Jost J. Partial Differential Equations

Подождите немного. Документ загружается.

9.5 Eigenvalues of Elliptic Operators 267

D|u

|,D|u

|

|u

|, |u

|

Du

,Du



u



= λ

Therefore, |u

| also minimizes

Du, Du

u, u

and by the reasoning leading to Theorem 9.5.1, it must also be an eigenfunc-

tion with eigenvalue λ

. Therefore, it is a nonnegative solution of

Δu + λu =0 inΩ,

and by the strong maximum principle (Theorem 9.1.2), it cannot assume a

nonpositive interior minimum. Thus, it cannot become 0 in Ω, and so it is

positive in Ω. This, however, implies that the original function u

cannot

become 0 either. Thus, u

is of a ﬁxed sign.

This argument applies to all eigenfunctions with eigenvalue λ

.Sincetwo

functions v

neither of which changes sign in Ω cannot satisfy



(x)v

(x)dx =0,

i.e., cannot be L

-orthogonal, the space of eigenfunctions for λ

is one-

dimensional. 

The classical text on eigenvalue problems is Courant–Hilbert [4].

Remark: More generally, Courant’s nodal set theorem holds: Let Ω ⊂ R

be open and bounded, with Dirichlet eigenvalues 0 <λ

<λ

≤ ... and

corresponding eigenfunctions u

,.... We call

:= {x ∈ Ω : u

(x)=0}

the nodal set of u

. The complement Ω \Γ

then has at most k components.

Summary

In this chapter we have introduced Sobolev spaces as spaces of integrable

functions that are not necessarily diﬀerentiable in the classical sense, but

do possess so-called generalized or weak derivatives that obey the rules for

integration by parts. Embedding theorems relate Sobolev spaces to spaces of

-functions or of continuous, H¨older continuous, or diﬀerentiable functions.

The weak solutions of the Laplace and Poisson equations, obtained in

Chapter 8 by Dirichlet’s principle, naturally lie in such Sobolev spaces. In

this chapter, embedding theorems allow us to show that weak solutions are

regular, i.e., diﬀerentiable of any order, and hence also solutions in the clas-

sical sense.

Based on Rellich’s theorem, we have treated the eigenvalue problem for

the Laplace operator and shown that any L

-function admits an expansion

in terms of eigenfunctions of the Laplace operator.

268 9. Sobolev Spaces and L

Regularity Theory

Exercises

9.1 Let u : Ω → R be integrable, and let α, β be multi-indices. Show that

if two of the weak derivatives D

α+β

u, D

u exist, then the

third one also exists, and all three of them coincide.

9.2 Let u, v ∈ W

1,1

(Ω)withuv, uDv + vDu ∈ L

(Ω). Then uv ∈ W

1,1

(Ω)

as well, and the weak derivative satisﬁes the product rule

D(uv)=uDv + vDu.

(For the proof, it is helpful to ﬁrst consider the case where one of the two

functions is of class C

(Ω).)

9.3 For m ≥ 2, 1 ≤ q ≤ m/2, u ∈ H

q+1

(Ω)∩L

q−1

(Ω)wehaveu ∈ H

(Ω)

and

Du

(Ω)

≤ const u

q−1

(Ω)

)

q+1

(Ω)

(Hint: For p =

= D

(uD

u|D

p−2

) − uD

u|D

p−2

The ﬁrst term on the right-hand side disappears upon integration over

Ω for u ∈ C

∞

(Ω) (approximation argument!), and for the second one,

we utilize the formula

(v|v|

p−2

)=(p − 1)(D

v)|v|

p−2

Finally, you need the following version of H¨older’s inequality

u



(Ω)

≤u



(Ω)

u



(Ω)

u



(Ω)

for u

∈ L

(Ω),

= 1 (proof!).)

9.4 Let

B(0, 1) ⊂ R

:= R

B(0, 1),

i.e., the d-dimensional unit ball and its complement. For which values of

k, p, d, α is

f(x):=|x|

in W

k,p

(Ω

)orW

k,p

(Ω

9.5 Prove the following version of the Sobolev embedding theorem:

Let u ∈ W

k,p

(Ω), Ω



⊂⊂ Ω ⊂ R

.Then

u ∈



d−kp

(Ω



)forkp < d,

(Ω



) for 0 ≤ m<k− d/p.

Exercises 269

9.6 State and prove a generalization of Corollary 9.1.5 for u ∈ W

k,p

(Ω)that

is analogous to Exercise 8.5.

9.7 Supply the details of the proof of Theorem 9.3.2 (This may sound like

a dull exercise after what has been said in the text, but in order to

understand the techniques for estimating solutions of PDEs, a certain

drill in handling additional lower-order terms and variable coeﬃcients

may be needed.)

9.8 Carry out the eigenvalue analysis for the Laplace operator under periodic

boundary conditions as deﬁned in §1.1. In particular, state and prove an

analogue of Theorems 9.5.1 and 9.5.2.

10. Strong Solutions

10.1 The Regularity Theory for Strong Solutions

We start with an elementary observation: Let v ∈ C

(Ω). Then

)

(Ω)





i,j=1

= −





i,j=1





i=1



j=1

= Δv

(Ω)

(10.1.1)

Thus, the L

-norm of Δv controls the L

-norms of all second derivatives of

v. Therefore, if v is a solution of the diﬀerential equation

Δv = f,

the L

-norm of f controls the L

-norm of the second derivatives of v.This

is a result in the spirit of elliptic regularity theory as encountered in Sec-

tion 9.2 (cf. Theorem 9.2.1). In the preceding computation, however, we have

assumed that, ﬁrstly, v is thrice continuously diﬀerentiable, and secondly,

that it has compact support. The aim of elliptic regularity theory, however,

is to deduce such regularity results, and also, one typically encounters non-

vanishing boundary terms on ∂Ω. Thus, our assumptions are inappropriate,

and we need to get rid of them. This is the content of this section.

We shall ﬁrst discuss an elementary special case of the Calderon-Zygmund

inequality. Let f ∈ L

(Ω), Ω open and bounded in R

. We deﬁne the Newton

potential of f as

w(x):=



Γ (x, y)f (y)dy (10.1.2)

using the fundamental solution constructed in Section 1.1,

Γ (x, y)=



2π

log |x − y| for d =2,

d(2−d)ω

|x − y|

2−d

for d>2.

272 10. Strong Solutions

Theorem 10.1.1: Let f ∈ L

(Ω) and let w be the Newton potential of f.

Then w ∈ W

2,2

(Ω), Δw = f almost everywhere in Ω,and

)

= f

(Ω)

(10.1.3)

(w is called a strong solution of Δw = f, because this equation holds almost

everywhere).

Proof: We ﬁrst assume f ∈ C

∞

(Ω). Then w ∈ C

∞

). Let Ω ⊂⊂ Ω

, Ω

bounded with a smooth boundary. We ﬁrst wish to show that for x ∈ Ω,

∂

∂x

w(x)=



∂

∂x

Γ (x, y)(f (y) −f(x))dy

+ f(x)



∂Ω

∂

∂x

Γ (x, y)ν

do(y),

(10.1.4)

where ν =(ν

,...,ν

) is the exterior normal and do(y) yields the induced

measure on ∂Ω

. This is an easy consequence of the fact that



∂

∂x

Γ (x, y)(f (y) −f(x))



≤ const

|x − y|

|f(y) − f(x)|

≤ const

|x − y|

d−1

f

In other words, the singularity under the integral sign is integrable. (Namely,

one simply considers

(x)=



∂

∂x

Γ (x, y)η

(y)f(y)dy,

with η

(y)=0for|y|≤ε, η

(y)=1for|y|≥2ε and |Dη

|≤

,andshows

that as ε → 0, D

converges to the right-hand side of (10.1.4).)

Remark: Equation (10.1.4) continues to hold for a H¨older continuous f,cf.

Section 11.1 below, since in that case, one can estimate the integrand by

const

|x − y|

d−α

f

(0 <α<1).

Since

ΔΓ (x, y) = 0 for all x = y,

for Ω

= B(x, R), R suﬃciently large, from (10.1.4) we obtain

10.1 The Regularity Theory for Strong Solutions 273

Δw(x)=

dω

d−1

f(x)



|x−y|=R



i=1

(y)ν

(y) do(y)=f(x). (10.1.5)

Thus, if f has compact support, so does Δw; let the latter be contained in

the interior of B(0,R). Then



B(0,R)



i,j=1



∂

∂x



= −



B(0,R)



∂

∂x

∂

∂x



∂B(0,R)

Dw ·

∂

∂ν

Dw do(y)



B(0,R)

(Δw)



∂B(0,R)

Dw ·

∂

∂ν

Dw do(y).

(10.1.6)

As R →∞, Dw behaves like R

1−d

, D

w like R

−d

, and therefore, the integral

on ∂B(0,R) converges to zero for R →∞. Because of (10.1.5), (10.1.6) then

yields (10.1.3).

In order to treat the general case f ∈ L

(Ω), we argue that by The-

orem 8.2.2, for f ∈ C

∞

(Ω)theW

1,2

-norm of w can be controlled by the

-norm of f .

We then approximate f ∈ L

(Ω)by(f

) ∈ C

∞

(Ω). Apply-

ing (10.1.3) to the diﬀerences (w

−w

) of the Newton potentials w

of f

we see that the latter constitute a Cauchy sequence in W

2,2

(Ω). The limit w

again satisﬁes (10.1.3), and since L

-functions are deﬁned almost everywhere,

Δw = f holds almost everywhere, too. 

The above considerations can also be used to provide a proof of Theo-

rem 9.2.1. We recall that result:

Theorem 10.1.2: Let u ∈ W

1,2

(Ω) be a weak solution of Δu = f ,with

f ∈ L

(Ω). Then u ∈ W

2,2

(Ω



), for every Ω



⊂⊂ Ω,and

u

2,2

(Ω



)

≤ const



u

(Ω)

+ f

(Ω)



, (10.1.7)

with a constant depending only on d, Ω,andΩ



. Moreover,

Δu = f almost everywhere in Ω.

Proof: As before, we ﬁrst consider the case u ∈ C

(Ω). Let B(x, R) ⊂ Ω,

σ ∈ (0, 1), and let η ∈ C

(B(x, R)) be a cutoﬀ function with

See the proof of Lemma 8.3.1.

274 10. Strong Solutions

0 ≤ η(y) ≤ 1,

η(y)=1 fory ∈ B(x, σR),

η(y)=0 fory ∈ R

\ B



1+σ

· R



|Dη|≤

(1 − σ)R



≤

(1 − σ)

We put

v := ηu.

Then v ∈ C

(B(x, R)), and (10.1.1) implies

)

(B(x,R))

= Δv

(B(x,R))

. (10.1.8)

Now,

Δv = ηΔu +2Du ·Dη + uΔη,

and thus

)

(B(x,σR))

≤

)

(B(x,R))

≤ const



f

(B(x,R))

(1 − σ)R

Du

(

1+σ

·R

))

(1 − σ)

u

(B(x,R))



(10.1.9)

Now let ξ ∈ C

(B(x, R)) be a cutoﬀ function with

0 ≤ ξ(y) ≤ 1,

ξ(y)=1 fory ∈ B



1+σ



|Dξ|≤

(1 − σ)R

Putting w = ξ

u and using that u is a weak solution of Δu = f, we obtain



B(x,R)

Du · D(ξ

u)=−



B(x,R)

fξ

hence

10.1 The Regularity Theory for Strong Solutions 275



B(x,R)

|Du|

= − 2



B(x,R)

ξuDu · Dξ −



B(x,R)

fξ

≤



B(x,R)

|Du|



B(x,R)

|Dξ|

+(1− σ)



B(x,R)

(1 − σ)



B(x,R)

Thus,wehaveanestimateforξDu

(B(x,R))

,andalso

Du

(

1+σ

))

≤ξDu

(B(x,R))

≤ const



(1 − σ)R

u

(B(x,R))

+(1− σ)R f

(B(x,R))



(10.1.10)

Inequalities (10.1.9) and (10.1.10) yield

)

(B(x,σR))

≤ const



f

(B(x,R))

(1 − σ)

u

(B(x,R))



(10.1.11)

In (10.1.11) we put σ =

,andwecoverΩ



by a ﬁnite number of balls

B(x, R/2) with R ≤ dist(Ω



,∂Ω) and obtain (10.1.7) for u ∈ C

(Ω).

For the general case u ∈ W

1,2

(Ω), we consider the molliﬁcations u

de-

ﬁned in Appendix 12.3. Thus, let 0 <h<dist(Ω



,∂Ω). Then



· Dv = −



v, for all v ∈ H

1,2

(Ω),

and since u

∈ C

∞

(Ω), also

Δu

= f

By Lemma A.3,

u

− u, f

− f

(Ω)

→ 0.

In particular, the u

and the f

satisfy the Cauchy property in L

(Ω). We

apply (10.1.7) for u

− u

to obtain

u

− u



2,2

(Ω



)

≤ const



u

− u



(Ω)

+ f

− f



(Ω)



Thus, the u

satisfy the Cauchy property in W

2,2

(Ω



). Consequently, the

limit u is in W

2,2

(Ω



) and satisﬁes (10.1.7). 

276 10. Strong Solutions

If now f ∈ W

1,2

(Ω), then, because u ∈ W

2,2

(Ω



) for all Ω



⊂⊂ Ω, D

u is

a weak solution of ΔD

u = D

f in Ω



. We then obtain D

u ∈ W

2,2

(Ω



)for

all Ω



⊂⊂ Ω



, i.e., u ∈ W

3,2

(Ω



). Iteratively, we thus obtain a new proof of

Theorem 9.2.2, which we now recall:

Theorem 10.1.3: Let u ∈ W

1,2

(Ω) be a weak solution of Δu = f . Then

u ∈ W

k+2,2

(Ω

) for all Ω

⊂⊂ Ω,and

u

k+2,2

(Ω

)

≤ const



u

(Ω)

+ f

k,2

(Ω)



with a constant depending on k, d, Ω,andΩ

. 

In the same manner, we also obtain a new proof of Corollary 9.2.1:

Corollary 10.1.1: Let u ∈ W

1,2

(Ω) be a weak solution of Δu = f ,for

f ∈ C

∞

(Ω). Then u ∈ C

∞

(Ω).

Proof: Theorems 10.1.3 and 9.1.2. 

10.2 A Survey of the L

-Regularity Theory and

Applications to Solutions of Semilinear Elliptic

Equations

The results of the preceding section are valid not only for the exponent p =2,

but in fact for any 1 <p<∞. We wish to explain this result in the present

section. The basis of this L

-regularity theory is the Calderon–Zygmund in-

equality, which we shall only quote here without proof:

Theorem 10.2.1: Let 1 <p<∞, f ∈ L

(Ω) (Ω ⊂ R

open and bounded),

and let w be the Newton potential (10.1.1) of f. Then w ∈ W

2,p

(Ω), Δw = f

almost everywhere in Ω,and

)

(Ω)

≤ c(d, p) f

(Ω)

, (10.2.1)

with the constant c(d, p) depending only on the space dimension d and the

exponent p.

In contrast to the case p = 2, i.e., Theorem 10.1.1 above, where c(d, 2) = 1

for all d and the proof is elementary, the proof of the general case is relatively

involved; we refer the reader to Bers–Schechter [1] or Gilbarg–Trudinger [9].

The Calderon–Zygmund inequality yields a generalization of Theorem 10.1.2:

Theorem 10.2.2: Let u ∈ W

1,1

(Ω) be a weak solution of Δu = f , f ∈

(Ω), 1 <p<∞, i.e.,