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

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170 N. Fusco

Let us assume that (H

) and (H

) hold in some open set U ⊂ IR

n−1

.Then,

for every connected open subset U

of U, E ∩(U

×IR ) is equivalent to E

∩

× IR ) , up to a translation in the y direction.

In particular, if (H

) and (H

) hold in a connected open set U such that

n−1

(π(E)

\ U)=0, then E is equivalent to E

, up to a translation in the

y direction.

As far as I know, this result was known in the literature only for convex sets,

where it can be proved with a simple argument. In fact, let us assume that E

is an open convex set such that P (E)=P (E

) < ∞. Then, π(E)isalsoan

open convex set and there exist two functions y

: π(E) → IR , y

convex,

and y

concave, such that

E = {(x



,y): x



∈ π(E),y



) <y<y



)}.

Let us now ﬁx an open set U ⊂⊂ π(E). From assumption (2.10) and from (2.1)

we have that P (E

; U × IR ) = P (E; U × IR). Since y

and y

are Lipschitz

continuous in U , we can write this equality as



|∇(y

− y







1+|∇y







1+|∇y



From this equality, the strict convexity of the function t →

√

1+t

and the

arbitrariness of U,wegetthat∇y

= −∇y

in π(E) and thus y

= −y

const.. This shows that E coincides with E

, up to a translation in y direction.

The proof of Theorem 2.3, for which we refer to [9], uses delicate tools from

Geometric Measure theory. However, in the special case considered below it

can be greatly simpliﬁed.

Proof of Theorem 2.3 in a Special Case. Let us assume that E is an

open set, that π(E) is connected and that E is bounded in y direction.

Notice that since E is open, then µ is a lower semicontinuous function and

for any open set U ⊂⊂ π(E) there exists a constant c(U) > 0 such that

µ(x



) ≥ c(U ) for all x



∈ U. (2.11)

Moreover, since E is bounded in the y direction, the function



∈ π(E) → m(x







ydy

is bounded in π(E). Then, the same arguments used in the proofs of Lemmas

1.1 and 1.3 yield that m ∈ BV

loc

(π(E)) and that for L

n−1

-a.e. x



∈ π(E),

i =1,...,n−1,

∂m

∂x





y∈∂

∗



yν



,y)

|ν



,y)|

, (2.12)

Geometrical Aspects of Symmetrization 171

wherewehavedenotedby∂m/∂x

the absolutely continuous part of the

derivative D

By Proposition 2.1 we have that (H

) implies that the distribution

function µ is a Sobolev function. The same assumption implies also that

m ∈ W

1,1

loc

(π(E)). In fact, the argument used to prove (1.14) shows that if

B ⊂ π(E) is a Borel set, then |Dm|(B) ≤ MP(E; B × IR), where M is a

constant such that E ⊂ IR

n−1

× (−M, M). Therefore, if L

n−1

(B) = 0, from

(2.1) and Proposition 2.1 we have that P (E; B × IR ) = P (E

; B × IR ) = 0 ,

hence Dm is absolutely continuous with respect to L

n−1

Let us now denote, for any x



∈ π(E)byb(x



) the baricenter of the section



, i.e.,

b(x







ydy

µ(x



)

From (2.11) and Proposition 2.1 we have that b too belongs to the space

1,1

loc

(π(E)). Thus, to prove the assertion, since π(E) is a connected open set,

it is enough to show that ∇b ≡ 0, hence b is constant on π(E). To this aim,

let us evaluate the partial derivatives of b, using the representation formulas

(2.12) and (1.21). We have, for any i =1,...,n−1 and for L

n−1

-a.e. x



∈ π(E),

∂b

∂x



µ(x



)





y∈∂

∗



yν



,y)

|ν



,y)|

−





ydy

µ(x



)



y∈∂

∗





,y)

|ν



,y)|



. (2.13)

Since for L

n−1

-a.e. x



∈ π(E) (2.4) and (2.5) hold, the right hand side of

(2.13) is equal to

µ(x



)







)+y



)





))

|ν



))|

−



)−y



)



)−y



)

2ν



))

|ν



))|



=0.

Hence, the assertion follows. !"

3TheP`olya–Szeg¨o Inequality

We are going to present the classical P`olya–Szeg¨o inequality for the spherical

rearrangement of a Sobolev function u and discuss what can be said about

the function u when the equality holds. In order to simplify the exposition,

we shall assume that u is a nonnegative measurable function from IR

, with

compact support. However, most of the results presented here, like Theorem

3.1, still hold with no restrictions on the support or on the sign of u.

Given u, we set, for any t ≥ 0,

(t)=L

({x ∈ IR

: u(x) >t}) .

172 N. Fusco

The function µ

is called the distribution function of u. Clearly µ

is a de-

creasing, right-continuous function such that

(0)= L

(suppu),µ

(esssup u)=0,µ

(t−)=L

({u ≥ t}) for all t>0 .

(3.1)

Notice that from the last equality we have that when t>0

is continuous in t iﬀ L

({u = t})=0.

Let us now introduce the decreasing rearrangement of u, that is the function

∗

:[0, +∞) → [0, +∞) deﬁned, for any s ≥ 0, by setting

∗

(s)=sup{t ≥ 0: µ

(t) >s}.

Clearly, u

∗

is a decreasing, right-continuous function. The following elemen-

tary properties of u

∗

are easily checked:

(j) u

∗

(µ

(t)) ≤ t ≤ u

∗

(µ

(t)−) for all 0 ≤ t<esssup u ;

(jj) µ

∗

(s)) ≤ s ≤ µ

∗

(s)−) for all 0 ≤ s<L

(suppu);

(jjj) L

({s : u

∗

(s) >t} = µ

(t) for all t ≥ 0 .

Notice that (jjj) states that the functions u and u

∗

are equi-distributed, i.e.,

= µ

∗

. Let us now deﬁne the spherical symmetric rearrangement of u,that

is the function u



:IR

→ [0, +∞), such that for all x ∈ IR



(x)=u

∗

(ω

|x|

) . (3.2)

By deﬁnition and by (jjj) we have

({u



>t})=L

({u>t}) for all t ≥ 0 ,

i.e., µ

= µ



. Thus also u and u



are equi-distributed. As a simple conse-

quence of this equality and Fubini’s theorem we have, for all p ≥ 1,





(x)|

dx =



|u(x)|

dx ,

and, letting p → +∞, esssup u = esssup u



If u is a smooth function, in general its symmetric rearrangement will be

no longer smooth (actually the best we may expect from Theorem 3.1 below

is that u



is Lipschitz continuous). However, the symmetric rearrangement

behaves nicely on Sobolev functions, as shown by the next result.

Theorem 3.1 (P`olya–Szeg¨o Inequality). Let u ∈ W

1,p

(IR

), p ≥ 1,bea

nonnegative function with compact support. Then u



∈ W

1,p

(IR

) and



|∇u



(x)|

dx ≤



|∇u(x)|

dx . (3.3)

Geometrical Aspects of Symmetrization 173

The proof of this result relies upon two main ingredients, the isoperimetric

inequality and the coarea formula for BV functions. Let us start by recalling

the latter.

Let u be a BV (Ω) function. Then, for L

-a.e. t ∈ IR, the set {u>t} has

ﬁnite perimeter in Ω. Moreover, for any Borel function g : Ω → [0, +∞], the

following formula holds (see [1, Theorem 3.40]).



Ω

g(x) d|Du| =



∞

−∞



∂

∗

{u>t}

g(x) dH

n−1

. (3.4)

In the special case u ∈ W

1,1

(Ω), it can be shown that for L

-a.e. t the reduced

boundary ∂

∗

{u>t} coincides, modulo a set of H

n−1

-measure zero, with the

level set {+u = t},where+u denotes the precise representative of u,whichis

deﬁned H

n−1

-a.e. in Ω as in (2.9). Therefore, if u ∈ W

1,1

(Ω), (3.4) becomes



Ω

g(x)|∇u(x)|dx =



∞

−∞



{˜u=t}

g(x) dH

n−1

(x) . (3.5)

The isoperimetric inequality states that if E be a set of ﬁnite perimeter, then



min {L

(E), L

(IR

\ E)}



n−1

≤

nω

1/n

P (E) . (3.6)

Moreover, the equality holds if and only if E is (equivalent to) a ball.

Next lemma shows that if u is a Sobolev function, then the same is also

true for u



Lemma 3.1. Let u be a nonnegative function with compact support from the

space W

1,1

(IR

). Then u



belongs to W

1,1

(IR

) and



|∇u



|dx ≤



|∇u|dx . (3.7)

Proof. Letusﬁrstprovethatforany0<a<b, the function u

∗

is absolutely

continuous in (a, b)and



∗

(s)|ds ≤

nω

1/n

n−1



∗

(b)<u<u

∗

(a)}

|∇u|dx . (3.8)

To this aim, we start by observing that from the third equality in (3.1) and

from inequality (jj) we have

({x ∈ IR

: u

∗

(b) <u(x) <u

∗

(a)})=µ

∗

(b)) − µ

∗

(a)−) ≤ b − a.

(3.9)

Let us denote by ω :[0, ∞) → [0, +∞) the modulus of continuity of the

integral of |∇u|, i.e., a continuous function, vanishing at zero and such that

for any set of ﬁnite measure E

174 N. Fusco



|∇u|dx ≤ ω(L

(E)) .

Using the coarea formula (3.5), the isoperimetric inequality (3.6) and (jj)

again, we obtain the following estimate for the integral of |∇u| between two

level sets,



∗

(b)<u<u

∗

(a)}

|∇u|dx =



∗

(a)

∗

(b)

P ({u>t}) dt

≥ nω

1/n



∗

(a)

∗

(b)



({u>t}



n−1

≥ nω

1/n

[µ

∗

(a)−)]

n−1

∗

(a) − u

∗

(b)) (3.10)

≥ nω

1/n

n−1

∗

(a) − u

∗

(b)) .

Let us now take a ﬁnite number of pairwise disjoint intervals (a

) ⊂ (a, b),

i =1,...,N. By applying (3.9) and (3.10) to each interval (a

), we get



i=1

∗

) − u

∗

)|≤

nω

1/n

n−1



i=1



∗

)<u<u

∗

)}

|∇u|dx

≤

nω

1/n

n−1





i=1

− a

)



. (3.11)

From this inequality it follows immediately that u is absolutely continuous in

(a, b), since the left hand side is smaller than a given ε>0assoonasthesum

of the lengths of the intervals (a

) is suﬃciently small. Moreover, by taking

the supremum of the left hand side of (3.11) over all possible partitions of the

interval (a, b), from the ﬁrst inequality in (3.11) we get immediately (3.8).

Notice that from (3.8) it follows that u



is in W

1,1

loc

(IR

\{0}). To prove the

assertion, we ﬁx σ>1 and estimate the integral of |∇u| in the annuli A

k,σ

{x ∈ IR

: ω

−1/n

k/n

< |x| <ω

−1/n

(k+1)/n

},fork ∈ ZZ. Using (3.8) again,

and recalling the deﬁnition (3.2), we get, for any k ∈ ZZ ,



k,σ

|∇u



|dx = nω



k,σ

|x|

n−1

∗

(ω

|x|

)|dx

= n



−1/n

(k+1)/n

−1/n

k/n

2n−2

∗

(ω

)|dr

= nω

1/n



k+1

n−1

∗

(s)|ds

≤ σ

n−1



∗

(σ

)<u<u

∗

(σ

k+1

)}

|∇u|dx .

Then the assertion immediately follows by summing up both sides of this

inequality over all k ∈ ZZ and then letting σ → 1

. !"

Geometrical Aspects of Symmetrization 175

Notice that the lemma we have just proved provides the P´olya–Szeg¨oin-

equality for p = 1. However, for the general case p ≥ 1 we present a diﬀerent

proof which has the advantage of giving better information when the inequal-

ity becomes an equality.

To this aim let us introduce a few quantities that will be useful later. If

u ∈ W

1,1

loc

(IR

), we set

= {x ∈ IR

: ∇u(x) =0}, D

=IR

We can now give a representation formula for the derivative of µ

. Notice that

the formula stated in (3.12) uses the fact that the |∇u



| is H

n−1

-a.e. constant

on L

-a.e. level set {u



= t}.

Lemma 3.2. Let u ∈ W

1,1

(IR

) be a nonnegative function with compact sup-

port. Then, for L

-a.e. t>0,



(t)=−

n−1

({u



= t})

|∇u



|{u



=t}

≤−



{˜u=t}

|∇u|

n−1

. (3.12)

Proof. First of all let us evaluate µ

(t) using the coarea formula (3.5). We

get, for all t ≥ 0,

(t)=L



{u>t}∩D



+ L



{u>t}∩D



= L



{u>t}∩D





{u>t}

(x) dx (3.13)

= L



{u>t}∩D





+∞



{˜u=s}

|∇u|

n−1

= L



{u>t}∩D





+∞



{˜u=s}

|∇u|

n−1

where the last equality follows by observing that coarea formula (3.5) implies

that H

n−1

({˜u = t}∩D

)=0forL

-a.e. t ≥ 0. By applying (3.13) to u



,we

get also that for all t ≥ 0

(t)=L





>t}∩D







+∞

n−1

({u



= t})

|∇u



|{u



=t}

(3.14)

Let us now recall a nice property of absolutely continuous functions (see for

instance [10, Lemma 2.4]).

If g is an absolutely continuous function in a bounded open interval I and,

for all t ∈ IR, we set φ

(t)=L

({g>t}∩D

), then φ

is a nondecreasing

function such that φ



(t)=0forL

-a.e. t.

By applying this result with g = u

∗

and observing that L





>t}∩D







∗

>t}∩D

∗



, for all t>0, from (3.14) we get immediately the equality

in (3.12). On the other hand, the inequality

176 N. Fusco



(t) ≤−



{˜u=t}

|∇u|

n−1

follows immediately from (3.13). !"

We are now ready to prove the P´olya–Szeg¨o inequality (3.3).

Proof of Theorem 3.1. Let us ﬁx a nonnegative function u ∈ W

1,p

(IR

)

with compact support and let us assume, without loss of generality, that u

coincides with its precise representative +u. From Lemma 3.1 we know already

that u



belongs to the space W

1,1

(IR

). Thus, using the coarea formula (3.5)

with u replaced by u



and recalling that |∇u



| is constant on the level sets of



,weget



|∇u



dx =



+∞





=t}

|∇u



p−1

n−1



+∞

n−1

({u



= t})|∇u



p−1

|{u



=t}

dt .

From this equation, using twice (3.12), the isoperimetric inequality (3.6),

H¨older’s inequality and coarea formula again, we get



|∇u



dx =



+∞

n−1

({u



= t})

[−µ



(t)]

p−1

≤



+∞

n−1

({u



= t})







=t}

n−1

|∇u|



p−1

dt (3.15)

≤



+∞

n−1

({u = t})





{u=t}

n−1

|∇u|



p−1

≤



+∞



{u=t}

|∇u|

p−1

n−1



|∇u|

dx .

Hence (3.3) follows. !"

Let us now discuss the equality case in (3.3). First, notice that if this is

the case, then all inequalities in (3.15) are in fact equalities. In particular, if

the second inequality in (3.15) holds as an equality, then we can conclude that

the set {u>t} is (equivalent to) a ball for L

-a.e. t ≥ 0. Moreover, if the

equality holds in the third inequality (where we have used H¨older inequality),

the conclusion is that for L

-a.e. t ≥ 0, |∇u| is H

n−1

-a.e. constant on the level

set {u = t}.

These are the immediate consequences of the equality case. However, with

some extra work, one can prove the following, more precise, result (see [5] or

[11, Theorem 2.3]).

Geometrical Aspects of Symmetrization 177

Proposition 3.1. Let u ∈ W

1,p

(IR

), p ≥ 1, a nonnegative function with

compact support such that



|∇u



dx =



|∇u|

dx . (3.16)

Then there exist a function v, equivalent to u, i.e. such that v(x)=u(x) for

-a.e. x ∈ IR

,andafamilyofopenballs{U

}

t≥0

such that:

(i) {v>t} = U

for t ∈ [0, esssup u);

(ii) {v = esssup u} =

0≤t<esssup u

, and is a closed ball (possibly a point);

(iii) v is lower semicontinuous in {v<esssup u};

(iv) if v(x) ∈ (0, esssup u) and L

({u = v(x)})=0, then x ∈ ∂U

v(x)

;

(v) for every t ∈ (0, esssup u) there exists at most one point x ∈ ∂U

such

that v(x) = t;

(vi) the coarea formula (3.5) holds with +u replaced by v;

(vii) for L

-a.e. t ∈ (0, esssup u), |∇v(x)| = |∇u



|{u



=t}

for H

n−1

-a.e.

x ∈ ∂U

This proposition contains all the information that we can extract from equality

(3.16). However it is not true in general that (3.16) implies that u coincides

with u



, up to a translation in x. This can be easily seen by considering any

spherically symmetric nonnegative function w, such that L

({w = t

}) > 0

for some t

∈ (0, esssup w) and another function u whose graph agrees with

the graph of w where u<t

and with a slight translated of the graph of

w where u>t

.Thenu



= w and (3.16) holds, but u is not spherically

symmetric. What goes wrong in this example is the fact that the gradient of

w (and of u) vanishes in a set of positive measure. Thus, this example suggests

to introduce the following assumption,

(H) L

({0 <u



< esssup u}∩D



)=0.

Notice that we are in a situation similar to the one we were in the previous

lecture when dealing with the assumption (H

). In fact, it can be proved (see

for instance [10, Lemma 3.3]) that (H) is implied by the stronger assumption



) L

({0 <u<esssup u}∩D

)=0.

Moreover, (H) is equivalent to the absolute continuity in (0, +∞) of the dis-

tribution function µ

and the two conditions (H)and(H



) are equivalent if

(3.16) holds (see [10, Lemma 3.3] again).

The following result was proved for the ﬁrst time in the Sobolev setting

by Brothers and Ziemer ([5]). It shows that when equality holds in (3.3),

assumption (H) guarantees that u and u



agree.

178 N. Fusco

Theorem 3.2. Let u ∈ W

1,p

(IR

), p>1, be a nonnegative function with

compact support such that (3.16) and (H) hold. Then u



= u,uptoatrans-

lation in x.

Notice that the above result is in general false if p = 1, even in one dimension.

To see this it is enough to take a function which is increasing in the interval

(−∞,a) and decreasing in (a, +∞).

The starting point of the proof of Brothers and Ziemer is to observe, as we have

done before, that the equality (3.1) yields that L

-a.e. set {u>t} is a ball and

that |∇u| is constant on the corresponding boundary. Then the diﬃcult part

of the proof consists in exploiting assumption (H) to deduce that all these

balls are concentric, i.e. u is spherically symmetric. To prove this, we shall not

follow the original argument contained in ([5]), but a somewhat simpler one

used in [11], which is in turn inspired to an alternative proof of Theorem 3.2

given in [17].

To this aim from now on we shall assume, without loss of generality, that u

agrees with the representative v provided by Proposition 3.1 and that U

{u>0} is a ball centered at the origin. Then, for all 0 <t<esssup u we

denote by R

the radius of the ball U

and set, for all x ∈ U

Φ(x)=



(u(x))



1/n

. (3.17)

To understand the role of the function Φ, observe that if x ∈ U

is a point

such that u(x)=t,thenµ

(u(x)) = L

). Therefore, Φ(x) is equal to the

radius of the ball U

The following lemma is a crucial step toward the proof of Theorem 3.2.

Lemma 3.3. Under the assumptions of Theorem 3.2, Φ ∈ W

1,∞

) and

|∇Φ(x)| =1 forL

-a.e. x ∈ U

\{u = esssup u}. (3.18)

Proof. We claim that µ

◦ u ∈ W

1,∞

)andthat

∇(µ

◦u)(x)=−H

n−1

({u= u(x)})

∇u(x)

|∇u|

|{u=u(x)}

(x)forL

-a.e. x ∈ U

(3.19)

From assumption (H), which is equivalent by (3.16) to (H



), using (3.13) and

Proposition 3.1 (vii), we get that for all 0 ≤ t ≤ esssup u,

(t)=L

({u = esssup u})+



+∞

n−1

({u = s})

|∇u|

|{u=s}

ds .

For ε>0, we set

u,ε

(t)=L

({u = esssup u})+



+∞

n−1

({u = s})

|∇u|

|{u=s}

+ ε

ds .

Geometrical Aspects of Symmetrization 179

Clearly, µ

u,ε

(t) ↑ µ

(t) for every t ≥ 0asε ↓ 0. Moreover, µ

u,ε

is Lipschitz

continuous in [0, +∞), and



u,ε

(t)=−

n−1

({u= t})

|∇u|

|{u=t}

+ ε

for L

-a.e. t ≥ 0 ,

whence

|µ



u,ε

(t) − µ



(t)|≤

εH

n−1

({u= t})

|∇u|

|{u=t}



|∇u|

|{u=t}

+ ε



for L

-a.e. t ≥ 0 .

Thus, µ



u,ε

(t) → µ



(t) L

-a.e. in [0, +∞)asε → 0, since |∇u|

|{u=t}

= 0 for

-a.e. t ≥ 0, in as much as

n−1

({u= t})

|∇u|

|{u=t}

∈ L

(0, ∞). This membership and

the fact that

|µ



u,ε

(t) − µ



(t)|≤

n−1

({u= t})

|∇u|

|{u=t}

for L

-a.e. t ≥ 0

entail that µ



u,ε

→ µ



in L

(0, ∞). Hence, µ

u,ε

→ µ

uniformly in (0, ∞).

Consequently, the functions µ

u,ε

◦u converge uniformly to µ

◦u. Furthermore,

by the chain rule for Sobolev functions (see e.g. [1, Theorem 3.96]),

∇(µ

u,ε

◦ u)(x)=µ



u,ε

(u(x))∇u(x)=−H

n−1

({u = u(x)})

∇u(x)

|∇u|

|{u=u(x)}

+ ε

for L

-a.e. x ∈ U

The last expression clearly converges to the right-hand side of (3.19). More-

over, from Theorem 3.1, we have that H

n−1

({u = t})=H

n−1

(∂U

) ≤

nω

n−1

for L

-a.e. t ∈ (0, ess sup u). Thus,

|∇(µ

u,ε

◦ u)(x)|≤nω

n−1

for L

-a.e. x ∈ U

By dominated convergence, ∇(µ

u,ε

◦ u) converges to the right-hand side of

(3.19) in L

). Hence, the claim follows.

To conclude the proof let us now observe that for all t ∈ (0, esssup u),

(u(x)) ≥ µ

(t) > 0 for all x ∈ U

\ U

. Therefore, we can compute the

derivatives of Φ in U

\ U

by the usual chain rule formula for Sobolev func-

tions, thus getting, from (3.19), that for L

-a.e. x ∈ U

\ U

∇Φ(x)=−

nω

1/n

(µ

(u(x)))

1−n

n−1

({u= u(x)})

∇u(x)

|∇u|

|{u=u(x)}

(x)

= −

∇u(x)

|∇u|

|{u=u(x)}

(x). (3.20)

Then, (3.18) follows immediately from Proposition 3.1 (ii). In particular, this

proves that Φ is a W

1,∞

function in the open set U

\{u = esssup u} (which