Jost J. Partial Differential Equations

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

satisfy an estimate of the form

sup

B(x

,R)

u ≤ const inf

B(x

,R)

in each ball B(x

,R) in the interior of their domain of deﬁnition Ω. Here,

the coeﬃcients a

need to satisfy only an ellipticity condition, and have to

be measurable and bounded, but they need not satisfy any further conditions

like continuity. Moser’s inequality yields a proof of the fundamental result of

de Giorgi and Nash about the H¨older continuity of weak solutions of linear

elliptic diﬀerential equations of second order with measurable and bounded

coeﬃcients. These assumptions are appropriate and useful for applications to

nonlinear elliptic equations of the type



i,j

∂

∂x



(u(x))

∂

∂x

u(x)



=0.

Namely, if one does not yet know any detailed properties of the solution u,

then,eveniftheA

themselves are smooth, one can work only with the

boundedness of the coeﬃcients

(x):=A

(u(x)).

Here, a nonlinear equation is treated as a linear equation with not necessarily

regular coeﬃcients.

An application is de Giorgi’s theorem on the regularity of minimizers of

variational problems of the form



F (Du(x)) dx → min

under the structural conditions

(i) |

∂F

∂p

(p)|≤K|p|,

(ii) λ|ξ|

≤



∂

F (p)

∂p

≤ Λ|ξ|

for all ξ ∈ R

with constants K, Λ < ∞,λ> 0.

Exercises

12.1 Formulate conditions on the coeﬃcients of a diﬀerential operator of the

form

Lu =



i,j=1

∂

∂x



(x)

∂

∂x

u(x)





i=1

∂

∂x

(x)u(x)) + c(x)u(x)

that imply a Harnack inequality of the type of Corollary 12.1.1. Carry

out the detailed proof.

Exercises 337

12.2 As in Lemma 12.1.4, let

φ(p, R)=



−



B(x



1/p

for a ﬁxed positive u : B(x

,R) → R.

Show that

lim

p→0

φ(p, R)=exp



−



B(x

,R)

log u(x) dx



Appendix. Banach and Hilbert Spaces.

The L

-Spaces

In the present appendix we shall ﬁrst recall some basic concepts from calculus

without proofs. After that, we shall prove some smoothing results for L

functions.

Deﬁnition A.1: A Banach space B is a real vector space that is equipped

withanorm· that satisﬁes the following properties:

(i) x > 0 for all x ∈ B, x =0.

(ii) αx = |α|·x for all α ∈ R, x ∈ B.

(iii) x + y≤x + y for all x, y ∈ B (triangle inequality).

(iv) B is complete with respect to · (i.e., every Cauchy sequence has a

limit in B).

We recall the Banach ﬁxed point theorem

Theorem A.1: Let (B,·) be a Banach space, A ⊂ B a closed subset,

f : A → B amapwithf (A) ⊂ A which satisﬁes the inequality

f(x) − f(

y)≤θx − y for all x, y ∈ A,

for some ﬁxed θ with 0 ≤ θ<1.

Then f has unique ﬁxed point in A, that is, a solution of f(x)=x.

For example, every Hilbert space is a Banach space. We also recall that

concept:

Deﬁnition A.2: A (real) Hilbert space H is a vector space over R, equipped

with a scalar product

(·, ·):H ×H → R

that satisﬁes the following properties:

(i) (x, y)=(y, x) for all x, y ∈ H.

(ii) (λ

+λ

,y)=λ

,y)+λ

,y) for all λ

,λ

∈ R, x

,y ∈ H.

(iii) (x, x) > 0 for all x =0, x ∈ H.

(iv) H is complete with respect to the norm

x := (x, x)

340 Appendix. Banach and Hilbert Spaces. The L

-Spaces

In a Hilbert space H, the following inequalities hold:

– Schwarz inequality:

|(x, y)|≤x·y, (A.1)

with equality precisely if x and y are linearly dependent.

– Triangle inequality:

x + y≤x + y. (A.2)

Likewise without proof, we state the Riesz representation theorem:

Let L be a bounded linear functional on the Hilbert space H, i.e., L : H → R

is linear with

L := sup

x=0

|Lx|

x

< ∞.

Then there exists a unique y ∈ H with L(x)=(x, y) for all x ∈ H,and

L = y.

The following extension is important, too:

Theorem of Lax–Milgram: Let B be a bilinear form on the Hilbert space

H that is bounded,

|B(x, y)|≤K xy for all x, y ∈ H with K<∞,

and elliptic, or, as this property is also called in the present context, coercive,

|B(x, x)|≥λ x

for all x ∈ H with λ>0.

For every bounded linear functional T on H, there then exists a unique y ∈ H

with

B(x, y)=Tx for all x ∈ H.

Proof: We consider

(x)=B(x, z).

By the Riesz representation theorem, there exists Sz ∈ H with

(x, Sz)=L

x = B(x, z).

Since B is bilinear, Sz depends linearly on z.Moreover,

Appendix. Banach and Hilbert Spaces. The L

-Spaces 341

Sz≤K z.

Thus, S is a bounded linear operator.

Because of

λ z

≤ B(z, z)=(z, Sz) ≤zSz

we have

Sz≥λ z.

So, S is injective. We shall show that S is surjective as well. In fact, there

exists x =0with

(x, Sz) = 0 for all z ∈ H.

With z = x,weget

(x, Sx)=0.

Since we have already proved the inequality

(x, Sx) ≥ λ x

we conclude that x = 0. This establishes the surjectivity of S.Bywhathas

already been shown, it follows that S

−1

likewise is a bounded linear functional

on H. By Riesz’s theorem, there exists v ∈ H with

Tx =(x, v)

=(x, Sz) for a unique z ∈ H,sinceS is bijective

= B(x, z)=B(x, S

−1

v).

Then y = S

−1

v satisﬁes our claim. 

The Banach spaces that are important for us here are the L

spaces:

For 1 ≤ p<∞, we put

(Ω):=



u : Ω → R measurable,

with u

:= u

(Ω)



|u|

< ∞



and

∞

(Ω):=



u : Ω → R measurable, u

∞

(Ω)

:= sup |u| < ∞



Here

342 Appendix. Banach and Hilbert Spaces. The L

-Spaces

sup |u| := inf{k ∈ R : {x ∈ Ω : |u(x)| >k} is a null set}

is the essential supremum of |u|.

Occasionally, we shall also need the space

loc

(Ω):={u : Ω → R measurable with u ∈ L

(Ω



) for all Ω



⊂⊂ Ω},

1 ≤ p ≤∞.

In those constructions, one always identiﬁes functions that diﬀer on a null

set. (This is necessary in order to guarantee (i) from Deﬁnition A.1.)

We recall the following facts:

Lemma A.1: The space L

(Ω) is complete with respect to ·

, and hence

is a Banach space, for 1 ≤ p ≤∞. L

(Ω) is a Hilbert space, with scalar

product

(u, v)

(Ω)



u(x)v(x)dx.

Any sequence that converges with respect to ·

contains a subsequence that

converges pointwise almost everywhere. For 1 ≤ p<∞, C

(Ω) is dense in

(Ω); i.e., for u ∈ L

(Ω) and ε>0, there exists w ∈ C

(Ω) with

u − w

<ε. (A.3)

H¨older’s inequality holds: If u ∈ L

(Ω), v ∈ L

(Ω), 1/p +1/q =1, then



uv ≤u

(Ω)

·v

(Ω)

. (A.4)

Inequality (A.4) follows from Young’s inequality

ab ≤

, if a, b ≥ 0,p,q>1,

=1. (A.5)

To demonstrate this, we put

A := u

,B:= v

and without loss of generality A, B = 0. With a :=

|u(x)|

, b :=

|v(x)|

, (A.5)

then implies



|u(x)v(x)|

≤

=1,

i.e., (A.4).

Inductively, (A.4) yield that if u

∈ L

,...,u

∈ L

Appendix. Banach and Hilbert Spaces. The L

-Spaces 343



i=1

=1,

then



···u

≤u



···u



. (A.6)

By Lemma A.1, for 1 ≤ p<∞, C

(Ω) is dense in L

(Ω) with respect to

the L

-norm. We now wish to show that even C

∞

(Ω) is dense in L

(Ω). For

that purpose, we shall use so-called molliﬁers, i.e., nonnegative functions 

from C

∞

(B(0, 1)) with



dx=1.

Here,

B(0, 1) :=



x ∈ R

: |x|≤1



The typical example is

(x):=



c exp



|x|

−1



for |x| < 1,

0for|x|≥1,

where c is chosen such that



dx=1.Foru ∈ L

(Ω), h>0, we deﬁne the

molliﬁcation of u as

(x):=







x − y



u(y)dy, (A.7)

wherewehaveputu(y)=0fory ∈ R

\ Ω. (We shall always use that

convention in the sequel.) The important property of the molliﬁcation is

∈ C

∞





Lemma A.2: For u ∈ C

(Ω),ash → 0, u

converges uniformly to u on

any Ω



⊂⊂ Ω.

Proof:

(x)=



|x−y|≤h





x − y



u(y)dy



|z|≤1

(z)u(x − hz)dz with z =

x − y

(A.8)

Thus, if Ω



⊂⊂ Ω and 2h<dist(Ω



,∂Ω), employing

344 Appendix. Banach and Hilbert Spaces. The L

-Spaces

u(x)=



|z|≤1

(z)u(x)dz

(this follows from



|z|≤1

(z)dz = 1), we obtain

sup



|u − u

|≤sup

x∈Ω





|z|≤1

(z) |u(x) − u(x − hz)|dz,

≤ sup

x∈Ω



sup

|z|≤1

|u(x) − u(x − hz)|.

Since u is uniformly continuous on the compact set {x :dist(x, Ω



) ≤ h},it

follows that

sup



|u − u

|→0forh → 0.



Lemma A.3: Let u ∈ L

(Ω), 1 ≤ p<∞.Forh → 0, we then have

u − u



(Ω)

→ 0.

Moreover, u

converges to u pointwise almost everywhere (again putting u =

0 outside of Ω).

Proof: We use H¨older’s inequality, writing in (A.8)

(z)u(x − hz)=(z)

(z)

u(x − hz)

with 1/p +1/q = 1, to obtain

(x)|

≤





|z|≤1

(z)dz





|z|≤1

(z) |u(x − hz)|



|z|≤1

(z) |u(x − hz)|

dz.

We choose a bounded Ω



with Ω ⊂⊂ Ω



If 2h<dist(Ω,∂Ω



), it follows that



(x)|

dx ≤



|z|≤1

(z) |u(x − hz)|

dz dx



|z|≤1



(z)



|u(x − hz)|



≤





|u(y)|

(A.9)

Appendix. Banach and Hilbert Spaces. The L

-Spaces 345

(with the substitution y = x − hz). For ε>0, we now choose w ∈ C

(Ω



)

with

u − w

(Ω



)

<ε

(compare Lemma A.1). By Lemma A.2, for suﬃciently small h,

w − w



(Ω



)

<ε.

Applying (A.9) to u − w, we now obtain



(x) − w

(x)|

dx ≤





|u(y) − w(y)|

and hence

u − u



(Ω)

≤u − w

(Ω)

+ w − w



(Ω)

+ u

− w



(Ω)

≤ 2ε + u −w

(Ω



)

≤ 3ε.

Thus u

converges to u with respect to ·

. By Lemma A.1, a subsequence

of u

then converges to u pointwise almost everywhere. By a more reﬁned

reasoning, in fact the entire sequence u

converges to u for h → 0. 

Remark: Mollifying kernels were introduced into PDE theory by K.O. Fried-

richs. Therefore, they are often called “Friedrichs molliﬁers”.

For the proofs of Lemmas A.2 and A.3, we did not need the smoothness of ρ

at all. Thus, these results also hold for other kernels, and in particular for

σ(x)=



for |x|≤1,

0 otherwise.

The corresponding convolution is

(x)=





x − y



u(y) dy =

|B(x, r)|



B(x,r)

u(y) dy =: −



B(x,r)

i.e., the average or mean integral of u on the ball B(x, r). Thus, analogously

to Lemma A.3, we obtain the following result:

Lemma A.4: Let u ∈ L

(Ω), 1 ≤ p<∞.Forr → 0, then

−



B(x,r)

converges to u(x), in the space L

(Ω) as well as pointwise almost everywhere.

For a detailed presentation of all the results that have been stated here with-

out proof, we refer to Jost [12].