Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon
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114
Chapter 6: Some Extremal Problems in Martingale Theory
References
[1] R. Baiiuelos and A. Lindeman, A martingale study of the Beurling-Ahifors
transform in R", J. Funct. Anal. 145 (1997), 224-265.
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cations to the Beurling-Ahifors and Riesz transforms, Duke Math. J. 30
(1995), 575-600.
[3] D. L. Burkholder, Boundary value problems and sharp inequalities for
martingale transforms, Ann. Probab. 12 (1984), 647-702.
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Sharp inequalities for martingales and stochastic integrals, Col-
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Differential subordination of harmonic functions and martin-
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Explorations in martingale theory and its applications, Ecole
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Strong differential subordination and stochastic integration, Ann.
Probab. 22 (1994), 995-1025.
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Sharp norm comparison of martingale maximal functions and
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in Applied Mathematics 52 (1997), 343-358.
[9] C. Choi, A submartingale inequality, Proc. Amer. Math. Soc. 124 (1996),
2549-2553.
[10] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer,
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[11] W. Hammack, Sharp maximal inequalities for stochastic integrals in
which the integrator is a submartingale, Proc. Amer. Math. Soc. 124
(1996),931-938.
[12] T. Iwaniec, private communication, June 1996.
[13] T. Iwaniec and G. Martin, Riesz transforms and related singular integrals,
J. reine angew. Math. 473 (1996), 25-57.
Burkholder 115
[14] G. Wang, Differential subordination and strong differential subordina-
tion for continuous-time martingales and related sharp inequalities, Ann.
Probab. 23 (1995), 522-551.
Chapter 7
The Monge Ampere Equation,
Allocation Problems, and
Elliptic Systems with Affine
Invariance
Luis A. Caffarelli
The Monge Ampere equation, a classical equation in differential geometry
and analysis, has been surfacing more and more often in different contexts of
applied mathematics in recent times, posing new and challenging problems.
In this paper I will try to present some of these problems, partial answers
and possible directions.
7.1
The Monge Ampere Equation as a Fully
Nonlinear Equation
The Monge Ampere equation consists of finding a convex function cp that
satisfies
det D2cp = f (x, Dip).
It has a rich history in differential geometry and analysis.
It appears locally when prescribing the density of the Gauss map, i.e.,
the map that assigns, to each point (X,cp(X)) on the surface S = graph (cp),
the unit normal vector v (convexity of co guarantees heuristically that v is a
nice oriented "one to one" map).
Research partially supported by an NSF grant.
117
118
Chapter 7: The Monge Ampere Equation
It also appears in problems of optimal design in optics (see Oliker [111)
in Riemannian geometry, etc. The convexity restriction on cp, that as we
will see later appears naturally in several contexts, has the virtue that it
almost puts the Monge Ampere equation in the context of solutions to fully
nonlinear equations:
F(D2w) = f.
Indeed, in this theory, one requires F to be strictly monotone and Lip-
schitz as a function of symmetric matrices M, i.e.,
F(M) + )IINlI < F(M + N) G F(M) + AIINUU
for any symmetric matrices M, N, with N positive.
In the case of det M, both monotonicity and Lipschitz regularity hold
as long as M is positive (cp convex) and M, N remain bounded, but it
deteriorates as k goes to infinity.
Our interest in fitting the Monge Ampere equation in the framework of
fully nonlinear elliptic equations is of course the powerful local regularity
theory available.
7.2
Regularity Theory of Fully Nonlinear
Equations
We point out at this time that in the study of second-order, nonlinear equa-
tions there are usually two approaches to regularity theory: boundary inher-
ited regularity and local regularity.
Boundary inherited regularity looks, for instance, at a Dirichlet problem,
F(D2u) = f (x) in V
u = g(x) along the smooth boundary, oD
and tries first to control first and second derivatives of u along 8D, by ap-
propriate barriers, and next to extend this control to the interior of V by
maximum principle techniques for Du and D2u. This method is convenient
when the geometry of the problem is good to start with (9(x) and 01D are
smooth and geometrically constrained).
Another type of theory, the one I want to address here, is that of interior
a priori estimates, in which we are given a bounded solution u of
F(D2u) = f (x)
in B1, where u is smooth in any bounded subdomain, B1_t.
Caffarelli
119
The advantage of this theory is in its compactness properties and, as
a consequence, that it occurs in the natural functional spaces (u is "two
derivatives" better than f both in Ca and L" spaces).
Since this last theory is available for solutions of uniformly elliptic, fully
nonlinear equation, one may hope to apply it with some modification to the
Monge Ampere equation. That this is not the case, at least in a straightfor-
ward way, is due to the invariances of the Monge Ampere equation.
7.3
The Monge Ampere Equation: Invariance
and Counterexamples
The Monge Ampere equation, say
det D2cp = 1,
is not only invariant under the action of rigid motions R (i.e., ;P(X) = rp(RX )
is again a solution of the same equation) and quadratic dilations (i.e., ;p(X) =
Ay W(AX)) but also under any affine transformation T, with det T = 1 (i.e.,
7(X) = cp(TX) solves the same equation).
This last property has as a consequence that we may submit the graph of
a solution cp to an enormous deformation %X) = W(Ex1i Ex2) and still obtain
a solution of the same equation.
This makes the existence of nonsmooth solutions almost unavoidable. For
instance (Pogorelov), for n > 2,
SP(X) = `(xl,... ,
xn-1)I2-2/n f(xn)
is a solution for
1
- .)f f" - (2
- 2
)
(f,)21
f--2
n-2 = constant
[(1
(an ODE with nice local solutions). This W is nonnegative, but D2<p degen-
erates along the line xn = 0, where tp =_ 0. Therefore, an absolute regularity
theory is out of the question.
On the other hand, the positive effect of the rich renormalization proper-
ties of the Monge Ampere equation is that, if we start from a "reasonable"
situation, the possibility of renormalizing at every scale should give us a way
of controlling the geometry of cp in an iterative fashion.
7.4
Some Local Theorems
In order to describe these ideas let me state three basic facts.
120
Chapter 7: The Monge Ampere Equation
Fact 1 (Pogorelov):
Let cp be a (smooth) solution of
J det D2cp = 1
in St,
W =_ 0
on 199.
Assume that B1 C Q C BK, then
D2cp(0) < C(K).
Fact 2 (Elementary comparison):
Let WO be as in Fact 1, and
detD2cp1
- 1I < s,
cp,Ia. = 0.
Then
IWo - W1I <- Cc
in Q.
Fact 3 (Global propagation of singularities):
(a) 0 < A < detD 2W <A<ooinB1i
(b) cp > 0, cp(0) = 0.
Then the set
{X:W =0}
is (i) a point or (ii) a convex set I' with no extremal points in B1, i.e., gener-
ated by convex combinations of points in F n 0B1.
Note that the three facts are renormalization invariant. Note also that
Fact 3 is by compactness a very strong statement of strict convexity.
Indeed, if Wn is a sequence of bounded nonnegative solutions to the Monge
Ampere equation 0 < ) < det D2Wn < A that slowly deteriorates around
the origin (say inf (WnlaB,) (0) goes to zero), a limiting cpo has a nontrivial
set {cpo = 0} and therefore, from Fact 3, a very large singular set {Wo = 0).
The combination of these facts allows us to prove through renormalization
of the sections S(2) = {x : W (x) - £(x) < 0), f a linear function, that the
following holds:
"Theorem": Unless p degenerates according to Fact 3, cp is, in any compact
subset of B1, two derivatives better than f, i.e.,
(a) If f is bounded, cp is Cl," for some a (Harnack inequality for the Monge
Ampere equation).
(b) If f is continuous, cp is W2,r for every p (Calderbn-Zygmund).
(c) If f is C°, cp is C2'° (Schauder).
We will now discuss some applications of this theory.
Caffarelli
121
7.5
The Monge Ampere Equation as a
Potential of an "Irrotational Map" in
Lagrangian Coordinates
Two concepts at the heart of modelization in continuum mechanics are com-
pression and rotation. In Eulerian coordinates, infinitesimal compression is
measured through the divergence of the infinitesimal displacement (velocity)
and rotation through the curl.
Mathematically, the important fact is that the prescription of div v =
Trace Dv and curl = Dv - (Dv)t constitutes the simplest linear elliptic sys-
tem, and thus we have the two typical theorems:
Theorem 7.1. If div, curl belong to a given function space (say Ck,O or
Wk.P), then all of Dv belongs to that space (with estimates), or
Theorem 7.2 (Helmholtz decomposition). Given v in a function space,
it can be decomposed as v = U, +U2, with U, incompressible and v2 irrotational
in the same function space (with estimates). In fact v2 is found as v2 = Vu,
with
Du = div v
boundary co
nditions
and iJ =v-v2.
In Lagrangian coordinates, the adequate tool to describe large deforma-
tions, it is clear what the notion of compression and incompressibility should
be: instead of just having a velocity field, our dependent variable Y(X, t)
is the position at time t of that particle that at time zero was at X, and
therefore the compression that a differential of volume given at time zero
has suffered at time t is simply det DxY (i.e., conservation of mass becomes
pdet DXY = constant).
Rotation is a more unsettled question, and I assume that, for different
purposes, different definitions occur.
We will discuss briefly later the Cauchy-Novozilov notion of rotation, but
one may look at Truesdell and Toupin (16].
Let us agree now that irrotational means DIY symmetric; that is, heuris-
tically, v is the gradient of a potential gyp, and then we can state Brenier's
factorization [1].
Theorem 7.3. Let f1i S22 be two bounded domains of R", U a measurable
map from S21 into 12 that does not collapse sets of positive measure (i.e., if
E C 92 has AEI = 0 then Jv-1(E)I = 0). Then v can be factorized as
v= T2 . V1,
122
Chapter 7: The Monge Ampere Equation
where
vr:S21-- ui
is incompressible, i.e., for any measurable E C SZr
I(v1)-1(E)I = IEI
and v2 is irrotational; that is,
v2=VV,
where cp is convex.
Given this decomposition, we note that the questions raised by Theorem
7.2 become relevant: Suppose that v is in some function space (Wkp or
Ck'°). Is it true that v1, v2 are in the same spaces? This is the content of
several papers by the author (see [2],
[3], [4], [5]) where this is shown to
hold under the necessary hypothesis that S21 and S22 are convex. This is
achieved through the local regularity theory for convex solutions 'p of the
Monge Ampere equation
det D2'p = f
provided the "cornpression" f is under control.
7.6
Optimal Allocation Problems
In fact, the map V2 above depends on v only through the density 5(y) =
det D2v, with which Sit is carried into S22i and it has a classic interpretation as
an "optimal allocation" problem between S21 and Sl2 with prescribed densities
61(X ), 62(Y):
Let us describe optimal allocations in a discrete setting: Assume we are
given k points X1, ... , Xk and Y1, ... , Yk in
' and we want to map the Xi
onto the Y's, minimizing a "transportation" cost
T(Y(X)) = EiC(Y(Xi) - Xi),
where C(X - Y) represents the cost of transporting X into Y.
In the finite case, an optimal map YO(X) clearly exists, and it is cyclically
monotone in the sense that, for any permutation 7r(i), EC(Y(Xi) - X,(i)) >
EC(Y(Xi) - Xi). If C(Y - X) is IY - X12 then this condition transforms
into
E(Y(Xi),Xx(o)) <- E(Y(Xi) - Xi),
the classical definition of cyclically monotone maps.
CaffarelIi
123
A theorem of Rockafellar [14) then proves that Y is the gradient of a
convex potential. The continuous situation can then be described by the
following theorem: Suppose we are given the cost function
C(Y-X)= IY-X}2
62(Y) withtwo domains Q1, Q2 and densities S1(X),JS2dY.
f
S1dX =
(a) There exists a unique map Yo(X) that preserves densities:
i.e., for
every continuous 77
f
ri(Y) 52(Y) dY =
f
i?(Yo(X )) 61(X) dYS22
and such that minimizes
fIY(X)-X1S1(X)dX
among all such maps.
(b) Y = V(p, cp convex and if S2i are convex, v is locally "two derivatives
better" than S1, 52.
A very beautiful problem arises when studying general strictly convex cost
functions C, with differential C5 Up Then (Yo(X) - X) is an irrotational
vector field VV. Heuristically cp satisfies then
def (X + B ( ) _
S2 (X + B; (Ow)
where B the inverse map to C;, is the gradient of B, the conjugate con-
vex function to C. Little is known about solving this Monge Ampere type
equation (see [8], [10)).
Note the analogy of this family of Monge Ampere problems with the Euler
equations coming from convex functionals:
div (B; (VV)) = 0.
These last are sort of a "linearization" of the ones above.
7.7
Optimal Allocations in Continuum
Mechanics
This is an area that is evolving, so I will just make a few remarks. We start
by noticing the relation between the functional (Si(X) = 1),
J =
f(Y(X)
- X)2 dX,