Jost J. Partial Differential Equations

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9.5 Eigenvalues of Elliptic Operators 257

We now start our eigenvalue search with

λ := inf

u∈H\{0}

Du, Du

u, u



=inf

u∈H\{0}

Du

(Ω)

u

(Ω)



. (9.5.8)

We wish to show that (because the expression in (9.5.8) is scaling invariant,

in the sense that it is not aﬀected by replacing u by cu for some nonzero

constant c) this inﬁmum is realized by some u ∈ H with

Δu + λu =0.

We ﬁrst observe that (because the expression in (9.5.8) is scaling invariant,

in the sense that it is not aﬀected by replacing u by cu for some constant c)

we may restrict our attention to those u that satisfy

u

(Ω)

(= u, u)=1. (9.5.9)

We then let (u

)

n∈N

⊂ H be a minimizing sequence with u

 =1,and

thus

λ = lim

n→∞

Du

,Du

. (9.5.10)

Thus, (u

)

n∈N

is bounded in H, and by the compactness theorem of Rellich

(Theorems 8.2.2 and 9.4.2), a subsequence, again denoted by u

,converges

to some limit u in L

(Ω)thatthenalsosatisﬁesu

(Ω)

= 1. In fact, since

D(u

− u

)

(Ω)

+ D(u

+ u

)

(Ω)

=2Du



(Ω)

+2Du



(Ω)

for all n, m ∈ N,

and

D(u

+ u

)

(Ω)

≥ λ u

+ u



(Ω)

by deﬁnition of λ,

we obtain

Du

− Du



(Ω)

≤ 2 Du



(Ω)

+2Du



(Ω)

− λ u

+ u



(Ω)

. (9.5.11)

Since by choice of the sequence (u

)

n∈N

, Du



(Ω)

and Du



(Ω)

con-

verge to λ,andu

+ u



(Ω)

converges to 4, since the u

converge in L

(Ω)

to an element u of norm 1, the right-hand side of (9.5.11) converges to 0, and

so then does the left-hand side. This, together with the L

-convergence, im-

plies that (u

)

n∈N

is a Cauchy sequence even in H, and so it also converges

to u in H.Thus

258 9. Sobolev Spaces and L

Regularity Theory

Du, Du

u, u

= λ. (9.5.12)

In the Dirichlet case, the Poincar´e inequality (Theorem 8.2.2) implies

λ>0.

At this point, the assumption enters that Ω as a domain is connected. In

the Neumann case, we simply take any nonzero constant c, which now is an

element of H \{0}, to see that

0 ≤ λ ≤

Dc, Dc

c, c

=0,

i.e.,

λ =0.

Following standard conventions for the enumeration of eigenvalues, we put

λ =: λ

in the Dirichlet case,

λ =: λ

(= 0) in the Neumann case,

and likewise u =: u

and u =: u

, respectively.

Let us now assume that we have iteratively determined ((λ

)), (λ

...,(λ

m−1

), with

(λ

≤) λ

≤···≤λ

m−1

∈ L

(Ω) ∩ C

∞

(Ω),

=0 on∂Ω in the Dirichlet case, and

∂u

∂n

=0 on∂Ω in the Neumann case,

u

 = δ

for all i, j ≤ m − 1

Δu

+ λ

=0 inΩ for i ≤ m − 1. (9.5.13)

We deﬁne

:= {v ∈ H : v, u

 =0 fori ≤ m − 1}

and

:= inf

u∈H

\{0}

Du, Du

u, u

. (9.5.14)

9.5 Eigenvalues of Elliptic Operators 259

Since H

⊂ H

m−1

, the inﬁmum over the former space cannot be smaller

than the one over the latter, i.e.,

≥ λ

m−1

. (9.5.15)

Note that H

is a Hilbert space itself, being the orthogonal complement of

a ﬁnite-dimensional subspace of the Hilbert space H. Therefore, with the

previous reasoning, we may ﬁnd u

∈ H

with u



(Ω)

= 1 and

Du

,Du



u



. (9.5.16)

We now want to verify the smoothness of u

and equation (9.5.13) for i = m.

From (9.5.14), (9.5.16), for all ϕ ∈ H

, t ∈ R,

D (u

+ tϕ),D(u

+ tϕ)

u

+ tϕ, u

+ tϕ

≥ λ

wherewechoose|t| so small that the denominator is bounded away from 0.

This expression then is diﬀerentiable w.r.t. t near t = 0 and has a minimum

at 0. Hence the derivative vanishes at t = 0, and we get

Du

,Dϕ

u



−

Du

,Du



u



u

,ϕ

u



= Du

,Dϕ−λ

u

,ϕ for all ϕ ∈ H

In fact, this relation even holds for all ϕ ∈ H, because for i ≤ m − 1,

u

 =0

and

Du

,Du

 = Du

,Du

 = λ

u

 =0,

since u

∈ H

.Thus,u

satisﬁes



· Dϕ − λ



ϕ =0 forallϕ ∈ H. (9.5.17)

By Theorem 9.3.1 and Corollary 9.4.1, respectively, u

is smooth, and so we

obtain from (9.5.17)

Δu

+ λ

=0 inΩ.

As explained in the preceding section, we also have

∂u

∂n

=0 on∂Ω

260 9. Sobolev Spaces and L

Regularity Theory

in the Neumann case. In the Dirichlet case, we have of course

=0 on∂Ω

(this holds pointwise if ∂Ω is smooth, as explained in Section 9.4; for a

general, not necessarily smooth, ∂Ω, this relation is valid in the sense of

Sobolev).

Theorem 9.5.1: Let Ω ⊂ R

be connected, open and bounded. Then the

eigenvalue problem

Δu + λu =0,u∈ H

1,2

(Ω)

has countably many eigenvalues

0 <λ

<λ

≤···≤λ

≤···

with

lim

m→∞

= ∞

and pairwise L

-orthonormal eigenfunctions u

and Du

,Du

 = λ

.Any

v ∈ L

(Ω) can be expanded in terms of these eigenfunctions,

v =

∞



i=1

v, u

u

(and thus v, v =

∞



i=1

v, u



), (9.5.18)

and if v ∈ H

1,2

(Ω), we also have

Dv, Dv =

∞



i=1

v, u



. (9.5.19)

Theorem 9.5.2: Let Ω ⊂ R

be bounded, open, and of class C

∞

. Then the

eigenvalue problem

Δu + λu =0,u∈ W

1,2

(Ω)

has countably many eigenvalues

0=λ

≤ λ

≤···≤λ

≤···

with

lim

n→∞

= ∞

and pairwise L

-orthonormal eigenfunctions u

that satisfy

9.5 Eigenvalues of Elliptic Operators 261

∂u

∂n

=0 on ∂Ω.

Any v ∈ L

(Ω) can be expanded in terms of these eigenfunctions

v =

∞



i=0

v, u

u

(and thus v, v =

∞



i=0

v, u



), (9.5.20)

and if v ∈ W

1,2

(Ω),also

Dv, Dv =

∞



i=1

v, u



. (9.5.21)

Remark: Those v ∈ L

(Ω)thatarenotcontainedinH can be characterized

by the fact that the expression on the right-hand side of (9.5.19) or (9.5.21)

diverges.

The Proofs of Theorems 9.5.1 and 9.5.2 are now easy: We ﬁrst check

lim

m→∞

= ∞.

Indeed, otherwise,

Du

≤c for all m and some constant c.

By Rellich’s theorem again, a subsequence of (u

) would then be a Cauchy

sequence in L

(Ω). This, however, is not possible, since the u

are pairwise

-orthonormal.

It remains to prove the expansion. For v ∈ H we put

:= v,u



and



i≤m

:= v − v

Thus, v

is the orthogonal projection of v onto H

,andw

then is orthog-

onal to H

; hence

w

 =0 fori ≤ m.

Thus also

Dw

,Dw

≥λ

m+1

w



and

262 9. Sobolev Spaces and L

Regularity Theory

Dw

,Du

 = λ

u

 =0.

These orthogonality relations imply

w

 = v, v−v

,

Dw

,Dw

 = Dv, Dv−Dv

,Dv

,

(9.5.22)

and then

w

≤

m+1

Dv, Dv,

which converges to 0 as the λ

tend to ∞. Thus, the remainder w

converges

to0inL

,andso

v = lim

m→∞



v, u

u

in L

(Ω).

Also,



i≤m

and hence

Dv

,Dv

 =



i≤m

Du

,Du

 (since Du

,Du

 =0 fori = j)



i≤m

Since Dv

,Dv

≤Dv, Dv by (9.5.22) and the λ

are nonnegative, this

series then converges, and then for m<n,

Dw

− Dw

,Dw

− Dw

 = Dv

− Dv

,Dv

− Dv





i=m+1

→ 0form, n →∞,

and so (Dw

)

m∈N

is a Cauchy sequence in L

,andsow

converges in H,

and the limit is the same as the L

-limit, namely 0. Therefore, we get (9.5.19)

and (9.5.21), namely

Dv, Dv = lim

m→∞

Dv

,Dv

 =



The eigenfunctions (u

)n ∈ N thus are an L

-orthonormal sequence. The

closure of the span of the u

then is a Hilbert space contained in L

(Ω)and

containing H.SinceH (in fact, even C

∞

(Ω) ∩H, see the Appendix) is dense

in L

(Ω), this Hilbert space then has to be all of L

(Ω). So, the expansions

9.5 Eigenvalues of Elliptic Operators 263

(9.5.18), (9.5.20) are valid for all v ∈ L

(Ω).

The strict inequality λ

<λ

in the Dirichlet case will be proved in Theorem

9.5.4 below. 

A moment’s reﬂection also shows that the above procedure produces all

the eigenvalues of Δ on H, and that any eigenfunction is a linear combination

of the u

An easy consequence of the theorems is the following sharp version of the

Poincar´e inequality (cf. Theorem 8.2.2).

Corollary 9.5.1: For v ∈ H

1,2

(Ω),

v, v≤Dv, Dv (9.5.23)

where λ

is the ﬁrst Dirichlet eigenvalue according to Theorem 9.5.1.

For v ∈ H

1,2

(Ω) with

∂v

∂ν

on ∂Ω

v − ¯v, v − ¯v≤Dv, Dv (9.5.24)

where λ

now is the ﬁrst Neumann eigenvalue according to Theorem 9.5.2,

and ¯v :=

Ω



v(x)dx is the average of v on Ω (Ω is the Lebesgue measure

of Ω). Moreover, if such a v with vanishing Neumann boundary values is of

class H

2,2

(Ω), then also

Dv, Dv≤Δv, Δv, (9.5.25)

again being the ﬁrst Neumann eigenvalue.

Proof: The inequalities (9.5.23), (9.5.24) readily follow from (9.5.14), noting

that in the second case, v−¯v is orthogonal to the constants, the eigenfunctions

for λ

=0,since



(v(x) − ¯v)dx =0. (9.5.26)

As an alternative, and in order to obtain also (9.5.25), we note that

Dv = D(v − ¯v), Δv = Δ(v − ¯v), and

v − ¯v, v − ¯v =

∞



i=1

v, u



, (9.5.27)

that is, the term for i = 0 disappears from the expansion because v − ¯v is

orthogonal to the constant eigenfunction u

.Using

Dv, Dv =

∞



i=1

v, u



Δv, Δv =

∞



i=1

v, u



and λ

≤ λ

then yields (9.5.24), (9.5.25). 

264 9. Sobolev Spaces and L

Regularity Theory

More generally, we can derive Courant’s minimax principle for the eigen-

values of Δ:

Theorem 9.5.3: Under the above assumptions, let P

be the collection of all

k-dimensional linear subspaces of the Hilbert space H. Then the kth eigen-

value of Δ (i.e., λ

in the Dirichlet case, λ

k−1

in the Neumann case) is

characterized as

max

L∈P

k−1

min



Du, Du

u, u

u =0,u orthogonal to L,

i.e., u, v =0 for all v ∈ L



, (9.5.28)

or dually as

min

L∈P

max



Du, Du

u, u

: u ∈ L \{0}



. (9.5.29)

Proof: We have seen that

=min



Du, Du

u, u

: u =0,u orthogonal to the u

with i ≤ m − 1



(9.5.30)

It is also clear that

=max



Du, Du

u, u

: u = 0 linear combination of u

with i ≤ m



(9.5.31)

and in fact, this minimum is realized if u is a multiple of the mth eigenfunc-

tion u

, because λ

Du

,Du



u



≤ λ

for i ≤ m and the u

are pairwise

orthogonal.

Now let L be another linear subspace of H of the same dimension as the

span of the u

, i ≤ m.LetL be spanned by vectors v

, i ≤ m.Wemaythen

ﬁnd some v =



∈ L with

v, u

 =



v

 =0 fori ≤ m − 1. (9.5.32)

(This is a system of homogeneous linearly independent equations for the α

with one fewer equation than unknowns, and so it can be solved.) Inserting

(9.5.32) into the expansion (9.5.19) or (9.5.21), we obtain

Dv, Dv

v, v



∞

j=m

v, u





∞

j=m

v, u



≥ λ

Therefore,

max

v∈L\{0}

Dv, Dv

v, v

≥ λ

and (9.5.29) follows. Suitably dualizing the preceding argument, which we

leave to the reader, yields (9.5.28). 

9.5 Eigenvalues of Elliptic Operators 265

While for certain geometrically simple domains, like balls and cubes, one

may determine the eigenvalues explicitly, for a general domain, it is a hopeless

endeavor to attempt an exact computation of its eigenvalues. One therefore

needs approximation schemes, and the minimax principle of Courant suggests

one such method, the Rayleigh–Ritz scheme. For that scheme, one selects

linearly independent functions w

,...,w

∈ H, which then span a linear

subspace L, and seeks the critical values, and in particular the maximum of

Dw, Dw

w, w

for w ∈ L.

With

:= Dw

,Dw

,A:= (a

)

i,j=1,...,k

:= w

,B:= (b

)

i,j=1,...,k

for

w =



j=1

then

Dw, Dw

w, w



i,j=1



i,j=1

and the critical values are given by the solutions μ

,...,μ

det(A − μB)=0.

These values μ

,...,μ

then are taken as approximations of the ﬁrst k eigen-

values; in particular, if they are ordered such that μ

is the largest among

them, that value is supposed to approximate the kth eigenvalue. One then

tries to optimize with respect to the choice of the functions w

,...,w

; i.e.,

one tries to make μ

as small as possible, according to (9.5.29), by suitably

choosing w

,...,w

The characerizations (9.5.28) and (9.5.29) of the eigenvalues have many

further useful applications. The basis of those applications is the following

simple remark: In (9.5.29), we take the maximum over all u ∈ H that are

contained in some subspace L.IfwethenenlargeH to some Hilbert space



,thenH



contains more such subspaces than H, and so the minimum over

all of them cannot increase.

Formally, if we put P

(H):={k − dimensional linear subspaces of H},

then, if H ⊂ H



, it follows that P

(H) ⊂ P



), and so

min

L∈P

(H)

max

u∈L\{0}

Du, Du

u, u

≥ min



∈P



)

max

u∈L



\{0}

Du, Du

u, u

. (9.5.33)

266 9. Sobolev Spaces and L

Regularity Theory

Corollary 9.5.2: Under the above assumptions, we let 0 <λ

≤ λ

≤···

be the Dirichlet eigenvalues, and 0=λ

<λ

≤ λ

≤··· be the Neumann

eigenvalues. Then

j−1

≤ λ

for all j.

Proof: The Hilbert space for the Dirichlet case, namely H

1,2

(Ω), is a subspace

of that for the Neumann case, namely W

1,2

(Ω), and so (9.5.33) applies. 

The next result states that the eigenvalues decrease if the domain is en-

larged:

Corollary 9.5.3: Let Ω

⊂ Ω

be bounded open subsets of R

. We denote

the eigenvalues for the Dirichlet case of the domain Ω by λ

(Ω). Then

(Ω

) ≤ λ

(Ω

) for all k. (9.5.34)

Proof: Any v ∈ H

1,2

(Ω

) can be extended to a function ˜v ∈ H

1,2

(Ω

), simply

by putting

˜v(x)=



v(x)forx ∈ Ω

0forx ∈ Ω

\ Ω

Lemma 8.2.2 tells us that indeed ˜v ∈ H

1,2

(Ω

). Thus, the Hilbert space

employed for Ω

is contained in that for Ω

, and the principle (9.5.33) again

implies the result for the Dirichlet case. 

Remark: Corollary 9.5.3 is not in general valid for the Neumann case. A ﬁrst

idea to show a result in that case is to extend functions v ∈ W

1,2

(Ω

)toΩ

the extension operator E constructed in Section 9.4. However, this operator

does not preserve the norm: In general, Ev

1,2

(Ω

)

> v

1,2

(Ω

)

,andso

this does not represent W

1,2

(Ω

) as a Hilbert subspace of W

1,2

(Ω

). This

diﬃculty makes the Neumann case more involved, and we omit it here.

The next result concerns the ﬁrst eigenvalue λ

of Δ with Dirichlet bound-

ary conditions:

Theorem 9.5.4: Let λ

be the ﬁrst eigenvalue of Δ on the open and bounded

domain Ω ⊂ R

with Dirichlet boundary conditions. Then λ

is a simple

eigenvalue, meaning that the corresponding eigenspace is one-dimensional.

Moreover, an eigenfunction u

for λ

has no zeros in Ω, and so it is either

everywhere positive or negative in Ω.

Proof: Let

Δu

+ λ

=0 inΩ.

By Corollary 8.2.2, we know that |u

|∈W

1,2

(Ω), and