Gradient Estimates

Section 2.4: Gradient Estimates

2.4 Gradient Estimates

The basic idea in the treatment of gradient estimates, due to Bernstein, involves

dierentiation of the equation with respect to x

k

, k = 1, . . . , n, followed by multiplication

by D

k

u and summation over k. The maximum principle is then applied to the resulting

equation in the function v = |Du|

2

, possibly with some modication. There are two

kinds of gradient estimates, global gradient estimates and interior gradient estimates.

We will use semilinear equations to illustrate the idea.

Suppose Ω is a bounded and connected domain in R

n

. Consider the equation

a

ij

(x)D

ij

u + b

i

(x)D

i

u = f(x, u) in Ω

for u ∈ C

2

(Ω) and f ∈ C(Ω × R). We always assume that a

ij

and b

i

are continuous

and hence bounded in Ω and that the equation is uniformly elliptic in Ω in the following

sense:

a

ij

(x)ξ

i

ξ

j

≥ λ|ξ|

2

for any x ∈ Ω and any ξ ∈ R

n

for some positive constant λ.

Proposition 2.18. Suppose u ∈ C

3

(Ω) ∩ C

1

(Ω) satises

n

a

ij

(x)D

ij

u + b

i

(x)D

i

u = f(x, u) in Ω

(2.1)

for a

ij

, b

i

∈ C

1

(Ω) and f ∈ C

1

(Ω × R). Then there holds

sup

Ω

|Du| ≤ sup

∂Ω

|Du| + C

where C is a positive constant depending only on λ, diam(Ω), ka

ij

, b

i

k

C

1

(Ω)

, M =

kuk

L

∞

(Ω)

, and kf k

C

1

(Ω×[−M,M ])

.

Proof. Set L ≡ a

ij

D

ij

+ b

i

D

i

. We calculate L( |Du|

2

) rst. Note

D

i

(|Du|

2

) = 2D

k

uD

ki

u,

D

ij

(|Du|

2

) = 2D

ki

uD

kj

u + 2D

k

uD

kij

u.

Dierentiating (2.1) with respect to x

k

, multiplying by D

k

u, and summing over k, we

have by (2.4)

a

ij

D

ij

(|Du|

2

) + b

i

D

i

(|Du|

2

)

= 2a

ij

D

ki

uD

kj

u − 2D

k

a

ij

D

k

uD

ij

u

− 2D

k

b

i

D

k

uD

i

u + 2D

x

f|Du|

2

+ 2D

k

fD

k

u.

43

Chapter 2: Maximum Principles

The ellipticity assumption implies

X

i,j,k

a

ij

D

ki

uD

kj

u ≥ λ|D

2

u|

2

.

By the Cauchy inequality, we have

L(|Du|

2

) ≥ λ|D

2

u|

2

− C|Du|

2

− C

with C a positive constant depending only on kf k

C

1

(Ω×[−M,M ])

, ka

ij

, b

i

k

C

1

(Ω)

, and λ.

We need to add another term u

2

. We have by the ellipticity assumption

L(u

2

) = 2a

ij

D

i

uD

j

u + 2u{a

ij

D

ij

u + b

i

D

i

u}

≥ 2λ|Du|

2

+ 2uf.

Therefore we obtain

L(|Du|

2

+ αu

2

) ≥ λ|D

2

u|

2

+ (2λα − C)|Du|

2

− C

≥ λ|D

2

u|

2

+ |Du|

2

− C

if we choose α > 0 large, with C depending in addition on M. In order to control the

constant term we may consider another function e

βx

1

for β > 0. Hence we get

L(|Du|

2

+ αu

2

+ e

βx

1

) ≥ λ|D

2

u|

2

+ |Du|

2

+ {β

2

a

11

e

βx

1

+ βb

1

e

βx

1

− C}.

If we put the domain Ω ⊂ {x

1

> 0}, then e

βx

1

≥ 1 for any x ∈ Ω. By choosing β large,

we may make the last term positive. Therefore, if we set w = |Du|

2

+α|u|

2

+e

βx

1

for large

α, β depending only on λ, diam(Ω), ka

ij

, b

i

k

C

1

(Ω)

, M = kuk

L

∞

(Ω)

, and kfk

C

1

(Ω×[−M,M ])

,

then we obtain

Lw ≥ 0 in Ω.

By the maximum principle we have

sup

Ω

w ≤ sup

∂Ω

w.

This nishes the proof.

Similarly, we can discuss the interior gradient bound. In this case, we just require

the bound of sup

Ω

|u|.

44

Section 2.4: Gradient Estimates

Proposition 2.19. Suppose u ∈ C

3

(Ω) satises

a

ij

(x)D

ij

u + b

i

(x)D

i

u = f(x, u) in Ω

for a

ij

, b

i

∈ C

1

(Ω) and f ∈ C

1

(Ω ×R). Then there holds for any compact subset Ω

′

⋐ Ω

sup

Ω

′

|Du| ≤ C

where C is a positive constant that depends only on λ, diam(Ω), dist(Ω

′

, ∂Ω), ka

ij

, b

i

k

C

1

(Ω)

,

M = kuk

L

∞

(Ω)

, and kf k

C

1

(Ω×[−M,M ])

.

Proof. We need to take a cuto function γ ∈ C

∞

0

(Ω) with γ ≥ 0 and consider the

auxiliary function with the following form:

w = γ|Du|

2

+ α|u|

2

+ e

βx

1

.

Set v = γ|Du|

2

. Then we have for operator L dened as before

Lv = (Lγ)|Du|

2

+ γL(|Du|

2

) + 2a

ij

D

i

γD

j

|Du|

2

.

Recall an inequality in the proof of Proposition 2.18,

L(|Du|

2

) ≥ λ|D

2

u|

2

− C|Du|

2

− C.

Hence we have

Lv ≥ λγ|D

2

u|

2

+ 2a

ij

D

k

uD

i

γD

kj

u − C|Du|

2

+ (Lγ)|Du|

2

− C.

The Cauchy inequality implies for any ε > 0

|2a

ij

D

k

uD

i

γD

kj

u| ≤ ε|Dγ|

2

|D

2

u|

2

+ c(ε)|Du|

2

.

For the cuto function γ, we require that

|Dγ|

2

≤ Cγ in Ω.

Therefore we have by taking ε > 0 small

Lv ≥ λγ|D

2

u|

2



1 − ε

|Dγ|

2

γ



− C|Du|

2

− C

≥

1

2

λγ|D

2

u|

2

− C|Du|

2

− C.

Now we may proceed as before.

45

Chapter 2: Maximum Principles

In the rest of this section we use barrier functions to derive the boundary gradient

estimates. We need to assume that the domain Ω satises the uniform exterior sphere

property.

Proposition 2.20. Suppose u ∈ C

2

(Ω) ∩ C(Ω) satises

a

ij

(x)D

ij

u + b

i

(x)D

i

u = f(x, u) in Ω

for a

ij

, b

i

∈ C(Ω) and f ∈ C(Ω × R). Then there holds

|u(x) − u(x

0

)| ≤ C|x − x

0

| for any x ∈ Ω and x

0

∈ ∂Ω

where C is a positive constant depending only on λ, Ω, ka

ij

, b

i

k

L

∞

(Ω)

, M = kuk

L

∞

(Ω)

,

kfk

L

∞

(Ω×[−M,M ])

, and kφk

C

2

(Ω)

for some φ ∈ C

2

(Ω) with φ = u on ∂Ω.

Proof. For simplicity we assume u = 0 on ∂Ω. As before, set L = a

ij

D

ij

+ b

i

D

i

. Then

we have

L(±u) = ±f ≥ −F in Ω

where we denote F = sup

Ω

|f(·, u)|. Now x x

0

∈ ∂Ω. We will construct a function w

such that

Lw ≤ −F in Ω, w(x

0

) = 0, w|

∂Ω

≥ 0.

Then by the maximum principle we have

−w ≤ u ≤ w in Ω.

Taking the normal derivative at x

0

, we have









∂u

∂n

(x

0

)









≤

∂w

∂n

(x

0

).

So we need to bound

∂w

∂n

(x

0

) independently of x

0

.

Consider the exterior ball B

R

(y) with B

R

(y) ∩Ω = {x

0

}. Dene d(x) as the distance

from x to ∂B

R

(y). Then we have

0 < d(x) < D ≡ diam(Ω) for any x ∈ Ω.

In fact, d(x) = |x − y| − R for any x ∈ Ω. Consider w = ψ(d) for some function ψ

dened in [0, ∞). Then we need

ψ(0) = 0 ( =⇒ w(x

0

) = 0)

ψ(d) > 0 for d > 0 ( =⇒ w|

∂Ω

≥ 0)

46

Section 2.4: Gradient Estimates

ψ

′

(0) is controlled.

From the rst two inequalities, it is natural to require that ψ

′

(d) > 0. Note

Lw = ψ

′′

a

ij

D

i

dD

j

d + ψ

′

a

ij

D

ij

d + ψ

′

b

i

D

i

d.

Direct calculation yields

D

i

d(x) =

x

i

− y

i

|x − y|

,

D

ij

d(x) =

δ

ij

|x − y|

−

(x

i

− y

i

)(x

j

− y

j

)

|x − y|

3

,

which imply |Dd| = 1 and with Λ = sup | a

ij

|

a

ij

D

ij

d =

a

ii

|x − y|

−

a

ij

|x − y|

D

i

dD

j

d

≤

nΛ

|x − y|

−

λ

|x − y|

=

nΛ − λ

|x − y|

≤

nΛ − λ

R

.

Therefore we obtain by ellipticity

Lw ≤ ψ

′′

a

ij

D

i

dD

j

d + ψ

′



nΛ − λ

R

+ |b|



≤ λψ

′′

+ ψ

′



nΛ − λ

R

+ |b|



if we require ψ

′′

< 0. Hence in order to have Lw ≤ −F we need

λψ

′′

+ ψ

′



nΛ − λ

R

+ |b|



+ F ≤ 0.

To this end, we study the equation for some positive constants a and b

ψ

′′

+ aψ

′

+ b = 0

whose solution is given by

ψ(d) = −

b

a

d +

C

1

a

−

C

2

a

e

−ad

for some constants C

1

and C

2

. For ψ(0) = 0, we need C

1

= C

2

. Hence we have for some

constant C

ψ(d) = −

b

a

d +

C

a

(1 − e

−ad

),

47

Chapter 2: Maximum Principles

which implies

ψ

′

(d) = Ce

−ad

−

b

a

= e

−ad



C −

b

a

e

ad



ψ

′′

(d) = −Cae

−ad

.

In order to have ψ

′

(d) > 0, we need C ≥

b

a

e

aD

. Since ψ

′

(d) > 0 for d > 0, so

ψ(d) > ψ(0) = 0 for any d > 0. Therefore we take

ψ(d) = −

b

a

d +

b

a

2

e

aD

(1 − e

−ad

)

=

b

a



1

a

e

aD

(1 − e

−ad

) − d



.

Such ψ satises all the requirements we imposed. This nishes the proof.

48