Jost J. Partial Differential Equations

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316 12. Moser Iteration Method and Regularity Theorem of de Giorgi and Nash

φ(μ, 3) ≤ β

φ(−μ, 3),

and hence (12.1.27), thus completing the proof. 

A reference for this section is Moser [18].

Krylov and Safonov have shown that solutions of elliptic equations that

are not of divergence type satisfy Harnack inequalities as well. In order to

describe their results in the simplest case, we again omit all lower-order terms

and consider solutions of

Mu :=



i,j=1

(x)

∂

∂x

u(x)=0.

Here the coeﬃcients a

(x) again need only be (measurable and) bounded

and satisfy the structural condition (12.1.1), i.e.,

λ|ξ|

≤



i,j=1

(x)ξ

for all x ∈ Ω, ξ ∈ R

and

sup

i,j,x

(x)|≤Λ

with constants 0 <λ<Λ<∞.

We then have the following theorem:

Theorem 12.1.3: Let u ∈ W

2,d

(Ω) be positive and satisfy Mu ≥ 0 almost

everywhere in B(x

, 4R) ⊂ R

. For any p>0, we then have

sup

B(x

,R)

u ≤ c



−



B(x

,2R)



1/p

with a constant c

depending on d,

,andp.

Theorem 12.1.4: Let u ∈ W

2,d

(Ω) be positive and satisfy Mu ≤ 0 almost

everywhere in B(x

, 4R) ⊂ R

. Then there exist p>0 and some constant c

depending only on d and

, such that



−



B(x

,R)



1/p

≤ c

inf

B(x

,R)

As in the case of divergence-type equations (see Section 12.2 below), these

results imply Harnack inequalities, maximum principles, and the H¨older con-

tinuity of solutions u ∈ W

2,d

(Ω)of

Mu = 0 almost everywhere Ω ⊂ R

Proofs of the results of Krylov–Safonov can be found in Gilbarg–Trudinger [9].

12.2 Properties of Solutions of Elliptic Equations 317

12.2 Properties of Solutions of Elliptic Equations

In this section we shall apply the Moser–Harnack inequality in order to de-

duce the H¨older continuity of weak solutions of Lu = 0 under the structural

condition (12.1.1). That result had originally been proved by E. de Giorgi

and J. Nash independently of each other, and with diﬀerent methods, before

J. Moser found the proof presented here, based on the Harnack inequality.

Lemma 12.2.1: Let u ∈ W

1,2

(Ω) be a weak solution of L, i.e.,

Lu =



i,j=1

∂

∂x



(x)

∂

∂x

u(x)



≥ 0 weakly,

with L satisfying the conditions stated in Section 12.1. Then u is bounded

from above on any Ω

⊂⊂ Ω. Thus, if u is a weak solution of Lu =0,itis

bounded from above and below on any such Ω

Proof: By Lemma 12.1.3, for any positive k,

v(x) := max(u(x),k)

is a positive subsolution (by the way, in place of v, one might also employ

the approximating subsolutions f

◦u from the proof of Lemma 12.1.3). The

local boundedness of v, hence of u, then follows from Theorem 12.1.1, using

a ball chain argument as in the proof of Corollary 12.1.2. 

Theorem 12.2.1: Let u ∈ W

1,2

(Ω) be a weak solution of

Lu =



i,j=1

∂

∂x



(x)

∂

∂x

u(x)



=0, (12.2.1)

assuming that the measurable and bounded coeﬃcients a

(x) satisfy the struc-

tural conditions

λ |ξ|

≤



i,j=1

(x)ξ



(x)



≤ Λ (12.2.2)

for all x ∈ Ω, ξ ∈ R

, with constants 0 <λ<Λ<∞. Then u is H¨older

continuous in Ω. More precisely, for any Ω

⊂⊂ Ω, there exist some α ∈

(0, 1) and a constant c with

|u(x) − u(y)|≤c |x − y|

(12.2.3)

for all x, y ∈ Ω

. α depends on d,

,andΩ

, c in addition on sup

u −

inf

318 12. Moser Iteration Method. Regularity Theorem of de Giorgi and Nash

Proof: Let x ∈ Ω.ForR>0andB(x, R) ⊂ Ω, we put

M(R):= sup

B(x,R)

u, m(R):= inf

B(x,R)

(By Lemma 12.2.1, −∞ <m(R) ≤ M(R) < ∞.)Then

ω(R):=M (R) − m(R)

is the oscillation of u in B(x, R), and we plan to prove the inequality

ω(r) ≤ c





ω(R) for 0 <r≤

(12.2.4)

for some α to be speciﬁed. This will then imply

u(x) − u(y) ≤ sup

B(x,r)

u − inf

B(x,r)

u = ω(r) ≤ c

ω(R)

|x − y|

. (12.2.5)

for all y with |x − y| = r. This, in turn, easily implies the claim.

We now turn to the proof of (12.2.4):

M(R) − u and u − m(R)

are positive solutions of Lu =0inB(x, R).

Thus, by Corollary 12.1.1,

M(R) − m





=sup

B(x,

)

(M(R) − u) ≤ c

inf

B(x,

)

(M(R) − u)

= c



M(R) − M





and analogously,





− m(R)= sup

B(x,

)

(u − m(R)) ≤ c

inf

B(x,

)

(u − m(R))

= c





− m(R)



(By Corollary 12.1.1, c

does not depend on R.) Adding these two inequalities

yields





− m





≤

− 1

(M(R) − m(R)). (12.2.6)

With ϑ :=

−1

< 1, thus

More precisely, these are nonnegative solutions, and as in the proof of Theo-

rem 12.1.2, one adds ε>0 and lets ε approach to 0.

12.2 Properties of Solutions of Elliptic Equations 319





≤ ϑω(R).

Iterating this inequality gives





≤ ϑ

ω(R)forn ∈ N. (12.2.7)

Now let

n+1

≤ r ≤

. (12.2.8)

We now choose α>0 such that

ϑ ≤





Then

ω(r) ≤ ω





since ω is obviously monotonically increasing

≤ ϑ

ω(R) by (12.2.7)

≤





ω(R)

≤





ω(R) by (12.2.8)

−α





ω(R),

whence (12.2.4). 

We now want to prove a strong maximum principle:

Theorem 12.2.2: Let u ∈ W

1,2

(Ω) satisfy Lu ≥ 0 weakly, the coeﬃcients

of L again satisfying

λ |ξ|

≤



i,j

(x)ξ



(x)



≤ Λ

for all x ∈ Ω, ξ ∈ R

. If for some ball B(y

,R) ⊂⊂ Ω,

sup

B(y

,R)

u =sup

u, (12.2.9)

then u is constant.

Proof: If (12.2.9) holds, we may ﬁnd some ball B(x

)withB(x

, 4R

)

⊂ Ω and

320 12. Moser Iteration Method. Regularity Theorem of de Giorgi and Nash

sup

B(x

)

u =sup

u. (12.2.10)

Without loss of generality sup

u<∞, because sup

B(y

,R)

u<∞ by

Lemma 12.2.1. For

M>sup

M − u then is a positive subsolution, and we may apply Theorem 12.1.2 to

it. Passing to the limit, the resulting inequalities then continue to hold for

M =sup

u. (12.2.11)

Thus, for p = 1, we get from Theorem 12.1.2

−



B(x

,2R

)

(M − u) ≤ c inf

B(x

)

(M − u)=0

by (12.2.10), (12.2.11). Since by choice of M,wealsohaveu ≤ M, it follows

that

u ≡ M (12.2.12)

in B(x

, 2R

Now let y ∈ Ω. We may ﬁnd a chain of balls B(x

), i =0,...,m,

with B(x

, 4R

) ⊂ Ω, B(x

i−1

) ∩ B(x

) =0fori =1,...,m,

y ∈ B(x

). We already know that u ≡ M on B(x

, 2R

). Because of

B(x

) ∩ B(x

) = 0, this implies

sup

B(x

)

u = M,

hence by our preceding reasoning

u ≡ M on B(x

, 2R

Iteratively, we obtain

u ≡ M on B(x

, 2R

and because of y ∈ B(x

u(y)=M.

Since y was arbitrary, it follows that

u ≡ M in Ω.



12.3 Regularity of Minimizers of Variational Problems 321

As another application of the Harnack inequality, we shall now demonstrate

a result of Liouville type:

Theorem 12.2.3: Any bounded (weak) solution of Lu =0that is deﬁned on

all of R

, where L has measurable bounded coeﬃcients a

(x) satisfying

λ |ξ|≤



i,j

(x)ξ



(x)



≤ Λ

for ﬁxed constants 0 <λ≤ Λ<∞ and all x ∈ R

, ξ ∈ R

, is constant.

Proof: Since u is bounded, inf

u and sup

u are ﬁnite. Thus, for any

μ<inf

u − μ is a positive solution of Lu =0onR

. Therefore, by Corollary 12.1.1

0 ≤ sup

B(0,R)

u − μ ≤ c



inf

B(0,R)

u − μ



for any R>0 and any μ<inf

u, and passing to the limit, then this also

holds for

μ =inf

Since c

does not depend on R, it follows that

0 ≤ sup

u − μ ≤ c



inf

u − μ



=0,

and hence

u ≡ const.



12.3 Regularity of Minimizers of Variational Problems

The aim of this section is the proof of (a special case of) the fundamental

result of de Giorgi on the regularity of minima of variational problems with

elliptic Euler–Lagrange equations:

Theorem 12.3.1: Let F : R

→ R be a function of class C

∞

satisfying

the following conditions: For some constants K, Λ < ∞,λ>0 and for all

p =(p

,...,p

) ∈ R

(i)



∂F

∂p

(p)



≤ K |p| (i =1,...,d).

322 12. Moser Iteration Method. Regularity Theorem of de Giorgi and Nash

(ii) λ |ξ|

≤



∂

F (p)

∂p

≤ Λ |ξ|

for all ξ ∈ R

Let Ω ⊂ R

be a bounded domain. Let u ∈ W

1,2

(Ω) be a minimizer of the

variational problem

I(v):=



F (Dv(x))dx,

i.e.,

I(u) ≤ I(u + ϕ) for all ϕ ∈ H

1,2

(Ω). (12.3.1)

Then u ∈ C

∞

(Ω).

Remark: Because of (i), there exist constants c

, c

with

|F (p)|≤c

+ c

|p|

. (12.3.2)

Since Ω is assumed to be bounded, this implies

I(v)=



F (Dv) < ∞

for all v ∈ W

1,2

(Ω). Therefore, our variational problem, namely to minimize

I in W

1,2

(Ω), is meaningful.

We shall ﬁrst derive the Euler–Lagrange equations for a minimizer of I:

Lemma 12.3.1: Suppose that the assumptions of Theorem 12.3.1 hold. We

then have for all ϕ ∈ H

1,2

(Ω),





i=1

(Du)D

ϕ = 0 (12.3.3)

(using the abbreviation F

∂F

∂p

Proof: By (i),





i=1

(Dv)D

ϕ ≤ dK



|Dv||Dϕ|≤dKDv

(Ω)

Dϕ

(Ω)

and this is ﬁnite for ϕ, v ∈ W

1,2

(Ω). By a standard result of Lebesgue inte-

gration theory, on the basis of this inequality we may compute

I(u + tϕ)

by diﬀerentiation under the integral sign:

12.3 Regularity of Minimizers of Variational Problems 323

I(u + tϕ)=





(Du + tDϕ)D

ϕ. (12.3.4)

In particular, I(u + tϕ) is a diﬀerentiable function of t ∈ R, and since u is a

minimizer,

I(u + tϕ)|

t=0

=0. (12.3.5)

Equation (12.3.4) for t = 0 then implies (12.3.3). 

Lemma 12.3.1 reduces Theorem 12.3.1 to the following:

Theorem 12.3.2: Let A

: R

→ R, i =1,...,d,beC

∞

-functions satisfying

the following conditions: There exist constants K, Λ < ∞, λ>0 such that

for all p ∈ R

(i)



(p)



≤ K |p| (i =1,...,d).

(ii) λ |ξ|

≤



i,j=1

∂A

(p)

∂p

for all ξ ∈ R

(iii)



∂A

(p)

∂p



≤ Λ.

Let u ∈ W

1,2

(Ω) be a weak solution of



i=1

∂

∂x

(Du)=0 in Ω ⊂ R

, (12.3.6)

i.e., for all ϕ ∈ H

1,2

(Ω),let





i=1

(Du)D

ϕ =0. (12.3.7)

Then u ∈ C

∞

(Ω).

The crucial step in the proof will be Theorem 12.2.1, of de Giorgi and

Nash. Important steps towards Theorem 12.3.2 had been obtained earlier

by S. Bernstein, L. Lichtenstein, E. Hopf, C. Morrey, and others.

We shall start with a lemma.

Lemma 12.3.2: Under the assumptions of Theorem 12.3.2, for any Ω



⊂⊂

Ω we have u ∈ W

2,2

(Ω



), and moreover, u

2,2

(Ω



)

≤ c u

1,2

(Ω)

, where

c = c(λ, Λ, dist(Ω



,∂Ω)).

Proof: We shall proceed as in the proof of Theorem 9.2.1. For

|h| < dist(supp ϕ, ∂Ω),

k,−h

(x):=ϕ(x − he

)(e

being the kth unit vector) is of class H

1,2

(Ω)as

well. Therefore,

324 12. Moser Iteration Method. Regularity Theorem of de Giorgi and Nash





i=1

(Du(x))D

k,−h

(x)dx





i=1

(Du(x))D

ϕ(x − he

)dx





i=1

(Du(y + he

))D

ϕ(y)dy





i=1

((Du)

k,h

) D

ϕ.

Subtracting (12.3.7), we obtain







(Du(x + he

)) − A

(Du(x))



ϕ(x)=0. (12.3.8)

For almost all x ∈ Ω

(Du(x + he

)) − A

(Du(x))



(tDu(x + he

)+(1− t)Du(x)) dt



⎛

⎝



j=1

(tDu(x + he

)+(1− t)Du(x)) D

(u(x + he

) − u(x))

⎞

⎠

dt.

(12.3.9)

We thus put

(x):=



(tDu(x + he

)+(1− t)Du(x)) dt,

and using (12.3.9), we rewrite (12.3.8) as





i,j

(x)D



u(x + he

) − u(x)



ϕ(x)dx =0. (12.3.10)

Here, because of (ii) and (iii),

λ |ξ|

≤



i,j

(x)ξ

≤ Λ |ξ|

for all ξ ∈ R

We may thus proceed as in Section 9.2 and put

ϕ =

(u(x + he

) − u(x)) η

12.3 Regularity of Minimizers of Variational Problems 325

with η ∈ C

(Ω



), where we choose Ω



satisfying



⊂⊂ Ω



⊂⊂ Ω,

dist(Ω



,∂Ω), dist(Ω



,∂Ω



) ≥

dist(Ω



,∂Ω), and require

0 ≤ η ≤ 1,

η(x)=1 forx ∈ Ω



|Dη|≤

dist(Ω



,∂Ω)

as well as

|2h| < dist(Ω



,∂Ω).

Using the notation

u(x)=

u(x + he

) − u(x)

(12.3.10) then implies





DΔ



≤





i,j







= −





i,j

u 2η(D

η)Δ

u by (12.3.10)

≤ εΛ





DΔ







|Dη|

for all ε>0,

and with ε =

2Λ





DΔ



≤ c







≤ c



|Du|

by Lemma 9.2.1, with c

independent of h. Hence

)

DΔ

)

(Ω



)

≤ c

Du

(Ω)

. (12.3.11)

Since the right hand side of (12.3.11) does not depend on h, from Lemma 9.2.2

we obtain D

u ∈ L

(Ω



) and the inequality

)

(Ω



)

≤ c

Du

(Ω)

. (12.3.12)

Consequently, u ∈ W

2,2

(Ω



). 