x)dx = 0 , for all HW1,p0 () .

is a convex functional on HW1,p, therefore weak solutions are minimiz-ers for . For m N define the approximating problems of minimizingfunctionals

m(u) =

1

m+ |Xu(x)|2

p2

and the corresponding Euler equations

(6.5)2i=1

Xi

1

m+ |Xu|2

p22

Xiu

= 0 , in .

The differentiated version of this equation has the form

(6.6)2

i,j=1

amij XiXju = 0 , in


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40 JUAN J. MANFREDI

where

amij (x) = ij + (p 2)Xiu(x) Xju(x)

1m + |Xu(x)|2

.

Consider a weak solution um HW1,p() of equation (6.5). Then coef-ficients amij are bounded. Define the mapping Lm : HW

2,20 () L2()

by

(6.7) Lm(v)(x) =2

i,j=1

amij (x)XiXjv(x) .

The Cordes condition is satisfied precisely when

p 2

1 52

,1 +

5

2

independently of m.

Theorem 20. The case p > 2: For 2 p < 2 + 1+

52 if u W1,p() is a

minimizer for the functional , then u HW2,2loc ().Proof. The case p = 2 it is well known, so let us suppose p = 2. Letu HW1,p() be a minimizer for . Consider x0 and r > 0 such thatB4r = B(x0, 4r)

. Choose a cut-off function

C0 (B2r) such that

= 1 on Br. Also consider minimizers um for m on HW1,p(B2r) subjectto um u HW1,p0 (B2r). Then um u in HW1,p(B2r) as m .

By [D1, Ma] for 2 p < 4 we have um HW2,2loc (), but with boundsdepending on m. Also that um satisfies equation Lm(um) = 0 a.e. in B2r.So, in B2r we have a.e.

XiXj(2um) = XiXj(

2)um + Xj(2)Xium + Xi(

2)Xjum + 2XiXjum

and hence

Lm(2um) = um Lm(

2) +2

i,j=1

amij (x)

Xj(2)Xium + Xi(

2)Xjum

.

From the Cordes estimate it follows thatX2umL2(Br) X2(2um)L2(B2r) cLm(2um)L2(B2r)

cumW1,p(B2r) cuHW1,p(B2r)where c is independent of m. Therefore, u HW2,2(Br).

For p < 2 a different argument is needed since we only get u HW2,ploc ().We finish by quoting the result obtained in [DM]:

Theorem 21. The case p < 2: For the range

1712 p 2 if u

HW1,p() is a minimizer for the functional , then u W2,2loc ().


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Department of Mathematics, University of Pittsburgh, Pittsburgh, PA15260, USA

E-mail address: [email protected]

arch ivo newton

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