Lemma 13.11. Under the hypotheses of Lemma 13.9, if G (u 0 ) exists and u 0 = v 0 +w0

is a saddle point, then

G (u 0 ) = G M (u 0 ) = G N (u 0 ) = 0.

Proof. By de¬nition,

G(v + w0 ) ¤ G(u 0 ) ¤ G(v 0 + w), v ∈ N, w ∈ M.

Since v 0 is a maximum point on N, we see that G N (u 0 ) = 0. Since w0 is a minimum

point on M, we have G M (u 0 ) = 0 for the same reason. We then apply Lemma 13.9.

Corollary 13.12. Under the hypotheses of Lemma 13.10, if G is either strictly convex

on M or strictly concave on N (or both), then G has exactly one saddle point.

Proof. This follows from inequality (13.25).

158 13. Semilinear Wave Equations

13.6 The theorems

In solving problem (13.1)“(13.3) we take p(x, t, ξ ) to be a Carath´ odory function on

e

— R which is 2π-periodic in t and satis¬es

θ’ ξ 2 ¤ ξ [ p(x, t, ξ1 ) ’ p(x, t, ξ0 )] ¤ θ+ ξ 2 , ξ = ξ1 ’ ξ2 ∈ R.

(13.26)

Let

μ = (μ1 , . . . , μn ), μ j ≥ 0, μ j ∈ Z, μ2 = |μ|2 = μ2 .

j

Let σ ( ) be the set of integers of the form » = μ2 ’ k 2 , and let ρ( ) = R\σ ( ).

We shall prove

Theorem 13.13. If

[θ’ , θ+ ] ‚ ρ( ),

(13.27)

then there is a unique weak solution of (13.1)“(13.3).

Theorem 13.14. If

(θ’ , θ+ ) ‚ ρ( ),

(13.28)

and there are constants β± such that θ’ ¤ β’ ¤ β+ ¤ θ+ , [β’ , β+ ] ‚ ρ( ) and

β’ ¤ lim inf 2P(x, t, ξ )/ξ 2 ¤ lim sup 2P(x, t, ξ )/ξ 2 ¤ β+

(13.29)

|ξ |’∞ |ξ |’∞

, where

uniformly in

ξ

P(x, t, ξ ) = p(x, t, s)ds,

0

then (13.1)“(13.3) has at least one weak solution.

Theorems 13.13 and 13.14 are proved in the next section.

13.7 The proofs

In this section we present the proof of Theorems 13.13 and 13.14. Let x = (x 1 , . . . , x n ),

•μ (x) = sin μ1 ξ1 · · · sin μn x n /(π/2)n/2 .

Note that

(•μ , •ν ) = δμν ,

where the scalar product is that of L 2 ([0, π]n ). We take

ek (t) = eikt /(2π)1/2, t ∈ [0, 2π].

Then

(ek , e ) = δk ,

13.7. The proofs 159

where the scalar product is that of L 2 ([0, 2π]). If

u= ±μk •μ (x)ek (t),

(13.30)

then

u ≡ ut t ’ u= (μ2 ’ k 2 )±μk •μ (x)ek (t).

Let γ be a ¬xed number in (θ’ , θ+ ) and de¬ne

G(u) = ∇u ’ ut ’2 u ∈ E,

2 2

P(x, t, u),

where the norm is that of L 2 ( ), and E is the set of all u ∈ L 2 ( ) of the form (13.30)

such that

u2= |μ2 ’ k 2 ’ γ | · |±μk |2 < ∞.

E

It is easily checked that G ∈ C 1 (E, R) and

(G (u), v)/2 = ( u, v) ’ ( p(u), v), u, v ∈ E,

where we write p(u) in place of p(x, t, u). Let m + be the smallest point of σ ( ) above

γ and m ’ the largest point of σ ( ) below γ .

Let

M = {u ∈ E : ±μk = 0 when μ2 ’ k 2 < γ },

N = {u ∈ E : ±μk = 0 when μ2 ’ k 2 > γ }.

We have

Lemma 13.15.

θ+ ’ γ

(G (v + w1 ) ’ G (v + w0 ), w)/2 ≥ 1 ’ w E,

2

m+ ’ γ

where w = w1 ’ w0 ∈ M, v ∈ N.

Proof. The left-hand side equals

( w, w) ’ ( p(v + w1 ) ’ p(v + w0 ), w)

≥w ’ (θ+ ’ γ ) w

2 2

E

θ+ ’ γ 2

≥ 1’ w E.

m+ ’ γ

Lemma 13.16.

γ ’ θ’

(G (v 1 + w) ’ G (v 0 + w), v)/2 ¤ ’ 1 ’ v E,

2

γ ’ m’

where v = v 1 ’ v 0 ∈ N, w ∈ M.

160 13. Semilinear Wave Equations

Proof. Now the left-hand side equals

( v, v) ’ ( p(v 1 + w) ’ p(v 0 + w), v)

¤’ v + (γ ’ θ’ ) v

2 2

E

γ ’ θ’ 2

¤’ 1’ v E.

γ ’ m’

We can now give the proof of Theorem 13.14.

Proof. By (13.28) and Lemmas 13.15 and 13.16, G(v +w) is convex in w and concave

in v. Moreover,

G(w) ’ ∞ as w ’ ∞, w ∈ M,

(13.31) E