Chapter 2: Maximum Principles
2.3 A Priori Estimates
In this section we derive a priori estimates for solutions to the Dirichlet problem and
the Neumann problem.
Suppose Ω is a bounded and connected domain in R
n
. Consider the operator L in Ω
Lu ≡ a
ij
(x)D
ij
u + b
i
(x)D
i
u + c(x)u
for u ∈ C
2
(Ω) ∩C(Ω). We assume that a
ij
, b
i
, and c are continuous and hence bounded
in Ω and that L is uniformly elliptic in Ω, that is,
a
ij
(x)ξ
i
ξ
j
≥ λ|ξ|
2
for any x ∈ Ω and any ξ ∈ R
n
where λ is a positive number. We denote by Λ the sup-norm of a
ij
and b
i
, that is,
max
Ω
|a
ij
| + max
Ω
|b
i
| ≤ Λ.
Proposition 2.15. Suppose u ∈ C
2
(Ω) ∩ C(Ω) satises
Lu = f in Ω,
u = φ on ∂Ω,
for some f ∈ C(Ω) and φ ∈ C(∂Ω). If c(x) ≤ 0 in Ω, then there holds
|u(x)| ≤ max
∂Ω
|φ| + C max
Ω
|f| for any x ∈ Ω
where C is a positive constant depending only on λ, Λ, and diam(Ω).
Proof. We will construct a function w in Ω such that
(i) L(w ± u) = Lw ± f ≤ 0 or Lw ≤ ∓ f in Ω,
(ii) w
±
u
=
w
±
φ
≥
0
or
w
≥ ∓
φ
on
∂
Ω
.
Denote F = max
Ω
|f| and Φ = max
∂Ω
|φ|. We need
Lw ≤ −F in Ω,
w ≥ Φ on ∂Ω.
Suppose the domain Ω lies in the set {0 < x
1
< d} for some d > 0. Set w =
Φ + (e
αd
− e
αx
1
)F with α > 0 to be chosen later. Then we have by direct calculation
−Lw = (a
11
α
2
+ b
1
α)F e
αx
1
− cΦ − c(e
αd
− e
αx
1
)F
40
Section 2.3: A Priori Estimates
≥ (a
11
α
2
+ b
1
α)F ≥ (α
2
λ + b
1
α)F ≥ F
by choosing α large such that α
2
λ + b
1
(x)α ≥ 1 for any x ∈ Ω. Hence w satises (i)
and (ii). By Corollary 2.8 (the comparison principle) we conclude −w ≤ u ≤ w in Ω; in
particular,
sup
Ω
|u| ≤ Φ + (e
αd
− 1)F
where α is a positive constant depending only on λ and Λ.
Proposition 2.16. Suppose u ∈ C
2
(Ω) ∩ C
1
(Ω) satises
Lu = f in Ω,
∂u
∂n
+ α(x)u = φ on ∂Ω,
where n is the outward normal direction to ∂Ω. If c(x) ≤ 0 in Ω and α(x) ≥ α
0
> 0 on
∂Ω, then there holds
|u(x)| ≤ C
max
∂
Ω
|φ| + max
Ω
|f|
for any x ∈ Ω
where C is a positive constant depending only on λ, Λ, α
0
, and diam(Ω).
Proof. We prove for a special case and the general case.
CASE 1. Special case: c(x) ≤ −c
0
< 0.
We will show
|u(x)| ≤
1
c
0
F +
1
α
0
Φ for any x ∈ Ω
where F = max
Ω
|f| and Φ = max
∂Ω
|φ|.
Dene v =
1
c
0
F +
1
α
0
Φ ± u. Then we have
Lv = c(x)
1
c
0
F +
1
α
0
Φ
± f ≤ −F ±f ≤ 0 in Ω,
∂v
∂n
+ αv = α
1
c
0
F +
1
α
0
Φ
± φ ≥ Φ ± φ ≥ 0 on ∂Ω.
If v has a negative minimum in Ω, then v attains it on ∂Ω by Theorem 2.5, say at
x
0
∈ ∂Ω. This implies
∂v
∂n
(x
0
) ≤ 0 for n = n(x
0
), the outward normal direction at x
0
.
Therefore we get
∂v
∂n
+ αv
(x
0
) ≤ αv(x
0
) < 0,
which is a contradiction. Hence we have v ≥ 0 in Ω, in particular,
|u(x)| ≤
1
c
0
F +
1
α
0
Φ for any x ∈ Ω.
41
Chapter 2: Maximum Principles
Note that for this special case c
0
and α
0
are independent of λ and Λ.
CASE 2. General case: c(x) ≤ 0 for any x ∈ Ω.
Consider the auxiliary function u(x) = z(x)w(x) where z is a positive function in Ω
to be determined. Direct calculation shows that w satises
a
ij
D
ij
w + B
i
D
i
w +
c +
a
ij
D
ij
z + b
i
D
i
z
z
w =
f
z
in Ω,
∂w
∂n
+
α +
1
z
∂z
∂n
w =
φ
z
on ∂Ω,
where B
i
=
1
z
(a
ij
+ a
ji
)D
j
z + b
i
. We need to choose the function z > 0 in Ω such that
there hold in
c +
a
ij
D
ij
z + b
i
D
i
z
z
≤ −c
0
(λ, Λ, d, α
0
) < 0 in Ω,
α +
1
z
∂z
∂n
≥
1
2
α
0
on ∂Ω,
or
a
ij
D
ij
z + b
i
D
i
z
z
≤ −c
0
< 0 in Ω,
1
z
∂z
∂n
≤
1
2
α
0
on ∂Ω.
Suppose the domain Ω lies in {0 < x
1
< d}. Choose z(x) = A + e
βd
− e
βx
1
for x ∈ Ω
for some positive A and β to be determined. Direct calculation shows
−
1
z
(a
ij
D
ij
z + b
i
D
i
z) =
(β
2
a
11
+ βb
1
)e
βx
1
A + e
βd
− e
βx
1
≥
β
2
a
11
+ βb
1
A + e
βd
≥
1
A + e
βd
> 0
if β is chosen such that β
2
a
11
+ βb
1
≥ 1. Then we have
1
z
∂z
∂n
≤
β
A
e
βd
≤
1
2
α
0
if A is chosen large. This reduces to the special case we just discussed. The new extra
rst-order term does not change the result. We may apply the special case to w.
Remark 2.17. The result fails if we just assume α(x) ≥ 0 on ∂Ω. In fact, we cannot
even get the uniqueness.
42