TECHNICAL LEMMAS.
1. Stampacchia’s lemma. Let f ≥ 0 be nonincreasing in [x̄, ∞).
Assume f satisfies, for some C > 0, p > 0, γ > 1:
(y − x)p f (y) ≤ Cf (x)γ , for y ≥ x ≥ x̄.
pγ
Then f (y) = 0 for y ≥ x̄ + d, where: dp = Cf (0)γ−1 2 γ−1 .
Proof. We may assume x̄ = 0. Letting g = (f /f (0))1/p and A =
(Cf (0)γ−1 )1/p we find:
(y − x)g(y) ≤ Ag(x)γ for y > x, g(0) = 1.
Fix y > 0, and let xn = y(1 − 21n ), n ≥ 0, so xn ↑ y. Then g(x0 ) = 1 and:
g(xn+1 ) ≤
A n+1
2
g(xn )γ .
y
Inductively we find, for n ≥ 1:
A
n−1
g(xn ) ≤ ( )1+γ+...+γ 2Sn ,
y
Sn = n + (n − 1)γ + . . . + γ n−1 .
It is an exercise to compute the sum:
Sn =
n−1
X
(n − j)γ j =
j=0
Suppose
y
A
γ n+1 + n − (n + 1)γ
, n ≥ 1.
(γ − 1)2
β
= 2 γ−1 . Then:
n
A γ n −1
−β γ −1
( ) γ−1 = 2 (γ−1)2 ,
y
while:
γ n+1 + n − (n + 1)γ − β(γ n − 1) = (γ n − 1)(γ − β) − n(γ − 1).
γ
This means that if we choose β = γ, we have, for the choice y = d = 2 γ−1 A:
0 ≤ g(d) ≤ g(xn ) ≤ 2
n
− γ−1
,
n ≥ 1.
pγ
Letting n → ∞, we see g(d) = 0, hence f (d) = 0, where dp = Cf (0)γ−1 2 γ−1 .
1
2. Almost positivity. Let K ⊂ Symn be a set of nonzero real symmetric n × n matrices, satisfying for some c0 > 0:
tr(A)
≥ c0 , ∀A ∈ K.
|A|
P
Here |A|2 = tr(A2 ) = i λ2i . Denote by sk : Symn → R the k th. elementary
symmetric function of the eigenvalues, for k = 0, . . . , n. So s0 ≡ 1, s1 (A) =
tr(A), sn (A) = det(A). Let λmin (A) be the smallest eigenvalue. Consider
the conditions:
(1)k : (∀η > 0)(∃C > 0)(∀A ∈ K)sk (A) ≥ −η(trA)k − C.
(2)k : ∀(Aj )j≥1 ∈ K, |Aj | → ∞ ⇒ lim inf sk (
j
Aj
) ≥ 0.
|Aj |
(1)min : (∀η > 0)(∃C > 0)(∀A ∈ K)λmin (A) ≥ −η(trA) − C.
(2)min : ∀(Aj )j≥1 ∈ K, |Aj | → ∞ ⇒ lim inf λmin (
j
Aj
) ≥ 0.
|Aj |
Lemma. We have: (1)k ⇔ (2)k for each k = 2, . . . , n; (1)min ⇔ (2)min ; and
(2)k for all k = 1, . . . , n ⇔ (2)min .
Proof. Assume (1)k holds. Let Aj ∈ K, |Aj | → ∞. Given an arbitrary
> 0, we find C > 0 so that for all j ≥ 1: sk (Aj ) ≥ −(trAj )k − C, hence:
sk (
Aj
Aj k
C
) ≥ −(tr
) −
.
|Aj |
|Aj |
|Aj |k
A
Hence lim inf j sk ( |Ajj | ) ≥ −nk/2 , proving (2)k since is arbitrary.
Now assume (2)k holds. If (1)k fails, there exists η0 > 0 so that for
every C > 0 there exists an A ∈ K with sk (A) ≤ −η0 (trA)k − C. Note that
|A| ≤ M implies |sk (A)+η0 (trA)k | ≤ c(n, k, η0 )M k ; so if we let C = Cj → ∞
the corresponding matrices Aj ∈ K cannot stay in a bounded set, and taking
a subsequence we may assume |Aj | → ∞. But then:
lim inf sk (
j
Aj
trAj k
) ≤ −η0 lim inf (
) ≤ −ck0 η0 ,
j
|Aj |
|Aj |
in contradiction with (2)k .
The proof that (1)min ⇔ (2)min is analogous.
That (2)k for all k = 1, . . . , n is equivalent with (2) follows from the
observation that, for symmetric matrices A, λmin (A) ≥ 0 is equivalent to
2
sk (A) ≥ 0 for all k. In one direction this is obvious; and conversely, if all
the sk (A) ≥ 0, the polynomial in t:
pA (t) = det(I + tA) =
n
X
tk sk (A)
k=0
takes positive values for each t > 0. But if λ < 0 is an eigenvalue of A, we
have pA (−1/λ) = λ−n det(λI − A) = 0.
Assuming (2)k holds for all k, let (Aj ) be a sequence in K, |Aj | → ∞.
A
A
Taking a subsequence, we may assume λmin ( |Ajj | ) → lim inf j λmin ( |Ajj | ) := λ̄;
and taking a further subsequence that
Aj
|Aj |
→ Ā. But then by (2)k we have
A
sk (Ā) = limj sk ( |Ajj | ) ≥ 0 for all k, hence λmin (Ā) ≥ 0. But λmin (Ā) = λ̄.
The proof that, conversely, (2)min implies (2)k for each k = 2, . . . , n is
very similar.
3
3. Perimeter of the union.
Proposition. Let E, F be Borel sets of locally finite perimeter in Rn .
Then for any open Ω ⊂ Rn :
PΩ (E ∪ F ) + PΩ (E ∩ F ) ≤ PΩ (E) + PΩ (F ).
Recall the definition:
Z
PΩ (E) = sup{
div(g)dx; g ∈ Cc1 (Ω) vector field , |g| ≤ 1}.
E
The easy case is when E ∩ F has zero Lebesgue measure
in Rn , or more
R
generally when PΩ (E ∩ F ) = 0, which is equivalent to E∩F div(g)dx = 0 for
all C 1 vector fields g with compact support in Ω. Then, for each such g:
Z
Z
Z
div(g)dx =
div(g)dx +
div(g)dx,
E∪F
E
F
and taking supremum over g establishes the inequality in the proposition.
In the general case, we consider first smoothly bounded domains E, F ,
then approximate. Denoting by |N |Ω the (n − 1)-dimensional Hausdorff
measure of N ∩ Ω, we have for bounded open sets E with“piecewise C 1 ”
boundary N = ∂E: PΩ (E) = |N |Ω . (It is enough for the divergence theorem
to apply to E).
It is easy to establish the topological fact: for any E, F (subsets of Rn ),
letting A = ∂(E ∪ F ), B = ∂(E ∩ F ), we have:
A ∪ B ⊂ ∂E ∪ ∂F,
A ∩ B ⊂ ∂E ∩ ∂F.
Now, for any Borel measure µ (even vector-valued), and any measurable
sets A, B, one has:
µ(A ∪ B) + µ(A ∩ B) = µ(A) + µ(B).
(This is easily checked by computing the measure of the disjoint unions:
A ∪ B = (A∆B) t (A ∩ B),
A = (A \ B) t (A ∩ B),
B = (B \ A) t (A ∩ B),
where A∆B = (A \ B) t (B \ A) is the symmetric difference.)
In particular, for the A, B defined above we have:
|A|Ω + |B|Ω = |A ∪ B|Ω + |A ∩ B|Ω
4
≤ |∂E ∪ ∂F |Ω + |∂E ∩ ∂F |Ω = |∂E|Ω + |∂F |Ω ,
and recalling the definition of A, B this means:
|∂(E ∪ F )|Ω + |∂(E ∩ F )|Ω ≤ |∂E|Ω + |∂F |Ω .
This establishes the proposition in the case of smoothly bounded open sets
E, F . To deal with the general case, observe that it is enough to assume E, F
are bounded. (Just let Ωi ↑ Ω with Ωi bounded and use PΩi (E) ↑ PΩ (E).)
For bounded Borel sets E ⊂ Rn of locally finite perimeter, one has the
approximation theorem [Giusti, 1.24]: there exist smoothly bounded open
sets Ei so that:
Ln (E∆Ei ) → 0,
PΩ (Ei ) → PΩ (E),
for each open set Ω ⊂ Rn (where Ln denotes Lebesgue measure in Rn ). In
the same way we find an approximating sequence (Fj ) for the (bounded) set
F.
It is easy to convince oneself of the set-theoretic facts:
(E∪F )∆(Ei ∪Fj ) ⊂ (E∆Ei )∪(F ∆Fj ),
(E∩F )∆(Ei ∩Fj ) ⊂ (E∆Ei )∪(F ∆Fj ).
Now let the vector fields g, h ∈ Cc1 (Ω) approximate PΩ (E ∪ F ), PΩ (E ∩ F )
(respectively), both with C 0 norm less than or equal to one pointwise. Then:
Z
PΩ (E ∪ F ) − ≤
div(g)dx
E∪F
Z
Z
Z
div(g)dx −
div(g)dx +
=
Ei ∪Fj
(E∪F )\(Ei ∪Fj )
div(g)dx
(Ei ∪Fj )\(E∪F )
≤ PΩ (Ei ∪ Fj ) + |g|C 1 (Ln (E∆Ei ) + Ln (F ∆Fj )).
And similarly for the intersection:
PΩ (E ∩ F ) − ≤ PΩ (Ei ∩ Fj ) + |h|C 1 (Ln (E∆Ei ) + Ln (F ∆Fj )).
We conclude, using the proposition in the case of smooth boundaries:
PΩ (E ∪ F ) + PΩ (E ∩ F ) − PΩ (E) − PΩ (F ) ≤ 2
+(|g|C 1 +|h|C 1 )(Ln (E∆Ei )+Ln (F ∆Fj ))+PΩ (Ei )−PΩ (E)+PΩ (Fj )−PΩ (F ),
and letting i, j → ∞ ends the proof.
5
4.The perimeter measure and the measure-theoretic unit normal.
Denote by χ0c (Ω), χ1c (Ω) the space of C 0 (resp. C 1 ) vector fields with
compact support in the open set Ω ⊂ Rn (we omit Ω from the notation if
Ω = Rn .)
4.1 Riesz representation theorem. Let L : χ0c → R be a linear functional
so that for each compact subset K ⊂ Rn :
sup{L[g]; g ∈ χ0c , |g| ≤ 1, spt(g) ⊂ K} := CK < ∞.
Then there exists a Borel regular Radon measure µ in Rn and a measurable
vector field ν so that:
Z
(i)L[g] =
g · νdµ, ∀g ∈ χ0c ;
Rn
(ii)|ν(x)| = 1, µ − a.e.(x).
Def. The measure µ is the variation measure of L, defined for each open
Ω ⊂ Rn by:
µ(Ω) := sup{L[g]; g ∈ χ0c , |g| ≤ 1, spt(g) ⊂ Ω}.
Proof. (outline) (ref. [Evans-Gariepy 1.8]). The vector-valued case and
the existence of ν are not commonly stated. (In the well-known scalar case,
ν is not needed.)
(i) Defining µ(A) = inf{µ(V ), A ⊂ V open} for arbitrary sets A, one first
shows µ is a Borel regular Radon measure.
Next we identify the linear functional defined by integration of positive
functions w.r.t. µ (in the scalar case, this would just be L itself). In the
cone Cc+ of nonnegative continuous functions with compact support in Rn ,
consider the functional:
λ(f ) = sup{|L[g]|, g ∈ χ0c , |g| ≤ f },
f ∈ Cc+ .
One shows λ is linear, and then that, for all f ∈ Cc+ :
Z
λ(f ) =
f dµ.
Rn
6
(ii) To find ν, consider for each unit vector e ∈ Rn the linear functional:
f ∈ Cc (Rn ).
Le [f ] = L[f e],
Then:
χ0c , |g|
|Le [f ]| = |L[f e]| ≤ sup{L[g]; g ∈
Z
|f |dµ.
≤ f } = λ(|f |) =
Rn
Thus Le extends to a bounded linear functional on L1 (µ), and hence there
exists a function νe ∈ L∞ (µ) so that:
Z
Le (f ) =
νe f dµ.
Rn
P
Now let (ei ) be an orthonormal basis of Rn , and let ν = i νei ei . Then for
any g ∈ χ0c :
Z
X
X
XZ
L[g] =
L[(g · ei )ei ] =
Lei (g · ei ) =
νei g · ei dµ =
g · νdµ.
i
i
Rn
i
Rn
(In particular, ν(x) 6= 0 for µ-a.e. x.)
(iii) To see that |ν| = 1 µ-a.e., note that for any Ω ⊂ Rn open, and any
g ∈ χ0c (Ω) with |g| ≤ 1:
Z
Z
g · νdµ ≤
|ν|dµ,
L[g] =
Ω
Ω
R
hence µ(Ω) ≤ Ω |ν|dµ (by definition of µ(Ω), see above.)
And conversely, for any Ω ⊂ Rn and any > 0 we may find (by an
application of Lusin’s theorem) a sequence gk ∈ χ0c (Ω) so that |gk | ≤ 1 + ν
and gk → |ν|
, µ-a.e in Ω. By dominated convergence we find:
Z
Z
|ν|dµ = lim gk · νdµ = lim L[gk ] ≤ (1 + )µ(Ω).
k
Ω
We conclude
R
Ω |ν|dµ
k
Ω
= µ(Ω) for all open Ω, and hence |ν| = 1, µ-a.e.
4.2 Structure Theorem for BV functions.([Evans-Gariepy, 5.1].)
Theorem. Let Ω ⊂ Rn open, f ∈ BVloc (Ω). Then there exists a Borelregular Radon measure µ in Ω and a measurable vector field σ in Ω (with
|σ| = 1, µ − a.e.) so that for any g ∈ χ1c (Ω):
Z
Z
f div(g)dx = − g · σdµ.
Ω
Ω
7
Proof. Clearly the idea is to apply the Riesz representation theorem,
and for that we must extend the functional L on χ1c (Ω):
Z
L[g] = − f div(g)dx
Ω
to a bounded linear functional on χ0c (Ω). This is standard, since for each
open V ⊂⊂ Ω, we have (by definition of BVloc ):
sup{L[g]; g ∈ χ1c (Ω), spt(g) ⊂ V, |g| ≤ 1} := CV < ∞,
hence |L[g]| ≤ CV ||g||∞ , for each V ⊂⊂ Ω, g ∈ χ1c (V ).
Thus we may extend L to a bounded linear functional L̄ on χ0c (Ω) in the
usual way. If g ∈ χ0c (Ω) with spt(g) ⊂ K (compact ) and V ⊃ K is open in
Ω, let gk ∈ χ1c (Ω) be a sequence converging to g uniformly in V , and define:
L̄[g] = lim L[gk ]
k
(one checks easily the limit is independent of the approximating sequence.)
Now apply the Riesz representation theorem to L̄ to conclude the proof.
Notation. (i) For f ∈ BVloc denote the variation measure µ by Vf (a
Borel-regular Radon measure) and let [Vf ] := σVf , a vector-valued Borel
measure. Thus, for g ∈ χ1c (Ω):
Z
Z
Z
f div(g)dx = − g · σdVf = − g · d[Vf ]
Ω
Ω
Ω
(“integration by parts”.)
(ii) If E is a Borel set of locally finite perimeter in Ω (so χE ∈ BVloc (Ω)),
we denote µ by PE , the perimeter measure, and let νE = −σ, [PE ] = νE PE
(vector-valued measure).
In particular PE (Ω) is the perimeter of E in Ω (denoted by PΩ (E) in
section 3. ) We have, for any g ∈ χ1c (Ω):
Z
Z
Z
div(g)dx =
g · νE dµ =
g · d[PE ]
E
Ω
Ω
(“divergence theorem.”)
Note in passing that, for any measurable sets A, B ⊂ Rn we have (in this
new notation):
PE (A ∪ B) + PE (A ∩ B) = PE (A) + PE (B).
8
Remark 1. If E ⊂ Rn is open with smooth boundary, we have:
PE (Ω) = Hn−1 (∂E ∩ Ω),
νE = unit outward normal of E, Hn−1 − a.e..
(Note however that the (n − 1)-dimensional Hausdorff measure Hn−1 is not
a Radon measure in Rn , unlike PE .)
Remark 2. It is easy to see that if E is of finite
R perimeter in Ω, then
spt(PE ) ⊂ ∂E. Indeed if U ⊂ Ω \ E is open, E div(g)dx = 0 for all
g ∈ χ1c (U ); so PE (U ) = 0.
And if V ⊂ int(Ω ∩ E) is open and g ∈ χ1c (V ), we have χE = 1 on V ,
hence:
Z
Z
Z
div(g)dx =
χE div(g)dx =
div(g)dx = 0,
E
V
V
so PE (V ) = 0.
4.3 Reduced boundary. Let E ⊂ Rn be a Borel set of locally finite
perimeter, with perimeter measure E and measure-theoretic outward unit
normal νE . A point x ∈ Rn is in the reduced boundary of E (x ∈ ∂ ∗ E) if the
following three conditions hold:
(i) PE (BR (x)) > 0∀R > 0. In particular, x ∈ ∂E.
(ii) For R > 0, let:
R
[PE ](BR (x))
B (x) νE dPE
νR (x) =
= R
∈ Rn .
PE (BR (x))
PE (BR (x))
Then limR→0 νR (x) exists (call it ν(x) ∈ Rn ).
(iii) |ν(x)| = 1.
Note that (by Lebesgue differentiation) conditions (ii) and (iii) hold for
PE -a.e. x in ∂E, with ν(x) = νE (x).
Main properties of ∂ ∗ E:[Giusti, Thm. 4.4],[Evans-Gariepy, Ch.5]
(i) ∂ ∗ E can be written as a countable union of compact subsets of C 1
hypersurfaces Nk , union a PE -null set:
∗
∂ E=(
∞
[
Ck ) ∪ Z,
Ck ⊂ Nk , PE (Z) = 0.
k=1
On Ck , ν(x) ⊥ Nk .
(ii) ∂ ∗ E is dense in ∂E;
(iii) For each open set Ω ⊂ Rn ;
PE (Ω) = Hn−1 (∂ ∗ E ∩ Ω).
For each measurable set B ⊂ ∂ ∗ E, PE (B) = Hn−1 (B).
9