Jost J. Partial Differential Equations

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

Performing the limit h → 0 in (12.3.10), with

(x):=A

(Du(x)),

v := D

(12.3.13)

we also obtain





i,j

(x)D

ϕ =0 forallϕ ∈ H

1,2

(Ω).

By (ii), (iii), (a

(x))

i,j=1,...,d

satisﬁes the assumptions of Theorem 12.2.1.

Applying that result to v = D

u then yields the following result:

Lemma 12.3.3: Under the assumptions of Theorem 12.2.1,

Du ∈ C

(Ω)

for some α ∈ (0, 1), i.e.,

u ∈ C

1,α

(Ω).



Thus v = D

u, k =1,...,d, is a weak solution of



i,j=r



(x)D



=0. (12.3.14)

Here, the coeﬃcients a

(x) satisfy not only the ellipticity condition

λ |ξ|

≤



i,j=1

(x)ξ



(x)



≤ Λ

for all ξ ∈ R

, x ∈ Ω, i, j =1,...,d, but by (12.3.13), they are also H¨older

continuous, since A

is smooth and Du is H¨older continuous by Lemma 12.3.3.

For the proof of Theorem 12.3.2, we thus need a regularity theory for such

equations. Equation (12.3.14) is of divergence type, in contrast to those

treated in Chapter 11, and therefore, we cannot apply the results of Schauder

directly. However, one can develop similar methods. For the sake of variety,

here, we shall present the method of Campanato as an alternative approach.

As a preparation, we shall now prove some auxiliary results for equations of

type (12.3.14) with constant coeﬃcients. (Of course, these results are already

essentially known from Chapter 9.)

The ﬁrst result is the Caccioppoli inequality:

12.3 Regularity of Minimizers of Variational Problems 327

Lemma 12.3.4: Let (A

)

i,j=1,...,d

be a matrix with



≤ Λ for all i, j,

and

λ |ξ|

≤



i,j=1

for all ξ ∈ R

with λ>0.Letu ∈ W

1,2

(Ω) be a weak solution of



i,j=1





=0 in Ω. (12.3.15)

We then have for all x

∈ Ω and 0 <r<R<dist(x

,∂Ω) and all μ ∈ R,



B(x

,r)

|Du|

≤

(R − r)



B(x

,R)\B(x

,r)

|u − μ|

. (12.3.16)

Proof: We choose η ∈ H

1,2

(B(x

,R)) with

0 ≤ η ≤ 1,

η ≡ 1onB(x

,r), hence Dη ≡ 0onB(x

,r),

|Dη|≤

R − r

As in Section 9.2, we employ the test function

ϕ =(u − μ)η

and obtain





i,j



(u − μ)η







i,j

uη





i,j

u(u − μ)ηD

η.

Using the ellipticity conditions, we deduce the inequality



B(x

,R)

|Du|

≤



B(x

,R)



uη

≤ εΛ d



B(x

,R)

|Du|



B(x

,R)\B(x

,r)

|Dη|

|u − μ|

since Dη =0onB(x

,r). Hence, with ε =

Λd

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



B(x

,R)

|Du|

≤

(R − r)



B(x

,R)\B(x

,r)

|u − μ|

and because of



B(x

,r)

|Du|

≤



B(x

,R)

|Du|

the claim results. 

The next lemma contains the Campanato estimates:

Lemma 12.3.5: Under the assumptions of Lemma 12.3.4, we have



B(x

,r)

|u|

≤ c







B(x

,R)

|u|

(12.3.17)

as well as



B(x

,r)



u − u

B(x

,r)



≤ c





d+2



B(x

,R)



u − u

B(x

,R)



. (12.3.18)

Proof: Without loss of generality r<

. We choose k>d.BytheSobolev

embedding theorem (Theorem 9.1.1) or an extension of this result analogous

to Corollary 9.1.3,

k,2

(B(x

,R)) ⊂ C

(B(x

,R)).

By Theorem 9.3.1, now u ∈ W

k,2





, with an estimate analogous to

Theorem 9.2.2. Therefore,



B(x

,r)

|u|

≤ c

sup

B(x

,r)

|u|

≤ c

d−2k

u

k,2

(

))

≤ c



B(x

,R)

|u|

(Concerning the dependence on the radius: The power r

is obvious. The

power R

can easily be derived from a scaling argument, instead of carefully

going through all the intermediate estimates.) This yields (12.3.17). Since

we are dealing with an equation with constant coeﬃcients, Du is a solution

along with u.Forr<

,wethusobtain



B(x

,r)

|Du|

≤ c



(

)

|Du|

. (12.3.19)

By the Poincar´e inequality (Corollary 9.1.4),



B(x

,r)



u − u

B(x

,r)



≤ c



B(x

,r)

|Du|

. (12.3.20)

12.3 Regularity of Minimizers of Variational Problems 329

By the Caccioppoli inequality (Lemma 12.3.4)



(

)

|Du|

≤



B(x

,R)



u − u

B(x

,R)



. (12.3.21)

Then (12.3.19)–(12.3.21) imply (12.3.18). 

We may now use Campanato’s method to derive the following regularity

result:

Theorem 12.3.3: Let a

(x), i, j =1,...,d, be functions of class C

, 0 <

α<1,onΩ ⊂ R

, satisfying the ellipticity condition

λ |ξ|

≤



i,j=1

(x)ξ

for all ξ ∈ R

,x∈ Ω (12.3.22)

and



(x)



≤ Λ for all x ∈ Ω,i, j =1,...,d, (12.3.23)

with ﬁxed constants 0 <λ≤ Λ<∞. Then any weak solution v of



i,j=1



(x)D



= 0 (12.3.24)

is of class C

1,α



(Ω) for any α



with 0 <α



<α.

Proof: For x

∈ Ω,wewrite

= a



(x) − a

)



Letting

:= a

(12.3.24) becomes



i,j=1







i,j=1



) − a

(x))D





j=1



(x)



with

(x):=



i=1



) − a

(x))D



. (12.3.25)

This means that

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





i,j=1

ϕ =





j=1

ϕ for all ϕ ∈ H

1,2

(Ω). (12.3.26)

For some ball B(x

,R) ⊂ Ω,let

w ∈ H

1,2

(B(x

,R))

be a weak solution of



i,j=1





=0 inB(x

,R),

w = v on ∂B(x

,R).

(12.3.27)

Thus w is a solution of



B(x

,R)



i,j=1

ϕ =0 forallϕ ∈ H

1,2

(B(x

,R)). (12.3.28)

Such a w exists by the Lax–Milgram theorem (see Appendix 12.3). Note that

we seek z = w − v with

B(ϕ, z):=





= −





=:F (ϕ) for all ϕ ∈ H

1,2

(B(x

,R)).

Since (12.3.27) is a linear equation with constant coeﬃcients, then if w is

a solution, so is D

w, k =1,...,d (with diﬀerent boundary conditions, of

course). We may thus apply (12.3.17) from Lemma 4.3.5 to u = D

w and

obtain



B(x

,r)

|Dw|

≤ c







B(x

,R)

|Dw|

. (12.3.29)

(Here, Dw stands for the vector (D

w,...,D

w).) Since w = v on ∂B(x

,R),

ϕ = v −w is an admissible test function in (12.3.28), and we obtain



B(x

,R)



i,j=1

w =



B(x

,R)



i,j=1

v. (12.3.30)

Using (12.3.27), (12.3.23) and the Cauchy–Schwarz inequality, this implies



B(x

,R)

|Dw|

≤



Λd





B(x

,R)

|Dv|

. (12.3.31)

12.3 Regularity of Minimizers of Variational Problems 331

Equations (12.3.26) and (12.3.28) imply



B(x

,R)



i,j=1

(v − w)D

ϕ =



B(x

,R)



i,j=1

for all ϕ ∈ H

1,2

(B(x

,R)). We utilize once more the test function ϕ = v −w

to obtain



B(x

,R)

|D(v − w)|

≤



B(x

,R)



i,j

(v − w)D

(v − w)



B(x

,R)



(v − w)

≤





B(x

,R)

|D(v − w)|



⎛

⎝



B(x

,R)





⎞

⎠

by the Cauchy–Schwarz inequality, i.e.,



B(x

,R)

|D(v − w)|

≤



B(x

,R)





. (12.3.32)

We now put the preceding estimates together. For 0 <r≤ R,wehave



B(x

,r)

|Dv|

≤ 2



B(x

,r)

|Dw|



B(x

,r)

|D(v − w)|

≤ c







B(x

,r)

|Dv|



B(x

,r)

|D(v − w)|

by (12.3.29), (12.3.31). Now



B(x

,r)

|D(v − w)|

≤



B(x

,R)

|D(v − w)|

, since r ≤ R

≤



B(x

,R)





by (12.3.32)

≤

sup

i,j

x∈B(x

,R)



) − a

(x)





B(x

,R)

|Dv|

by (12.3.25)

≤ c

2α



B(x

,R)

|Dv|

(12.3.33)

since the a

are of class C

. Altogether, we obtain

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



B(x

,r)

|Dv|

≤ γ







+ R

2α





B(x

,R)

|Dv|

(12.3.34)

with some constant γ. If (12.3.34) did not contain the term R

2α

(which is

present solely for the reason that the a

(x), while H¨older continuous, are not

necessarily constant), we would have a useful inequality. That term, however,

can be made to disappear by a simple trick. For later purposes, we formulate

a somewhat more general result:

Lemma 12.3.6: Let σ(r) be a nonnegative, monotonically increasing func-

tion satisfying

σ(r) ≤ γ





+ δ



σ(R)+κR

for all 0 <r≤ R ≤ R

,withμ>ν and δ ≤ δ

(γ,μ,ν).Ifδ

is suﬃciently

small, for 0 <r≤ R ≤ R

, we then have

σ(r) ≤ γ





σ(R)+κ

with γ

depending on γ,μ,ν,andκ

depending in addition on κ (κ

=0if

κ =0).

Proof: Let 0 <τ <1, R<R

.Thenbyassumption

σ(τR) ≤ γτ



1+δτ

−μ



σ(R)+κR

We choose 0 <τ<1 such that

2γτ

= τ

with ν<λ<μ(without loss of generality 2γ>1), and assume that

−μ

≤ 1.

It follows that

σ(τR) ≤ τ

σ(R)+κR

and thus iteratively for k ∈ N,

σ(τ

k+1

R) ≤ τ

σ(τ

R)+κτ

kν

≤ τ

(k+1)λ

σ(R)+κτ

kν



j=0

j(λ−ν)

≤ γ

(k+1)ν

(σ(R)+κR

)

(where γ

, as well as the subsequent γ

, contains a factor

). We now choose

k ∈ N such that

12.3 Regularity of Minimizers of Variational Problems 333

k+2

R<r≤ τ

k+1

and obtain

σ(r) ≤ σ(τ

k+1

R) ≤ γ





σ(R)+κ



Continuing with the proof of Theorem 12.3.3, applying Lemma 12.3.6 to

(12.3.34), where we have to require 0 <r≤ R ≤ R

with R

2α

≤ δ

,we

obtain the inequality



B(x

,r)

|Dv|

≤ c





d−ε



B(x

,R)

|Dv|

(12.3.35)

for each ε>0, where c

and R

depend on ε. We repeat this procedure,

but this time applying (12.3.18) from Lemma 12.3.5 in place of (12.3.17).

Analogously to (12.3.29), we obtain



B(x

,r)



Dw − (Dw)

B(x

,r)



≤ c





d+2



B(x

,R)



Dw − (Dw)

B(x

,R)



(12.3.36)

We also have



B(x

,R)



Dw − (Dw)

B(x

,R)



≤



B(x

,R)



Dw − (Dv)

B(x

,R)



because for any L

-function g, the following relation holds:



B(x

,R)



g − g

B(x

,R)



=inf

κ∈R



B(x

,R)

|g − κ|

. (12.3.37)

(Proof: For g ∈ L

(Ω),F(κ):=



|g − κ|

is convex and diﬀerentiable with

respect to κ,and



(κ)=



2(κ − g);

hence F



(κ) = 0 precisely for

κ =

|Ω|



and since F is convex, a critical point has to be a minimizer.)

Moreover,

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



B(x

,R)



Dw − (Dv)

B(x

,R)



≤



B(x

,R)



i,j



w − (D

B(x

,R)



w − (D

B(x

,R)





B(x

,R)



i,j



w − (D

B(x

,R)



v − (D

B(x

,R)





B(x

,R)



i,j

B(x

,R)

v − D

by (12.3.30). The last integral vanishes, since A

B(x

,R)

is constant and

v − w ∈ H

1,2

(B(x

,R)). Applying the Cauchy–Schwarz inequality as usual,

we altogether obtain



B(x

,R)



Dw − (Dw)

B(x

,R)



≤



B(x

,R)



Dv − (Dv)

B(x

,R)



(12.3.38)

Finally,



B(x

,r)



Dv − (Dv)

B(x

,r)



≤ 3



B(x

,r)



Dw − (Dw)

B(x

,R)





B(x

,r)

|Dv − Dw|



B(x

,r)



(Dv)

B(x

,r)

− (Dw)

B(x

,r)



The last expression can be estimated by H¨older’s inequality:



B(x

,r)



|B(x

,r)|



B(x

,r)

(Dv − Dw)



≤ 3



B(x

,r)

(Dv − Dw)

Thus



B(x

,r)



Dv − (Dv)

B(x

,r)



≤ 3



B(x

,r)



Dw − (Dw)

B(x

,r)





B(x

,r)

|Dv − Dw|

≤ 3



B(x

,r)



Dw − (Dw)

B(x

,r)



+ c

2α



B(x

,r)

|Dv|

(12.3.39)

by (12.3.33). From (12.3.39), (12.3.36), (12.3.38), we obtain

12.3 Regularity of Minimizers of Variational Problems 335



B(x

,r)



Dv − (Dv)

B(x

,r)



≤ c





d+2



B(x

,r)



Dv − (Dv)

B(x

,R)



+ c

2α



B(x

,R)

|Dv|

≤ c





d+2



B(x

,R)



Dv − (Dv)

B(x

,R)



+ c

d−ε+2α

(12.3.40)

applying (12.3.35) for 0 <R≤ R

in place of 0 <r≤ R. Lemma 12.3.6

implies



B(x

,r)



Dv − (Dv)

B(x

,r)



≤ c





d+2α−ε



B(x

,R)



Dv − (Dv)

B(x

,R)



+ c

d+2α−ε

The claim now follows from Campanato’s theorem (Corollary 9.1.7). 

It is now easy to prove Theorem 12.3.2:

Proof of Theorem 12.3.2: We apply Theorem 12.3.3 to v = Du and obtain

v ∈ C

1,α



, hence u ∈ C

2,α



. We may then diﬀerentiate the equation with

respect to x

and observe that the second derivatives D

u, j, k =1,...,d,

again satisfy equations of the same type. By Theorem 12.3.3, then D

u ∈

1,α



; hence u ∈ C

3,α



. Iteratively, we obtain u ∈ C

m,α

for all m ∈ N with

0 <α

< 1. Therefore, u ∈ C

∞

. 

Remark: The regularity Theorem 12.3.1 of de Giorgi more generally applies

to minimizers of variational problems of the form

I(v):=



F (x, v(x),Dv(x))dx,

where F ∈ C

∞

(Ω ×R×R ×R

) again satisﬁes conditions like (i), (ii) of Theo-

rem 12.3.1 with respect to p,and

|p|

F (x, v, p) satisﬁes smoothness conditions

with respect to the variables x and v uniformly in p.

References for this section are Giaquinta [7],[8].

Summary

Moser’s Harnack inequality says that positive weak solutions u of

Lu =



i,j

∂

∂x



(x)

∂

∂x

u(x)

