¯

Combining these with (13.15), we have

A ¤ G(v 0 , w R ) ¤ G(v R , w R ) ¤ G(v R , w0 ) ¤ B.

¯ ¯¯ ¯

By (13.9) and (13.10), the sequences {v R }, {w R } are bounded. Hence, there are

¯ ¯

renamed subsequences such that

vR

¯ v, w R

¯¯ w as R ’ ∞

¯

13.4. Criteria for convexity 155

and

G(v R , w R ) ’ » as R ’ ∞.

¯¯

In view of (13.15), we have in the limit

G(v, w) ¤ » ¤ G(v, w),

¯ ¯ v ∈ N, w ∈ M.

This shows that » = G(v, w), and (v, w) is a saddle point. The theorem is completely

¯¯ ¯¯

proved.

13.4 Criteria for convexity

If G is a differentiable functional on a Hilbert space E, there are simple criteria that

can be used to verify the convexity of G. We have

Theorem 13.7. Let G be a differentiable functional on a closed, convex subset M of E.

Then G is convex on E iff it satis¬es any of the following inequalities for u 0 , u 1 ∈ M :

(G (u 0 ), u 1 ’ u 0 ) ¤G(u 1 ) ’ G(u 0 )

(13.16)

(G (u 1 ), u 1 ’ u 0 ) ≥G(u 1 ) ’ G(u 0 ),

(13.17)

(G (u 1 ) ’ G (u 0 ), u 1 ’ u 0 ) ≥0.

(13.18)

Moreover, it will be strictly convex iff there is strict inequality in any of them when

u 0 = u 1.

Proof. Let u t = (1 ’ t)u 0 + tu 1 , 0 ¤ t ¤ 1, and •(t) = G(u t ). If G is convex, then

G(u t ) ¤ (1 ’ t)G(u 0 ) + t G(u 1 )

(13.19)

or

•(t) ¤ (1 ’ t)•(0) + t•(1), 0 ¤ t ¤ 1.

(13.20)

In particular, the slope of • at t = 0 is less than or equal to the slope of the straight

line connecting (0, •(0)) and (1, •(1)). Thus, • (0) ¤ •(1) ’ •(0), and this is merely

(13.16). Reversing the roles of u 0 , u 1 produces (13.17). We obtain (13.18) by subtract-

ing (13.16) from (13.17). Conversely, (13.18) implies

• (t) ’ • (s) = (G (u t ) ’ G (u s ), u 1 ’ u 0 )

= (G (u t ) ’ G (u s ), u t ’ u s )/(t ’ s) ≥ 0, 0 ¤ s < t ¤ 1.

Thus,

• (t) ≥ • (s), 0 ¤ s ¤ t ¤ 1,

which implies (13.20). Since this is equivalent to (13.19), we see that G is convex. If G

is strictly convex, we obtain strict inequalities in (13.16)“(13.18), and strict inequalities

in any of them implies strict inequalities in (13.20) and (13.19).

156 13. Semilinear Wave Equations

Corollary 13.8. Let G be a differentiable functional on a closed, convex subset M of E.

Then G is concave on E iff it satis¬es any of the following inequalities for u 0 , u 1 ∈ M :

(G (u 0 ), u 1 ’ u 0 ) ≥ G(u 1 ) ’ G(u 0 ),

(13.21)

(G (u 1 ), u 1 ’ u 0 ) ¤ G(u 1 ) ’ G(u 0 ),

(13.22)

(G (u 1 ) ’ G (u 0 ), u 1 ’ u 0 ) ¤ 0.

(13.23)

Moreover, it will be strictly concave iff there is strict inequality in any of them when

u0 = u1.

Proof. Note that G(u) is concave iff ’G(u) is convex.

13.5 Partial derivatives

Let M, N be closed subspaces of a Hilbert space H satisfying H = M • N. Let G(u)

be a functional on H. We can consider “partial” derivatives of G in the same way we

considered total derivatives. We keep w = w0 ∈ M ¬xed and consider G(u) as a

functional on N, where u = v + w0 , v ∈ N. If the derivative of this functional exists

at v = v 0 ∈ N, we call it the partial derivative of G at u 0 = v 0 + w0 with respect

to v ∈ N and denote it by G N (u 0 ). Similarly, we can de¬ne the partial derivative

G M (u 0 ). We have

Lemma 13.9. If G exists at u 0 = v 0 + w0 , then G M (u 0 ) and G N (u 0 ) exist and satisfy

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

(13.24)

Proof. By de¬nition,

G(u 0 + u) = G(u 0 ) + (G (u 0 ), u) + o( u ), u ∈ H.

Therefore,

G(u 0 + v) = G(u 0 ) + (G (u 0 ), v) + o( v ), v∈N

and

G(u 0 + w) = G(u 0 ) + (G (u 0 ), w) + o( w ), w ∈ M.

But

G(u 0 + v) = G(u 0 ) + (G N (u 0 ), v) + o( v ), v∈N

and

G(u 0 + w) = G(u 0 ) + (G M (u 0 ), w) + o( w ), w ∈ M.

In particular, we have

(G (u 0 ) ’ G N (u 0 ), v) = o( v ) as v ’ 0, v ∈ N.

Thus,

(G (u 0 ) ’ G N (u 0 ), tv) = o(|t|) as |t| ’ 0

13.5. Partial derivatives 157

for each ¬xed v ∈ N. This means that

o(|t|)

(G (u 0 ) ’ G N (u 0 ), v) = ’ 0 as t ’ 0.

t

Hence,

(G (u 0 ), v) = (G N (u 0 ), v), v ∈ N.

Similarly,

(G (u 0 ), w) = (G M (u 0 ), w), w ∈ M.

These two identities combine to give (13.24).

Lemma 13.10. Under the hypotheses of Lemma 13.9, assume that G is differentiable

on H , convex on M, and concave on N. Then,

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

(13.25)

u = v + w, u 0 = v 0 + w0 , v, v 0 ∈ N, w, w0 ∈ M.

If G is either strictly convex on M or strictly concave on N (or both), then one has strict

inequality in (13.25) when u = u 0 .

Proof. This follows from Theorem 13.7 and its corollary. In fact, we have

G(u) ’ G(u 0 ) = G(u) ’ G(v + w0 ) + G(v + w0 ) ’ G(u 0 )

¤ (G (u), w ’ w0 ) + (G (u 0 ), v ’ v 0 ).

Apply Lemma 13.9.

We also have