Ambrosio L., Caffarelli L., Crandall M.G., Evans L.C., Fusco N. Calculus of Variations and Nonlinear Partial Differential Equations

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150 L.C. Evans

is periodic (even though D

= x + D

is not). Hence



H(Du

,x) · D

wσ

=2πi



2πim·D

m · D

=2πi



2πim·D

m · D

(P )σ

+2πi



2πim·D

m(D

− D

)σ

=: A + B.

Using formula (4.10), we can estimate

|B|≤C(

(P )|)

1/2

= o(1) as k →∞.

Consequently A → 0. Since

m · D

(P ) → m · V =0,

we deduce that



2πim·D

dx → 0.

This proves (4.12) for all ﬁnite trigonometric polynomials Φ, and then, by the

density of such trig polynomials, for all continuous Φ. !"

5 Some Other Viewpoints and Open Questions

This concluding section collects together some comments about other work

on, and extending, weak KAM theory and about possible future progress.

• Geometric properties of the eﬀective Hamiltonian

Concordel in [C1], [C2] initiated the systematic study of the geometric

properties of the eﬀective Hamiltonian

H, but many questions are still open.

Consider, say, the basic example H(p, x)=

|p|

+ W (x)andaskhow

the geometric properties of the periodic potential W inﬂuence the geometric

properties of

H, and vice versa. For example, if we know that

H has a “ﬂat

spot” at its minimum, what does this imply about W ?

It would be interesting to have some more careful numerical studies here,

as for instance in Gomes-Oberman [G-O].

Weak KAM Theory and Partial Diﬀerential Equations 151

• Nonresonance

Given the importance of nonresonance assumptions for the perturbative

classical KAM theory, it is a critically important task to understand conse-

quences of this condition for weak KAM in the large. Theorem 4.3 is a tiny

step towards understanding this fundamental issue. See also Gomes [G1], [G2]

for some more developments.

• Aubry and Mather sets

See Fathi’s book [F5] for a more detailed discussion of the Mather set

(only mentioned above), the larger Aubry set, and their dynamical systems

interpretations. One fundamental question is just how, and if, Mather sets can

act as global replacements for the classical KAM invariant tori.

• Weak KAM and mass transport

Bernard and Buﬀoni [B-B1], [B-B2], [B-B3] have rigorously worked out

some of the fascinating interconnections between weak KAM theory and opti-

mal mass transport theory (see Ambrosio [Am] and Villani [V]). Some formal

relationships are sketched in my expository paper [E5].

• Stochastic and quantum analogs

My paper [E4] discusses the prospects of ﬁnding some sort of quantum

version of weak KAM, meaning ideally to understand possible connections

with solutions of Schr¨odinger’s equation in the semiclassical limit h → 0. This

all of course sounds good, but it is currently quite unclear if any nontrivial

such connections really exist.

N. Anantharaman’s interesting paper [An] shows that a natural approx-

imation scheme (independently proposed in [E4]) gives rise to Mather mini-

mizing measures which additionally extremize an entropy functional.

Gomes in [G3] and Iturriaga and Sanchez-Morgado in [I-SM] have dis-

cussed stochastic versions of weak KAM theory, but here too many key ques-

tions are open.

• Nonconvex Hamiltonians

It is, I think, very signiﬁcant that the theory [L-P-V] of Lions, Papanico-

laou and Varadhan leads to the existence of solutions to the generalized eikonal

equation (E) even if the Hamiltonian H is nonconvex in the momenta p:all

that is really needed is the coercivity condition that lim

|p|→∞

H(p, x)=∞,

uniformly for x ∈ T

. In this case it remains a major problem to interpret

in terms of dynamics.

Fathi and Siconolﬁ [F-S2] have made great progress here, constructing

much of the previously discussed theory under the hypothesis that p →

H(p, x)begeometrically quasiconvex, meaning that for each real number λ

and x ∈ T

, the sublevel set {p | H(p, x) ≤ λ} is convex.

The case of Hamiltonians which are coercive, but nonconvex and nonqua-

siconvex in p, is completely open.

152 L.C. Evans

• Aronsson’s PDE.

I mention in closing one ﬁnal mystery: does Aronsson’s PDE (4.8) have

anything whatsoever to do with the Hamiltonian dynamics? This highly de-

generate, highly nonlinear elliptic equation occurs quite naturally from the

variational construction in Section 4, but to my knowledge has no interpre-

tation in terms of dynamical systems. (See Yu [Y] and Fathi–Siconolﬁ [F-S3]

for more on this strange PDE.)

References

[Am] L. Ambrosio, Lecture notes on optimal transport problems, in Mathemat-

ical Aspects of Evolving Interfaces 1–52, Lecture Notes in Mathematics

1812, Springer, 2003.

[An] N. Anantharaman, Gibbs measures and semiclassical approximation to

action-minimizing measures, Transactions AMS, to appear.

[B] E. N. Barron, Viscosity solutions and analysis in L

∞

,inNonlinear Analy-

sis, Diﬀerential Equations and Control, Dordrecht, 1999.

[B-B1] P. Bernard and B. Buﬀoni, The Monge problem for supercritical Ma˜n´e

potentials on compact manifolds, to appear.

[B-B2] P. Bernard and B. Buﬀoni, Optimal mass transportation and Mather

theory, to appear.

[B-B3] P. Bernard and B. Buﬀoni, The Mather-Fathi duality as limit of optimal

transportation problems, to appear.

[C1] M. Concordel, Periodic homogenization of Hamilton-Jacobi equations I:

additive eigenvalues and variational formula, Indiana Univ. Math. J. 45

(1996), 1095–1117.

[C2] M. Concordel, Periodic homogenization of Hamilton-Jacobi equations II:

eikonal equations, Proc. Roy. Soc. Edinburgh 127 (1997), 665–689.

[E1] L. C. Evans, Partial Diﬀerential Equations, American Mathematical So-

ciety, 1998. Third printing, 2002.

[E2] L. C. Evans, Periodic homogenization of certain fully nonlinear PDE,

Proc Royal Society Edinburgh 120 (1992), 245–265.

[E3] L. C. Evans, Some new PDE methods for weak KAM theory, Calculus of

Variations and Partial Diﬀerential Equations, 17 (2003), 159–177.

[E4] L. C. Evans, Towards a quantum analogue of weak KAM theory, Com-

munications in Mathematical Physics, 244 (2004), 311–334.

[E5] L. C. Evans, Three singular variational problems, in Viscosity Solutions

of Diﬀerential Equations and Related Topics, Research Institute for the

Mathematical Sciences, RIMS Kokyuroku 1323, 2003.

[E6] L. C. Evans, A survey of partial diﬀerential equations methods in weak

KAM theory, Communications in Pure and Applied Mathematics 57

(2004), 445–480.

[E-G1] L. C. Evans and D. Gomes, Eﬀective Hamiltonians and averaging for

Hamiltonian dynamics I, Archive Rational Mech and Analysis 157 (2001),

1–33.

[E-G2] L. C. Evans and D. Gomes, Eﬀective Hamiltonians and averaging for

Hamiltonian dynamics II, Archive Rational Mech and Analysis 161

(2002), 271–305.

Weak KAM Theory and Partial Diﬀerential Equations 153

[F1] A. Fathi, Th´eor`eme KAM faible et theorie de Mather sur les systemes

lagrangiens, C. R. Acad. Sci. Paris Sr. I Math. 324 (1997), 1043–1046.

[F2] A. Fathi, Solutions KAM faibles conjuguees et barrieres de Peierls, C. R.

Acad. Sci. Paris Sr. I Math. 325 (1997), 649–652.

[F3] A. Fathi, Orbites heteroclines et ensemble de Peierls, C. R. Acad. Sci.

Paris Sr. I Math. 326 (1998), 1213–1216.

[F4] A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad.

Sci. Paris Sr. I Math. 327 (1998), 267–270.

[F5] A. Fathi, The Weak KAM Theorem in Lagrangian Dynamics (Cambridge

Studies in Advanced Mathematics), to appear.

[F-S1] A. Fathi and A. Siconolﬁ, Existence of C

critical subsolutions of the

Hamilton-Jacobi equation, Invent. Math. 155 (2004), 363–388.

[F-S2] A. Fathi and A. Siconolﬁ, PDE aspects of Aubry-Mather theory for quasi-

convex Hamiltonians, Calculus of Variations and PDE 22 (2005), 185–228.

[F-S3] A. Fathi and A. Siconolﬁ, Existence of solutions for the Aronsson–Euler

equation, to appear.

[Fo-M] G. Forni and J. Mather, Action minimizing orbits in Hamiltonian systems,

in Transition to Chaos in Classical and Quantum Mechanics, Lecture

Notes in Math 1589, edited by S. Graﬃ, Springer, 1994.

[G] H. Goldstein, Classical mechanics (2nd ed), Addison-Wesley, 1980.

[G1] D. Gomes, Viscosity solutions of Hamilton-Jacobi equations and asymp-

totics for Hamiltonian systems, Calculus of Variations and PDE 14 (2002),

345–357.

[G2] D. Gomes, Perturbation theory for Hamilton-Jacobi equations and sta-

bility of Aubry-Mather sets, SIAM J. Math. Analysis 35 (2003), 135–147.

[G3] D. Gomes, A stochastic analog of Aubry-Mather theory, Nonlinearity 10

(2002), 271-305.

[G-O] D. Gomes and A. Oberman, Computing the eﬀective Hamiltonian using

a variational approach, SIAM J. Control Optim. 43 (2004), 792–812.

[I-SM] R. Iturriaga and H. Sanchez-Morgado, On the stochastic Aubry-Mather

theory, to appear.

[K] V. Kaloshin, Mather theory, weak KAM and viscosity solutions of

Hamilton-Jacobi PDE, preprint.

[L-P-V] P.-L. Lions, G. Papanicolaou, and S. R. S. Varadhan, Homogenization of

Hamilton–Jacobi equations, unpublished, circa 1988.

[Mn] R. Ma˜n´e, Global Variational Methods in Conservative Dynamics, Insti-

tuto de Matem´atica Pura e Aplicada, Rio de Janeiro.

[M1] P. Marcellini, Regularity for some scalar variational problems under

general growth conditions, J Optimization Theory and Applications 90

(1996), 161–181.

[M2] P. Marcellini, Everywhere regularity for a class of elliptic systems without

growth conditions, Annali Scuola Normale di Pisa 23 (1996), 1–25.

[M-M] E. Mascolo and A. P. Migliorini, Everywhere regularity for vectorial func-

tionals with general growth, ESAIM: Control, Optimization and Calculus

of Variations 9 (2003), 399-418.

[Mt1] J. Mather, Minimal measures, Comment. Math Helvetici 64 (1989),

375–394.

[Mt2] J. Mather, Action minimizing invariant measures for positive deﬁnite La-

grangian systems, Math. Zeitschrift 207 (1991), 169–207.

154 L.C. Evans

[P-R] I. Percival and D. Richards, Introduction to Dynamics, Cambridge Uni-

versity Press, 1982.

[V] C. Villani, Topics in Optimal Transportation, American Math Society,

2003.

[W] C. E. Wayne, An introduction to KAM theory, in Dynamical Systems and

Probabilistic Methods in Partial Diﬀerential Equations, 3–29, Lectures in

Applied Math 31, American Math Society, 1996.

[Y] Y. Yu, L

∞

Variational Problems, Aronsson Equations and Weak KAM

Theory, thesis, University of California, Berkeley, 2005.

Geometrical Aspects of Symmetrization

Nicola Fusco

Dipartimento di Matematica e Applicazioni

Universit`a di Napoli “Federico II” Via Cintia, 80126 Napoli, Italy

n.fusco@unina.it

1 Sets of ﬁnite perimeter

Symmetrization is one of the most powerful mathematical tools with several

applications both in Analysis and Geometry. Probably the most remarkable

application of Steiner symmetrization of sets is the De Giorgi proof (see [14],

[25]) of the isoperimetric property of the sphere, while the spherical sym-

metrization of functions has several applications to PDEs and Calculus of

Variations and to integral inequalities of Poincar´e and Sobolev type (see for

instance [23], [24], [19], [20]).

The two model functionals that we shall consider in the sequel are: the

perimeter of a set E in IR

and the Dirichlet integral of a scalar function

u. It is well known that on replacing E or u by its Steiner symmetral or its

spherical symmetrization, respectively, both these quantities decrease. This

fact is classical when E is a smooth open set and u is a C

function ([22],

[21]). Moreover, on approximating a set of ﬁnite perimeter with smooth open

sets or a Sobolev function by C

functions, these inequalities can be easily

extended by lower semicontinuity to the general setting ([19], [25], [2], [4]).

However, an approximation argument gives no information about the equality

case. Thus, if one is interested in understanding when equality occurs, one has

to carry on a deeper analysis, based on ﬁne properties of sets of ﬁnite perimeter

and Sobolev functions.

Let us start by recalling what the Steiner symmetrization of a measurable

set E is. For simplicity, and without loss of generality, in the sequel we shall

always consider the symmetrization of E in the vertical direction. To this aim,

it is convenient to denote the points x in IR

also by (x



,y), where x



∈ IR

n−1

and y ∈ IR. Thus, given x



∈ IR

n−1

, we shall denote by E



the corresponding

one-dimensional section of E



= {y ∈ IR : ( x



,y) ∈ E} .

The distribution function µ of E is deﬁned by setting for all x



∈ IR

n−1

156 N. Fusco

µ(x



)=L



) .

Here and in the sequel we denote by L

the Lebesgue measure in IR

. Then,

denoting the essential projection of E by π(E)

= {x



∈ IR

n−1

: µ(x



) > 0},

the Steiner symmetral of E with respect to the hyperplane {y =0} is the set

= {(x



,y): x



∈ π(E)

, |y| <µ(x



)/2}.

Notice that by Fubini’s theorem we get immediately that µ is a L

n−1

measurable function in IR

n−1

, hence E

is a measurable set in IR

and

(E)=L

). Moreover, it is not hard to see that the diameter of E de-

creases under Steiner symmetrization, i.e., diam(E

) ≤ diam(E), an inequal-

ity which in turn implies ([1, Proposition 2.52]) the well known isodiametric

inequality

(E) ≤ ω



diam(E)



where ω

denotes the measure of the unit ball in IR

Denoting by P (E)theperimeter of a measurable set in IR

, the following

result states that the perimeter too decreases under Steiner symmetrization.

Theorem 1.1. Let E ⊂ IR

be a measurable set. Then,

P (E

) ≤ P (E) . (1.1)

As we said before, inequality (1.1) is classic when E is a smooth set and can

be proved by a simple approximation argument in the general case of a set of

ﬁnite perimeter. However, following [9], we shall give here a diﬀerent proof of

Theorem 1.1, which has the advantage of providing valuable information in

the case when (1.1) reduces to an equality.

Let us now recall the deﬁnition of perimeter. If E is a measurable set in

and Ω ⊂ IR

is an open set, we say that E is a set of ﬁnite perimeter

in Ω if the distributional derivative of the characteristic function of E, Dχ

is a vector-valued Radon measure in Ω, with ﬁnite total variation |Dχ

|(Ω).

Thus, denoting by (D

,...,D

) the components of Dχ

,wehavethat

for all i =1,...,n and all test functions ϕ ∈ C

(Ω)



Ω

(x)

∂ϕ

∂x

(x) dx = −



Ω

ϕ(x) dD

(x) . (1.2)

From this formula it follows that the total variation of Dχ

in Ω can be

expressed as

|Dχ

|(Ω)=sup





i=1



Ω

(x) dD

: ψ ∈ C

(Ω;IR

), ψ

∞

≤ 1



(1.3)

=sup





divψ(x) dx : ψ ∈ C

(Ω;IR

), ψ

∞

≤ 1



Geometrical Aspects of Symmetrization 157

Notice that, if E is a smooth bounded open set, equation (1.2) reduces to



E∩Ω

∂ϕ

∂x

(x) dx = −



∂E∩Ω

ϕ(x)ν

(x) dH

n−1

(x) ,

where ν

denotes the inner normal to the boundary of E. Here and in the

sequel H

,1≤ k ≤ n −1, stands for the Hausdorﬀ k-dimensional measure in

. Thus, for a smooth set E,

= ν

n−1

∂E i =1,...,n,

|Dχ

|(Ω)=H

n−1

(∂E ∩ Ω) .

Last equation suggests to deﬁne the perimeter of E in Ω by setting P (E; Ω)=

|Dχ

|(Ω). More generally, if B ⊂ Ω is any Borel subset of Ω, we set

P (E; B)=|Dχ

|(B) .

If Ω =IR

, the perimeter of E in IR

will be denoted simply by P (E).

Notice that the last supremum in (1.3) makes sense for any measurable set E.

Indeed, if for some E that supremum is ﬁnite, then an application of Riesz’s

theorem on functionals on C

(Ω;IR

) yields that Dχ

is a Radon measure

and (1.3) holds. Thus, we may set for any measurable set E ⊂ IR

and any

open set Ω

P (E; Ω)=sup





divψ(x) dx : ψ ∈ C

(Ω;IR

), ψ

∞

≤ 1



. (1.4)

Clearly E a set of ﬁnite perimeter according to the deﬁnition given above if

and only if the right hand side of (1.4) is ﬁnite. Notice also that from (1.4)

it follows that if E

is a sequence of measurable sets converging locally in

measure to E in Ω, i.e., such that χ

→ χ

in L

loc

(Ω), then

P (E; Ω) ≤ lim inf

h→∞

P (E

; Ω) . (1.5)

Another immediate consequence of the deﬁnition of perimeter is that P (E; Ω)

does not change if we modify E by a set of zero Lebesgue measure. Moreover,

it is straightforward to check that

P (E; Ω \ ∂E)=|Dχ

|(Ω \ ∂E)=0,

i.e., Dχ

is concentrated on the topological boundary of E.

Next example shows that in general Dχ

may be concentrated on a much

smaller set. Let us denote by B

(x) the ball with center x and radius r and

set E = ∪

∞

i=1

1/2

), where {q

} a dense sequence in IR

.ThenE is an

open set of ﬁnite measure such that L

(∂E)=∞. However, E is a set of

ﬁnite perimeter in IR

. In fact, given ψ ∈ C

(IR

;IR

) with ψ

∞

≤ 1, by

158 N. Fusco

applying the classical divergence theorem to the Lipschitz open sets E

∪

i=1

1/2

), with k ≥ 1, we have



divψdx = lim

k→∞



divψdx= − lim

k→∞



∂E

ψ, ν

dH

n−1

≤ lim

k→∞

n−1

(∂E

) ≤

∞



i=1

nω

i(n−1)

< ∞.

To identify the set of points where the measure “perimeter” P (E; ·) is con-

centrated, we may use the Besicovitch derivation theorem (see [1, Theorem

2.22]), which guarantees that if E is a set of ﬁnite perimeter in IR

,thenfor

|Dχ

|-a.e. point x ∈ supp|Dχ

| (the support of the total variation of Dχ

)

there exists the derivative of Dχ

with respect to |Dχ

lim

r→0

Dχ

(x))

|Dχ

|(B

(x))

= ν

(x) , (1.6)

and that

|ν

(x)| =1. (1.7)

The set of points where (1.6) and (1.7) hold is called the reduced boundary

of E and denoted by ∂

∗

E.Ifx ∈ ∂

∗

E, ν

(x) is called the generalized inner

normal to E at x. Since from Besicovitch theorem we have that Dχ

|Dχ

| ∂

∗

E, formula (1.2) can be written as



∂ϕ

∂x

dx = −



∂

∗

ϕν

d|Dχ

| for all ϕ ∈ C

(IR

)andi =1,...,n.

(1.8)

The following theorem ([13] or [1, Theorem 3.59]) describes the structure of

the reduced boundary of a set of ﬁnite perimeter.

Theorem 1.2 (De Giorgi). Let E ⊂ IR

be a set of ﬁnite perimeter in IR

n ≥ 2. Then

(i) ∂

∗

E =

∞



h=1

∪ N

where each K

is a compact subset of a C

manifold M

and H

n−1

)=0;

(ii) |Dχ

| = H

n−1

∂

∗

E ;

for H

n−1

-a.e. x ∈ K

, ν

(x) is orthogonal to the tangent plane to M

at x.

From this theorem it is clear that for a set of ﬁnite perimeter the reduced

boundary plays the same role of the topological boundary for smooth sets. In

particular, the integration by parts formula (1.8) becomes

Geometrical Aspects of Symmetrization 159



∂ϕ

∂x

dx = −



∂

∗

ϕν

n−1

for all ϕ ∈ C

(IR

)andi =1,...,n,

(1.9)

an equation very similar to the one we have when E is a smooth open set.

In the one-dimensional case sets of ﬁnite perimeter are completely charac-

terized by the following result (see [1, Proposition 3.52]).

Proposition 1.1. Let E ⊂ IR be a measurable set. Then E has ﬁnite perime-

ter in IR if and only if there exist −∞ ≤ a

< ···<b

N−1

≤ +∞, such that E is equivalent to



i=1

). Moreover, if Ω is an open

set in IR ,

P (E; Ω)=#{i : a

∈ Ω} +#{i : b

∈ Ω}.

Notice that from this characterization we have that if E ⊂ IR is a set of ﬁnite

perimeter with ﬁnite measure, then P (E) ≥ 2. Moreover, P (E)=2ifand

only if E is equivalent to a bounded interval. Notice also that Proposition 1.1

yields immediately Theorem 1.1 and the characterization of the equality case

in (1.1).

If we translate Theorem 1.2 in the language of Geometric Measure theory,

then assertion (i) says that the reduced boundary ∂

∗

E of a set of ﬁnite perime-

ter E in IR

is a countably H

n−1

-rectiﬁable set (see [1, Deﬁnition 2.57]), while

(iii) states that for H

n−1

-a.e. x ∈ ∂

∗

E the approximate tangent plane to ∂

∗

at x (see [1, Section 2.11]) is orthogonal to ν

(x). Therefore, from the coarea

formula for rectiﬁable sets ([1, Remark 2.94]), we get that if g :IR

→ [0, +∞]

is a Borel function, then



∂

∗

g(x)|ν

(x)|dH

n−1

(x)=



n−1





(∂

∗



g(x



,y) dH

(y) , (1.10)

where H

denotes the counting measure.

Setting V = {x ∈ ∂E

∗

: ν

(x)=0} and g(x)=χ

(x), from (1.10) we get

that



n−1





(∂

∗





,y) dH

(y)=



∂

∗

(x)|ν

(x)|dH

n−1

(x)=0.

Therefore, if E is a set of ﬁnite perimeter, then V



= ∅ for L

n−1

-a.e. x



∈

n−1

, i.e.,

for L

n−1

-a.e. x



∈ IR

n−1

, ν



,y) = 0 for all y such that (x



,y) ∈ ∂

∗

(1.11)

Let Ω be an open subset of IR

and u ∈ L

(Ω). We say that u is a function of

bounded variation (shortly, a BV -function)inΩ, if the distributional deriv-

ative Du is a vector-valued Radon measure in Ω with ﬁnite total variation.

Thus, denoting by D

u, i =1,...,n, the components of Du we have that