Section 2.5: Alexandroff Maximum Principle
2.5 Alexandro Maximum Principle
Suppose Ω is a bounded domain in R
n
and consider a second-order elliptic operator
L in Ω
L ≡ a
ij
(x)D
ij
+ b
i
(x)D
i
+ c(x)
where coecients a
ij
, b
i
, c are at least continuous in Ω. Ellipticity means that the
coecient matrix A = (a
ij
) is positive denite everywhere in Ω. We set D = det(A)
and D
∗
= D
1/n
so that D
∗
is the geometric mean of the eigenvalues of A. Throughout
this section we assume
0 < λ ≤ D
∗
≤ Λ
where λ and Λ are two positive constants, which denote, respectively, the minimal and
maximal eigenvalues of A.
Before stating the main theorem, we rst introduce the concept of contact sets. For
u ∈ C
2
(Ω) we dene
Γ
+
= {y ∈ Ω : u(x) ≤ u(y) + Du(y) · (x − y) for any x ∈ Ω}.
The set Γ
+
is called the upper contact set of u, and the Hessian matrix D
2
u = (D
ij
u)
is nonpositive on Γ
+
. In fact, the upper contact set can also be dened for continuous
function u by the following:
Γ
+
= {y ∈ Ω :u(x) ≤ u(y) + p · (x − y) for any x ∈ Ω
and some p = p(y) ∈ R
n
}.
Clearly, u is concave if and only if Γ
+
= Ω. If u ∈ C
1
(Ω), then p(y) = Du(y), and any
support hyperplane must then be a tangent plane to the graph.
Now we consider the equation of the form
Lu = f in Ω
for some f ∈ C(Ω).
Theorem 2.21. Suppose u ∈ C(Ω) ∩ C
2
(Ω) satises Lu ≥ f in Ω with the following
conditions:
|b|
D
∗
,
f
−
D
∗
∈ L
n
(Ω) and c ≤ 0 in Ω,
where f
−
= max{−f, 0}. Then there holds
sup
Ω
u ≤ sup
∂Ω
u
+
+ C
f
−
D
∗
L
n
(Γ
+
)
49
Chapter 2: Maximum Principles
where Γ
+
is the upper contact set of u and C is a constant depending only on n, diam(Ω),
and
b
D
∗
L
n
(Γ
+
)
. In fact, C can be written as
d ·
(
exp
(
2
n−2
ω
n
n
n
b
D
∗
n
L
n
(Γ
+
)
+ 1
!)
− 1
)
with ω
n
as the volume of the unit ball in R
n
. Here b = (b
1
, b
2
, . . . , b
n
).
Remark 2.22. The integral domain Γ
+
can be replaced by
Γ
+
∩
x ∈ Ω : u(x) > sup
∂Ω
u
+
.
Remark 2.23. There is no assumption on uniform ellipticity. Compare with Hopf’s
maximum principle in Section 1.
We need a lemma rst.
Lemma 2.24. Suppose g ∈ L
1
loc
(R
n
) is nonnegative. Then for any u ∈ C(Ω) ∩ C
2
(Ω)
there holds
Z
B
M
(0)
g ≤
Z
Γ
+
g(Du)|det D
2
u|
where Γ
+
is the upper contact set of u and
˜
M = (sup
Ω
u − sup
∂Ω
u
+
)/d with d =
diam(Ω).
Remark 2.25. For any positive denite matrix A = (a
ij
) we have
det(−D
2
u) ≤
1
D
−a
ij
D
ij
u
n
n
on Γ
+
.
Hence we have another form for Lemma 2.24
Z
B
˜
M
(0)
g
≤
Z
Γ
+
g
(
Du
)
−a
ij
D
ij
u
nD
∗
n
.
Remark 2.26. A special case corresponds to g = 1:
sup
Ω
sup u ≤ sup
∂Ω
u
+
+
d
ω
1/n
n
Z
Γ
+
|det D
2
u|
1/n
≤ sup
∂Ω
u
+
+
d
ω
1/n
n
Z
Γ
+
−a
ij
D
ij
u
nD
∗
n
1/n
.
Note that this is Theorem 2.21 if b
i
≡ 0 and c ≡ 0.
Proof of Lemma 2.24. Without loss of generality we assume u ≤ 0 on ∂Ω. Set Ω
+
=
50
Section 2.5: Alexandroff Maximum Principle
{u > 0}. By the area formula for Du in Γ
+
∩ Ω
+
⊂ Ω, we have
Z
Du(Γ
+
∩Ω
+
)
g ≤
Z
Γ
+
∩Ω
+
g(Du)|det(D
2
u)|, (2.3)
where |det(D
2
u)| is the Jacobian of the map Du : Ω → R
n
. In fact, we may consider
χ
ε
= Du − εId : Ω → R
n
. Then Dχ
ε
= D
2
u − εI, which is negative denite in Γ
+
.
Hence by the change-of-variable formula we have
Z
χ
ε
(Γ
+
∩Ω
+
)
g =
Z
Γ
+
∩Ω
+
g(χ
ε
)|det(D
2
u − εI)|,
which implies (2.3) if we let ε → 0.
Now we claim B
˜
M
(0) ⊂ Du(Γ
+
∩ Ω
+
), that is, for any a ∈ R
n
with |a| <
˜
M there
exists x ∈ Γ
+
∩ Ω
+
such that a = Du(x).
We may assume u attains its maximum m > 0 at 0 ∈ Ω, that is,
u(0) = m = sup
Ω
u.
Consider an ane function for |a| <
m
d
(≡
˜
M)
L(x) = m + a · x.
Then L(x) > 0 for any x ∈ Ω and L(0) = m. Since u assumes its maximum at 0, then
Du(0) = 0. Hence there exists an x
1
close to 0 such that u(x
1
) > L(x
1
) > 0. Note that
u ≤ 0 < L on ∂Ω. Hence there exists an ˜x ∈ Ω such that Du(˜x) = DL(˜x) = a. Now
we may translate vertically the plane y = L(x) to the highest such position, that is, the
whole surface y = u(x) lies below the plane. Clearly at such a point, the function u is
positive.
Proof of Theorem 2.21. We should choose g appropriately in order to apply Lemma 2.24.
Note if f ≡ 0 and c ≡ 0 then (−a
ij
D
ij
u)
n
≤ |b|
n
|Du|
n
in Ω. This suggests that we
should take g(p) = |p|
−n
. However, such a function is not locally integrable (at the
origin). Hence we will choose g(p) = (|p|
n
+ µ
n
)
−
1
and then let µ → 0
+
.
First we have by the Cauchy inequality
−a
ij
D
ij
u ≤ b
i
D
i
u + cu − f
≤ b
i
D
i
u − f in Ω
+
= {x : u(x) > 0}
≤ |b| · |Du| + f
−
≤
|b|
n
+
(f
−
)
n
µ
n
1/n
· (|Du|
n
+ µ
n
)
1/n
· (1 + 1)
n−2
n
;
51
Chapter 2: Maximum Principles
in particular,
(−a
ij
D
ij
u)
n
≤
|b|
n
+
f
−
µ
n
(|Du|
n
+ µ
n
) · 2
n−2
.
Now we choose
g(p) =
1
|p|
n
+ µ
n
.
By Lemma 2.24 we have
Z
B
˜
M
(0)
g ≤
2
n−2
n
n
Z
Γ
+
∩Ω
+
|b|
n
+ µ
−n
(f
−
)
n
D
.
We evaluate the integral in the left-hand side in the following way:
Z
B
˜
M
(0)
g = ω
n
Z
˜
M
0
r
n−1
r
n
+ µ
n
dr =
ω
n
n
log
˜
M
n
+ µ
n
µ
n
=
ω
n
n
log
˜
M
n
µ
n
+ 1
!
.
Therefore we obtain
˜
M
n
≤ µ
n
(
exp
(
2
n−2
ω
n
n
n
"
b
D
∗
n
L
n
(Γ
+
∩Ω
+
)
+ µ
−n
f
−
D
∗
n
L
n
(Γ
+
∩Ω
+
)
#)
− 1
)
.
If f 6≡ 0, we choose µ =
f
−
D
∗
L
n
(Γ
+
∩Ω
+
)
. If f 6≡ 0, we may choose any µ > 0 and
then let µ → 0.
In what follows we use Theorem 2.21 and Lemma 2.24 to derive some a priori esti-
mates for solutions to quasi-linear equations and fully nonlinear equations. In the next
result we do not assume uniform ellipticity.
Proposition 2.27. Suppose that u ∈ C(Ω) ∩ C
2
(Ω) satises
Qu ≡ a
ij
(x, u, Du)D
ij
u + b(x, u, Du) = 0 in Ω
where a
ij
∈ C(Ω × R × R
n
) satises
a
ij
(x, z, p)ξ
i
ξ
j
> 0 for any (x, z, p) ∈ Ω × R × R
n
and ξ ∈ R
n
.
Suppose there exist nonnegative functions g ∈ L
n
loc
(R
n
) and h ∈ L
n
(Ω) such that
|b(x, z, p)|
nD
∗
≤
h(x)
g(p)
for any (x, z, p) ∈ Ω × R × R
n
,
52
Section 2.5: Alexandroff Maximum Principle
Z
Ω
h
n
(x)dx <
Z
R
n
g
n
(p)dp ≡ g
∞
.
Then there holds sup
Ω
|u| ≤ sup
∂Ω
|u| + C diam(Ω) where C is a positive constant
depending only on g and h.
EXAMPLE. The prescribed mean curvature equation is given by
(1 + |Du|
2
)∆u − D
i
uD
j
uD
ij
u = nH(x)(1 + |Du|
2
)
3/2
for some H ∈ C(Ω). We have
a
ij
(x, z, p) = (1 + |p|
2
)δ
ij
− p
i
p
j
=⇒ D = (1 + |p|
2
)
n−1
,
b = −nH(x)(1 + |p|
2
)
3/2
.
This implies
|b(x, z, p)|
nD
∗
≤
|H(x)|(1 + |p|
2
)
3/2
(1 + |p|
2
)
n−1
n
= |H(x)|(1 + |p|
2
)
n+2
2n
and in particular
g
∞
=
Z
R
n
g
n
(p)dp =
Z
R
n
d
p
(1 + |p|
2
)
n+2
2
= ω
n
.
Corollary 2.28. Suppose u ∈ C(Ω) ∩C
2
(Ω) satises
(1 + |Du|
2
)∆u − D
i
uD
j
uD
ij
u = nH(x)(1 + |Du|
2
)
3/2
in Ω
for some H ∈ C(Ω). Then if
H
0
≡
Z
Ω
|H(x)|
n
dx < ω
n
,
we have
sup
Ω
|u| ≤ sup
∂Ω
|u| + C diam(Ω)
where C is a positive constant depending only on n and H
0
.
Proof of Proposition 2.27. We prove for subsolutions. Assume Qu ≥ 0 in Ω. Then we
have
−a
ij
D
ij
u ≤ b in Ω.
Note that {D
ij
u} is nonpositive in Γ
+
. Hence −a
ij
D
ij
u ≥ 0, which implies b(x, u, Du) ≥
53
Chapter 2: Maximum Principles
0 in Γ
+
. Then in Γ
+
∩ Ω
+
there holds
b(x, z, Du)
nD
∗
≤
h(x)
g(Du)
.
We may apply Lemma 2.24 to g
n
and get
Z
B
˜g
(0)
g
n
≤
Z
Γ
+
∩Ω
+
g
n
(Du)
−a
ij
D
ij
u
nD
∗
n
≤
Z
Γ
+
∩Ω
+
g
n
(Du)
b
nD
∗
n
≤
Z
Γ
+
∩Ω
+
h
n
≤
Z
Ω
h
n
<
Z
R
n
g
n
.
Therefore there exists a positive constant C, depending only on g and h, such that
˜
M ≤ C. This implies
sup
Ω
u ≤ sup
∂Ω
u
+
+ C diam(Ω).
Next we discuss Monge-Ampère equations.
Corollary 2.29. Suppose u ∈ C(Ω) ∩C
2
(Ω) satises
det(D
2
u) = f(x, u, Du) in Ω
for some f ∈ C(Ω × R × R
n
). Suppose there exist nonnegative functions g ∈ L
1
loc
(R
n
)
and h ∈ L
1
(Ω) such that
|f(x, z, p)| ≤
h(x)
g(p)
for any (x, z, p) ∈ Ω × R × R
n
,
Z
Ω
h(x)dx <
Z
R
n
g(p)dp ≡ g
∞
.
Then there holds
sup
Ω
|u| ≤ sup
∂Ω
|u| + C diam(Ω)
where C is a positive constant depending only on g and h.
The proof is similar to that of Proposition 2.27. There are two special cases. The
rst case is given by f = f(x). We may take g ≡ 1 and hence g
∞
= ∞. So we obtain
the following:
Corollary 2.30. Let u ∈ C(Ω) ∩C
2
(Ω) satisfy
det(D
2
u) = f(x) in Ω
54
Section 2.5: Alexandroff Maximum Principle
for some f ∈ C(Ω). Then there holds
sup
Ω
|u| ≤ sup
∂Ω
|u| +
diam(Ω)
ω
1/n
n
Z
Ω
|f|
n
1/n
.
The second case is about the prescribed Gaussian curvature equations.
Corollary 2.31. Let u ∈ C(Ω) ∩C
2
(Ω) satisfy
det(D
2
u) = K(x)(1 + |Du|
2
)
n+2
2
in Ω
for some K ∈ C(Ω). Then if
K
0
≡
Z
Ω
|K(x)| < ω
n
,
we have
sup
Ω
|u| ≤ sup
∂Ω
|u| + C diam(Ω)
where C is a positive constant depending only on n and K
0
.
We nish this section by proving a maximum principle in a domain with small volume
that is due to Varadhan.
Consider
Lu ≡ a
ij
D
ij
u + b
i
D
i
u + cu in Ω
where {a
ij
} is positive denite pointwise in Ω and
|b
i
| + |c| ≤ Λ and det(a
ij
) ≥ λ
for some positive constants λ and Λ.
Theorem 2.32. Suppose u ∈ C(Ω) ∩ C
2
(Ω) satises Lu ≥ 0 in Ω with u ≤ 0 on ∂Ω.
Assume diam(Ω) ≤ d. Then there is a positive constant δ = δ(n, λ, Λ, d) > 0 such that
if |Ω| ≤ δ then u ≤ 0 in Ω.
Proof of Theorem 2.32. If c ≤ 0, then u ≤ 0 by Theorem 2.21. In general, write c =
c
+
− c
−
. Then
a
ij
D
ij
u + b
i
D
i
u − c
−
u ≥ −c
+
u ≡ f).
By Theorem 2.21 we have
sup
Ω
u ≤ c(n, λ, Λ, d)kc
+
u
+
k
L
n
(Ω)
≤ c(n, λ, Λ, d)kc
+
k
L
∞
|Ω|
1/
n
sup
Ω
u ≤
1
2
sup
Ω
u
55
